NCAR/TN-331+STR NCAR TECHNICAL NOTE
i I -
May
1989
Signal Processing for Atmospheric Radars
R. Jeffrey Keeler Richard E. Passarelli
ATMOSPHERIC TECHNOLOGY DIVISION NATIONAL
CENTER FOR ATMOSPHERIC RESEARCH BOULDER, COLORADO
TBSIE OF COTENTS
TABLE OF CONTENTS .....................
.
iii
LIST OF FIGURES ......................... LIST OF TABLES ................... PREFACE.
..
v .
..
.vii
..
i.......................
1.
Purpose and scope .................
2. 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6
General characteristics of atmospheric radars. Characteristics of processing .......... Sampling ................... Noise ...... ............... Scattering ............... Signal to noise ratio (SNR) .......... Types of atmospheric radars .......... Microwave radars . ........... ST/MST radars or wind profilers ....... FM-CW radars .. .............. Mobile radars ............ .. Lidar ............ ....... Acoustic sounders . ..........
3 3 3 4 5 6 6 7 8 8 9 10 11
3. 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.3.2 3.3.3
Doppler power spectrum moment estimation . .... General features of the Doppler power spectrum. Frequency domain spectral moment estimation . Fast Fourier transform techniques . .... Maximum entropy techniques .......... Maximum likelihood techniques . ....... Classical spectral moment computation ..... Time domain spectral moment estimation. ..... Geometric interpretations ........... "Pulse pair" estimators ........... Circular spectral moment computation for
13 14 18 18 20 23 25 27 27 28
3.3.4 3.4 3.4.1 3.4.2 3.4.3 4. 4.1 4.1.1 4.1.2 4.1.3 4.2 4.2.1
1
sampled data. . ............. Poly pulse pair techniques ..... Uncertainties in spectrum moment estimators . Reflectivity. ... ............ Velocity. . . ..... . ...... Velocity spectrum width ...........
31 33 .
Signal processing to eliminate bias and artifacts. Doppler techniques for ground clutter suppression Antenna and analog signal considerations. ... Frequency domain filtering. ......... Time domain filtering ............. Range/velocity ambiguity resolution ....... Resolution of velocity ambiguities ...... iii
35 35 36 37 43 43 44 45 46 50 51
4.2.2 4.3
Resolution of range ambiguities ....... Polarization switching consequences .......
55 56
5. 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2 5.3 5.4
Exploratory signal processing techniques . .... Pulse compression .... .......... Advantages of pulse compression . ...... Disadvantages of pulse compression. ...... Ambiguity function. .. Comparison with multiple frequency scheme . Adaptive filtering algorithms ......... Adaptive filtering applications ....... Adaptive antenna applications .. ..... Multi-channel processing. ............ A priori information. .............
57 57 58 59 61 63 63 64 68 69 70
6. 6.1 6.2 6.3 6.4
Signal processor implementation ......... Signal processing control functions ..... Signal Z?D conversion and calibration ...... Reflectivity processing ... .......... Thresholding for data quality .........
71 71 74 76 78
7. 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.2 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5
Trends in signal processing. ............ Realization factors ............... Digital signal processor chips ....... Storage media ................. Display technology . .............. Commercial radar processors .......... Trends in programmability of DSP. ........ Short term expectations .......... .... Range/velocity ambiguities ......... Ground clutter filtering .......... Waveforms for fast scanning radars ...... Data compression. ............. Artificial intelligence based feature extraction Real time 3D weather image processing .. ... Long term expectations . ............ Advanced hardware ....... .. Optical interconnects and processing ..... Communications . . . . . . . . . . . . . . . . Electronically scanned array antennas ..... Adaptive systems ...............
81 81 81 82 83 83 84 85 85 86 86 87 87 87 87 88 88 88 88 89
8. 8.1 8.2 8.3 8.4
Conclusions. . ................... Assessment of our past. ............. Recommendations for our future . ........ Acceptance of new techniques ........... Acknowledgements. ..............
91 91 92 93 93
ACRONYM LIST ........................... BIBLIOGRAPHY .......
95
...........
....
iv
.
.
97
TIST OF FJIGRES
Fig 3.1
of Doppler power spectrum (128 point periodogram) Estimated typical weather echo in white noise. parameters are velocity ~ 0.4 Vax velocity spectrum 10 dB. width ~ .04 Vmax, and SNR
15
Fig 3.2
of the complex representation dimensional Three Radius of helix autocorrelation function as a helix. Ps; proportional to total signal power, Rs(0) is rotation rate of helix is proportional to velocity, V; width of envelope is inversely proportional to velocity Delta function Rn(0) represents spectrum width, W. noise power.
29
Fig 3.3
Periodogram power spectrum plotted on unit circle in the Note velocity aliasing point, the Nyquist z-plane. velocity, at z=-l.
32
Fig 3.4
Comparison of classical and circular (pulse pair) first Classical estimate is determined by moment estimators. linear weighting of spectrum estimate and circular estimate, by sinusoidal weighting.
34
Fig 3.5
Velocity error as function of spectrum width and SNR. normalized to Nyquist interval, Spectrum width is M is number of sample pairs and vn=W/2Vmx=2WTs/X. error is normalized to Nyquist velocity interval, 2va = Small circles represent simulation values 2Vmax. (Doviak and Zrnic, 1984).
39
Fig 3.6
Width error as a function of spectrum width and SNR. normalized to Nyquist interval, Spectrum width is
42
vn=W/2Vmax=2Wrs/X.
M
is
number of sample pairs
and
error is normalized to Nyquist interval, 2Vmax. Small circles represent simulation values (Doviak and Zrnic, 1984). Fig 4.la
Clutter filter frequency response for a 3 pole infinite For impulse response (IIR) high pass elliptic filter. 1 5 rpm and scan rate of ground clutter width of 0.6 ms stop this filter gives about 40 dB suppression. V = 16 ms - (Hamidi band. Vp = pass band cutoff, Vmax = and Zrnic, 1981).
47
Fig 4.lb
Implementation of 3rd order IIR clutter suppression filter; z- 1 is 1 PRT delay. K1 - K4 are filter coefficients (Hamidi and Zrnic, 1981).
48
v
Fig 5.1
Ambiguity diagram for single FM chirped pulse waveform with TB=10. T is range dimension. 0 is velocity dimension. Targets distributed in (r,q) space contribute to the filter output proportional to the ambiguity function. For atmospheric targets, Doppler shift frequencies are typically very small relative to pulse bandwidth (Rihaczek, 1969).
62
Fig 5.2
Prediction error surface for 2 weight adaptive filter. The LMS algorithm estimates the negative gradient of the quadratic error and steps toward the minimum mean square error (mse). The optimum weight vector is W* = (0.65, -2.10). If the input statistics change so that the error surface varies with time, the adaptive weights will track this change (Widrow and Stearns, 1985).
65
Fig 5.3
Adaptive filter structure. The desired response (dk) is determined by the application. The adaptive filter. coefficients (Wk) and/or the output signal (Yk) are the parameters used for spectrum moment estimation (Widrow and Stearns, 1985).
66
Fig 6.1
Block diagram of a typical signal processor.
26
vi
LSTr OF TAHBI Table 1
Comparison of remote sensor sampling schemes and rates.
Table 2
Characteristics of several popular windows when applied to time series data analysis (Marple, 1987).
20
Table 3
Expressions for variance of velocity estimators at high SNR. Assumes Gaussian spectra in white noise, low normalized velocity width (Wn=W/2Vmx) and large M. Expressions apply to both pulse pair and Fourier transform estimators.
38
Table 4
Expressions for variance of width estimators at high SNR. Assumes Gaussian spectra in white noise, low normalized velocity width (Wn=W/2Vmax) and large M. Expressions apply to both pulse pair and Fourier transform estimators.
41
vii
7
PiRFACE
This review of signal processing for atmospheric radars was originally written as Chapter 20 of the book Radar in Meteorology, edited by Dave Atlas (1989) for the Proceedings of the 40th Anniversary and Louis Battan Memorial Radar Meteorology Conference. overview
of
signal
processing
We have attempted to give the reader an techniques
and
the
technology
that
are
applicable to the atmospheric remote sensing tools of weather radar, lidar, ST/MST radars and wind profilers.
This NCAR Technical Note includes the signal processing
relevant references in a single document.
chapter and the
The text has had minor editing
and the references have been slightly expanded over the version published in Radar in Meteorology.
We hope that this Technical Note will assist the many individuals who want a
better understanding of signal processing to achieve that goal.
R. Jeffrey Keeler
Richard E. Passarelli
March 1989
ix
1.
PURPOSE AND SODFE
Signal processing is perhaps the area of atmospheric remote sensing where science and engineering make their point of closest contact. Signal processing offers challenges to engineers who enjoy developing state-of-theart systems and to scientists who enjoy being at the crest of the wave in
observing atmospheric phenomena in unique ways. The primary function of radar signal processing is the accurate, efficient
extraction of information from radar echoes.
A typical pulsed Doppler radar
system samples data at 1000 range bins at 1 kilohertz pulse repetition
frequency
(PRF), generating approximately 3 million samples per second
(typically in-phase (I) and quadrature phase (Q) components from a linear channel and often a log receiver). These "time series", in their raw form, convey little information that is of direct use in determining the state of
the atmosphere.
The volume of time series data is sufficiently large that
storage for later analysis is impractical except for limited regions of time
and space.
The data must be processed in real time to reduce its volume and to convert it to more useful form. In this paper the current state of signal processing for atmospheric radars (weather radars, ST/MST radars or wind profilers, and lidars) shall be discussed along with how signal processing is currently optimized for various applications and remote sensors. The focus shall be on signal processing for weather radar systems but the techniques and conclusions apply equally well to ST/MST radars and lidars. Zrnic (1979a) has given an excellent review of spectral moment estimation for weather radars and Woodman (1985) has done the same for MST radars. Problem areas and promising avenues for future research shall be identified. Finally, we shall discuss the scientific and technological forces that are likely to shape the future of atmospheric radar signal processing. We will differentiate between "signal processing" (the topic of this review) and "data processing" in the following way. "Signal processing" is that set 1
of operations performed on the analog or digital signals for efficiently extracting desired information or measuring some attribute of the signal. For atmospheric radars this information is often referred to as the "base parameter estimates". Fundamental base parameters are: Radar reflectivity factor
Z
dBZ
Radial velocity Velocity spectrum width1
V W
ms- 1 ms-l
In the course of extracting these estimates, signal processing algorithms will improve the signal to noise ratio (SNR) through filtering or averaging,
mitigate the effects of interfering echoes such as ground clutter, remove ambiguities such as range or velocity aliasing, and reduce the input data The end result of an effective signal rate by a significant factor. processing scheme is to provide minimum mean squared error estimates of the base parameters along with the expected error or a measure of the degree of
confidence that can be placed on the estimates (e.g., the SNR). Note that signal processing is primarily used in atmospheric remote sensing as an estimation procedure as well as a detection process as in some aviation applications. The emphasis is on making estimates of atmospheric parameters or meteorological events. "Data processing", on the other hand, takes up where signal processing leaves off -- although the line of demarcation is not razor sharp. Data processing algorithms take the base parameter estimates and further process
them so that they convey information that is of direct use to the radar For example, data processing techniques imply display generation, user. data navigation to a desired coordinate system, wind profile analyses, data syntheses from several Doppler radars or other sensors, applying physical constraints to the measured data, and forecasts or "nowcasts" of severe weather hazards. Many aspects of data processing are covered in other chapters.
1 The width is defined as the square root of the second central moment of the spectral power distribution. 2
2.
There are two main
classes of
Electromagnetic radars systems.
CIRACJERSLR'LCS OF AICMY4SERIC RADARS
GENRA
"radar"
--
electromagnetic
and acoustic.
include microwave, UHF, VHF, infrared and optical
Acoustic radars
The signal
are only briefly described here.
processing techniques employed for all these systems are similar (Serafin and Strauch, 1978).
2.1
ClARACTERISTICS OF IRDCESSING are nearly identical
Although the processing techniques atmospheric
radars,
the
way
in
which
this
for the various
backscattered
or
partially
reflected radiation is sampled, the principle noise sources, and the nature of the scattering mechanisms are different.
2.1.1 Sampling Because electromagnetic less than 1 im, constraints
radars employ wavelengths
from several
meters
to
they must use different sampling techniques. There are two (Ts)
on the sample time spacing
of the backscattered signal.
The first is that the backscattered signal should be coherent from sampleto-sample, i.e., the motion among the scatterers should be small compared to the wavelength so that their relative positions produce highly correlated echoes from sample-to-sample.
The nominal duration of this correlation is
called the coherence time (Nathanson, 1969), i.e.,
T s < tcoh =
(2.1)
/4rW
where the true velocity spectrum width W in ms-1 is a direct measure of the The coherence time is a measure of the
relative motions of the scatterers.
maximum time between successive samples for coherent phase measurements. Thus, signal
for
short wavelength systems,
must be
sampled much more
microwave system.
such as
a
rapidly than
lidar, the backscattered for a
longer wavelength
The autocorrelation function (defined later) can provide
a direct measure of the coherence time of a fluctuating target echo.
3
The second constraint on sampling is that for regularly spaced pulses, the sampling frequency must be at least twice the maximum desired Doppler shift
frequency which reduces the occurrence of velocity aliasing.
In this case
the time between samples is governed by,
Ts < tNyq =
(2.2)
4V'
where tNyq is the minimum time between samples such that the desired velocity V' is at least the so-called Nyquist velocity. Since V' is typically much larger than W, the latter constraint usually dominates the sampling requirement. + 25 ms-1 , then T s
In fact, if we assume the desired maximum velocity is
V/100 or PRF = 100/ \ is a useful rule of thumb.
2.1.2 Noise One of the goals of signal processing is to suppress the effects of noise.
The main source of noise in microwave radar is thermal in nature. This noise power is simply
Pn = k Tsys Bsys
(2.3)
where k is Boltzman's constant (1.38 x 10-23 W/Hz/°K), Tsys is the total
system temperature, and Bsys is the total system bandwidth including effects of preselector filters, IF filters, and all other amplifiers in the signal path
(Skolnik, 1970, 1980;
Paczowski and Whelehan, 1988).
With recent
improvements in low noise amplifiers (INA's), little room is left for sensitivity improvement in conventional radar receivers. Presently, most microwave radar systems are sufficiently sensitive that thermal radiation from the earth makes a strong contribution to the receiver input at low elevation angles. ST/MST radar noise, because of its lower frequency, has a large contribution
from environmental,
cosmic and atmospheric
sources,
and
is not easily
quantified (Rottger and Larsen, Chap 21A). Therefore, antenna design and the specific radar location and frequency band of operation define the system noise. 4
Coherent lidar systems utilize detection schemes using optical heterodyning
onto cryogenic detectors with a local oscillator laser having relatively high power mixing with the weak atmospheric return (Jelalian, 1980, 1981a,b). Because of the small wavelengths, quantum effects dominate the detection process associated with random photon arrivals impacting the LD laser. This "shot noise" contribution is a fundamental physical limitation of lidar sensitivity. 2.1.3 Scattering
Atmospheric radars respond to a variety of scattering targets-precipitation, cloud particles, aerosols, refractive index variations, chaff, insects, birds, and ground targets.
Probert-Jones (1962) derived the
familiar radar equation most often used by radar meteorologists for precipitation scattering. A detailed derivation can be found in Doviak and Zrnic (1984), Battan (1973), or Atlas (1964). The received power is Pt G2 02 cTr
3
1k12
Ze L
(2.4)
Pr=
1024 ln2 X 2 R 2 This equation includes L, the product of several small but significant loss terms which are necessary to accurately estimate radar reflectivity factor,
e.g. receiver filter loss, propagation loss, blockage loss, and processing bias. Zric (1978) defines the receiver filter loss as that portion of the input signal frequencies not passed by the finite receiver bandwidth, typically 1-3 dB. The other losses depend on atmospheric conditions and antenna pointing and are enumerated in Skolnik (1980). This equation is correct for Rayleigh scattering of a distributed target that completely fills the resolution volume. Non-Rayleigh targets or partially filled resolution volumes will give received power estimates that cannot accurately
be related to precipitation rate.
Rottger and Larsen (Chap. 21A) and
Huffaker, et al. (1976, 1984) give similar received power expressions for
returns from refractive index variations and from lidar aerosol returns, respectively.
5
required
The
dynamic
range
for
measuring the
backscattered power from
atmospheric targets is very large because:
The effective backscatter cross-sections of atmospheric scatterers span
1.
dynamic ranges of approximately 60 dB for precipitation but much larger if cloud particle, "clear air", and ground target returns are included.
The R
2.
2
dependence of the received power for distributed targets spans
a range of 50 dB between 1 and 300 km.
Microwave
systems
should accommodate the
sum of these
two effects
and
typically can achieve a dynamic range of order 100 dB for power measurements using either a log receiver,
linear receiver with AGC,
or some combination
of these.
2.1.4
Signal to noise ratio (SNR)
The ratio
of the received
signal
power
to the measured
noise power
is
defined to be the signal to noise ratio (SNR):
(2.5)
SNR = Pr/Pn
The SNR is extremely important for analyzing tradeoffs in signal processing. It
is
a key term along with spectrum width and integration time in
analytic
evaluation of spectrum moment errors.
2.2
YPES OF AM[SHFERIC RAIDRS
A summary of the characteristics of the different types of electromagnetic radars in use today for atmospheric research is discussed below. assembles these differences.
6
Table 1
Table 1 Remote Sensor Sampling Comparison
Sensor
Pulse Beamwidth Duration (deg) (~sec)
Wavelength Scatterers
Radar S-band Ka-band mm-band
10cm 1 cm 1 mm
Precipitation Precipitation Cloud
ST/MST (profilers) UHF 75 cm Refractive VHF 6 m index
0.5-3 0.5-2 0.2-1
3-10 3-10
Sample Rate (Hz)
0.25-4 0.25-1 0.25-1
103 104 105
0.2-5 0.2-5
104->102 103->10
Lidar IR Optical
2.2.1
10 gmi <1 im
Aerosols Molecules
0.01 0.1-3 (near field) <1
107
Microwave radars
Microwave pulsed radars radiate fields with wavelengths between 20 cm and 1 mm and are commonly used as "weather radars" et al.,
1979).
precipitation,
(Smith, et al., 1974; Doviak,
Depending on the wavelength, primary scattering is from insects
cloud particles.
(Vaughan,1985), refractive index
fluctuations, and
Beams are typically circular in cross section with widths
0.5 to 3 degrees and the maximum usable ranges for storm observation is 200-
500 km. After a few kilometers range, the pulse volume is "pancake" shaped, i.e., the pulse depth in beam.
range is small compared to the distance across the
Attenuation effects
range
from
severe
for millimeter wavelength
systems, to nearly insignificant for 10 cm S-band systems.
Most centimeter wavelength microwave systems collect coherent samples over several milliseconds.
Millimeter wavelength radars can make use of the
double pulsing technique
(Campbell and Strauch, 1976) to assure coherence
and to reduce an otherwise intolerable range ambiguity problem.
Doviak and
Zrnic (1984) and Strauch (1988) have shown that since only the second pulse of a double pulsing radar may be contaminated by overlaid echo from the first pulse of the pair, only random errors occur in the pulse to pulse correlations.
These random errors may change very slowly with time so they
would appear to be systematic (bias) errors at a given time. 7
2.2.2
ST/ST radars or wind profilers
VHF and UHF radars which probe the mesosphere, stratosphere
and/or the
troposphere are called ST/MST radars and sometimes known as wind profilers,
observe radial winds at wavelengths between 30 cm and 6 m at near vertical
incidence (Gage and Balsley, 1978; Rottger, et al., 1978). Scattering is from atmospheric refractive index fluctuations in space, analogous to Bragg scattering.
Beamwidths may be as large as several degrees for tropospheric sounding, but much narrower beams are used for longer stratospheric and mesospheric ranges (Rottger and Larsen, Chap. 21A; Gage, Chap. 28A). For a nominal 1 m wavelength, the atmospheric coherence time is typically
large fractions of a second. Consequently, the sampling rate to achieve coherence is of order 10 Hz. Because of this and the typically weak clear air returns, it is advantageous to perform time domain averaging of the samples from pulse-to-pulse, e.g., at a given range, N successive complex
samples are averaged to yield a single complex pair. This operation effectively reduces the sampling frequency and the unambiguous velocity interval by a factor of N, but the fundamental interval is usually so large that this reduction is of little consequence. The main feature is that the data rate is reduced by a factor of N while the SNR is improved N times compared to the SNR of a data set sampled N times slower. The reduced data rate permits computationally intensive processing such as FFT analysis so that artifacts can be more easily eliminated. Doviak, et al. (1983) and Smith (1987) describe the optimum number of samples to average given the expected radial velocities and dispersions. Otherwise, the processing is similar to microwave radars following conventional techniques. Rottger and Larsen (Chap. 21A) techniques.
describe the
details
of
ST/MST
radar processing
2.2.3
FM-CW radars
FM-CW
(frequency modulated continuous wave) radars have also played an
important role in boundary layer remote sensing (Richter, 1969; Chadwick, et al.,
1976; Ligthart, et al., 1984).
Using an FM chirp waveform to obtain
range resolution of order 1 m and a continuous wave 8
(CW) to achieve
sensitivity 30 dB greater than a comparably chirped pulse system having the same peak power, this system has given high resolution information on the detailed structure of the boundary layer. Individual insects are apparently discernable, and can be differentiated from atmospheric refractive index variations. Strauch, et al. (1975) and Chadwick and Strauch (1979) have demonstrated both theoretically and experimentally that Doppler, as well as reflectivity, information can be extracted from a distributed target using this pulse compression waveform at microwave wavelengths. Any pulse compression waveform with range-time sidelobes limits the radar's performance in strong reflectivity gradients.
Alternatively, one can use
continuous, periodic, pseudo-random phase coding in a bistatic configuration with similar advantages as Woodman (1980b) describes for the Arecibo S-band
planetary radar. 2.2.4 Mobile radars Airborne and spaceborne radars are an important class of atmospheric remote sensors covered by Hildebrand and Moore (Chap. 22A). Special problems are evident when a moving platform supports the remote sensor. Many of the signal processing problems have well known solutions but have not been field
tested.
The basic processing algorithms are similar to those employed with
ground based sensors, but special processing techniques must be employed to
suppress moving ground clutter and to obtain adequate resolution and sensitivity from spaceborne instruments. Synthetic aperture radar (SAR) techniques can be used only if the platform moves rapidly so that atmospheric targets remain coherent during a "dwell time", thereby giving a synthetic aperture yielding the desired along-track resolution. SAR mapping of precipitation is possible from space vehicles because of the great distance traversed by the antenna during the coherency
time of the targets (Atlas and Moore, 1987). Quantitative measurements of precipitation from space involve a broad range of signal processing problems to achieve both maximum sensitivity and a sufficiently large number of independent samples. Obtaining reliable average echo power from individual storm cells while covering a large cross-track swath in the short times available to traverse a typical along-track beam width requires extremely 9
Research concerning atmospheric target measurements
high processing rates.
is just beginning in this important field (Li, et al., 1987).
Lidar
2.2.5
or
Optical
infrared
radars,
cammonly
known
as
lidars,
scatter
atmospheric aerosols at wavelengths between 10 and 0.3 microns 1974-75; Huffaker,
et al.,
1976; Jelalian, 1977; Bilbro, et al.,
dominates
scattering
1984 and
are the highest.
of the atmosphere where aerosol concentrations
Molecular
(Huffaker,
This makes them most useful in the lower
1986; and McCaul, et al., 1986). regions
from
at
the
shorter
Lidar
is
most useful
in
wavelengths.
severely attenuated by cloud and precipitation so it
is
"clear air" applications (Lawrence, et al., 1972; McWhirter and Pike, 1978).
Lidar requires a receiving aperture several thousand wavelengths in diameter to
achieve
atmospheric antenna
the
necessary
lidars,
gain
both ground
(or telescope)
and
Consequently,
sensitivity.
based and airborne,
"near field" range.
many
operate within the
A distinct advantage of this
near field operation is the collimation of the optical energy into the "near field tube" with minimal "sidelobe" radiation. beamwidths are measured in milliradians.
When in the far field, the
Maximum ranges are a few tens of
kilometers, and pulse volumes are usually elongated.
The expected Doppler shifts and coherence times require sampling at rates of 10 -
100 MHz.
This means that all
the information necessary for complete
spectral processing is acquired from a single pulse.
This makes lidar, by
its very nature, a "fast scanning" atmospheric remote sensor.
Current laser
duty cycle constraints limit PRF's to about 100 Hz, which produces data rates that can easily be processed and recorded (Hardesty, et al.,
1988;
Alldritt, et al., 1978).
An important characteristic of acquiring the data in
a single pulse is
the
degraded range resolution that results when the pulse propagates outward during the data collection interval.
During the sampling interval, "new"
particles are appearing at the leading edge of the illuminated volume, while
"old" particles are disappearing at the trailing edge.
10
This creates an
additional contribution to the spectrum width similar to that caused by antenna scanning for microwave radars.
2.2.6
Acoustic sounders
Acoustic radars, also known as echosondes, sodars, or acdars, are important sensors
for
the
boundary
layer
(Little,
1969).
longitudinal in nature and propagate at about 340 ms
Acoustic -1
.
waves
are
Scattering is from
temperature and velocity fluctuations caused by turbulent motion in the atmosphere.
The processing techniques, while at audio frequencies,
are
similar to those employed by lidar since spectral data representative of the scattering medium are obtained from a single pulse rather than pulse-topulse sampling. shifts, source
Because of the slow propagation speed and small Doppler
sampling the echoes obtained is
avoided.
possible.
Thus,
complex
from a real
(single channel)
(dual channel)
data
data
processing
is
Moreover, the real echoes are sampled at a rate substantially less
than the carrier frequency of the sodar so that zero Doppler shift is offset from
zero
frequency.
In
this
manner unambiguous
estimates can be made.
11
and
signed
velocity
DOPLER PE
3.
SPBCRlM MMENT ErAMHATICN
It is well established that the first three moments of the Doppler power spectral density or the "power spectrum" (incorrectly termed the "Doppler spectrum" in the community) are directly related to the desired atmospheric base parameters:
radar reflectivity, radial velocity, and velocity spectrum
width (Rogers and Chimera, 1960; Groginsky, 1966).
Before we discuss the
power spectrum and moment estimation, we shall find it useful to define the input waveform.
Since
the return
scattering
from
refractive index
from a
individual
range cells
typically
is generated by
large number of randomly distributed particles and/or
inhomogeneities, the received signal process is
central
limit
process
(Parzen, 1957;
theorem)
a very
good
Swerling,
approximation to a
1960;
Mitchell,
(by the
Gaussian
random
Thus,
signal
1976).
processing techniques should be assessed in the context of a statistical
estimation theory framework wherein one seeks to make the best estimate of the ensemble parameters given a particular sample function Davenport and Root, 1958).
(Wiener, 1949;
This statistical estimation framework becomes of
particular importance when one wishes to scan a phenomenon quickly since the random process nature of
the weather signal will necessitate
a certain
amount of averaging if the desired accuracies are to be achieved.
A
single
stationary point target at range R
reproduces
the transmitted
waveform after it has been filtered by the receiver
z(t,R) = A exp[j2rf(t-2R/c)
where
A
function
is
the complex voltage
that
depends
on
the
W(t-2t-2R/c)
amplitude and W(t)is transmit
pulse
length
(3.1)
a range weighting and
the
receiver
bandwidth (Doviak and Zrnic, 1984).
Actual targets in
the atmosphere are composed of many individual scatterers,
distributed over range, radar cross section, and velocity. 13
The received
waveform for a particular distributed target then is a sample function of
the random process which produces the atmospheric return.
We desire to
estimate the mean characteristics of the random target over an ensemble of
sample functions. The vector sum of the return complex voltage from the individual scatterers is z(t,R) = Z Ai exp[j2fi(t-2Ri/c)] W(t-t2Ri/c) i
(3.2)
where the subscript i represents the individual particle. Each particle has a complex voltage return (Ai), a Doppler shifted frequency (fi), and a range (Ri). At any given sampling instant for the kth pulse the received waveform can be represented in the complex signal plane by a vector (or "phasor")
which has an instantaneous amplitude or voltage IVk(R) and phase Ek(R) determined by the instantaneous vector sum of the individual scatterers. The complex signal is then Zk(R) =
Ik(R) + j Qk(R)
(3.3)
where Ik(R)=IVk(R) Icos ek(R) is the in-phase and Qk(R)=|Vk(R) Isin Ek(R) is the quadrature phase component (Rader, 1984).
that
These expressions illustrate
(for a specific received polarization) measurable, the complex amplitude and phase.
only
two quantities are All other quantities are
derived from these based on physical models. 3.1
GENERAL FEURES OF THE DOFPPER POWER SECRM
The concept of the Doppler power spectrum is fundamental in radar signal processing (Haykin, 1985b). A typical power spectrum, shown in Figure 3.1, is a plot of the returned power as a function of the Doppler shifted frequency components in the target resolution volume. The usual sign convention (taken from spherical coordinates) is that a positive Doppler velocity corresponds to a velocity away from the radar; the rate of change in range is positive. This corresponds to a negative Doppler frequency shift. The velocity limits ±Vmax are determined by the Nyquist constraint that two samples per wavelength or period are required to unambiguously measure a frequency (Whittaker, 1915; Nyquist, 1928; Shannon, 1949). For a 14
0 -10
dB -20 -30 -40 -0.5 -0.4 -0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
VELO)CITY/2 Vmax
Fig 3.1
Doppler power spectrum (128 point periodogram) of typical weather echo in white noise. Estimated parameters are velocity 0.4 Vmax, velocity spectrum width ~ .04 Vax, and SNR ~ 10 dB.
15
uniform pulse repetition time T s "Nyquist velocity" is
(equally spaced samples)
Vmax = \/4Ts
.
the so called
(3.4)
The interval [-Vmax, +Vma] is called the "unambiguous velocity interval" or
commonly the "Nyquist velocity interval" and all possible velocities are measured within this interval. The reality of sampling theory dictates that sampled Doppler spectra exist on a circular frequency domain rather than a frequency line extending both directions from zero (Gold and Rader, 1969). Thus, as a target velocity increases beyond Vmax, it aliases or "folds" onto
the negative velocity region of the Nyquist velocity interval (Passarelli, et al., 1984). The signal power spectrum rests on a platform of "white noise", so called because the noise power spectral density is independent of frequency. White noise is caused by several factors including thermal noise from the receiver, phase noise from the transmitter/receiver system, artifacts from the spectrum estimation algorithm, artifacts from receiver non-linearities, and quantization noise from the A/D converters. It is convenient to approximate the signal portion of the power spectrum with a Gaussian shape having some mean velocity and width. The area under the signal portion of the spectrum, not including the contribution of white noise, is the returned power. Depending on the distribution of velocities in the pulse volume and the scattering mechanism, asymmetric spectra and/or multi-modal spectra may occur. Second trip echoes are a common cause of bimodal spectra in klystron systems. Janssen and Van der Spek (1985) found that only about 75%
of observed precipitation spectra had the assumed
Gaussian shape.
For ST/MST radars the spectrum is often assumed to be Gaussian, but spectra
measured at near vertical antenna beam directions (zenith angles less than about 10°) very regularly show one or more strong spectral spikes superposed on a Gaussian shaped base. The spikes result from a corresponding number of 16
quasi-horizontal laminar refractive index structures producing partial reflections while the Gaussian floor results from scattering by turbulent refractive index structures. Moreover, the aspect sensitivity due to the quasi-horizontal laminar structures may produce strongly asymmetric mean power spectra if several single power spectra are averaged for oblique antenna beam directions. The width of the velocity spectrum has a number of contributions including
wind shear, turbulence, particle fallspeed dispersion, antenna rotation (Nathanson, 1969) and, in the case of lidar, range propagation of the pulse during sarpling. It is difficult to separate instrumental effects from the desired signal contributions. The goal of signal processing is to deduce the characteristics of the signal
portion of the spectrum. This means that the other contributions from clutter, noise, and artifacts must be either minimized or removed by the various steps of processing.
There are two basic approaches:
frequency
domain processing using the power spectrum, and time domain processing using
the
autocorrelation
function.
Each
approach
has
its
advantages
and
disadvantages but the essential information available from each is identical
since the power spectrum of the sampled signal and its autocorrelation function comprise a Discrete Fourier Transform (DFT) pair, (Oppenheim & Schafer, 1975; Tretter, 1976): N-1
S(nfo) = Z R(mTs) exp [-j2mnm/N] m=0 R(miTs) = N- 1
(3.5a)
N-l Z S(nfo) exp [+j27rmn/N]
(3.5b)
n=0 where S(nfo) is the Doppler spectrum in multiples of the fundamental frequency shift fo=l/NTs and R(rTs) is the autocorrelation function in multiples of the sample time Ts . This is the discrete version of the celebrated Wiener-Khinchine theorem (Wiener, 1930; Khinchine, 1934).
The
information content is identical in the two approaches. difference between time and frequency domain processing
the
17
The primary is that
information concerning the lower spectral moments is distributed over several frequencies of the power spectrum, while it is concentrated in the small lags of the autocorrelation function. It is important to realize that sampling theory dictates that both S(nfo) and R(mrs) be periodic. That is, the spectrum repeats at multiples of the sampling frequency and the correlation function repeats at multiples of N times the sampling period (NTS). When highly coherent spectral components (e.g. clutter) are present, the correlation usually will not decay to zero Thus, the periodicity requirement of R(mTs) will produce a biased spectrum estimate. Care must be exercised in these cases. within the N/2 samples.
3.2
FREQUENCY DOMAIN SPECTRAL MEMENT ESTIMATION
Estimating the Doppler power spectrum and its moments directly are straightforward techniques (Haykin and Cadzow, 1982). However, some basic questions must be answered first. We implicitly assume a data model for weather and clutter spectra when we choose a spectrum estimation technique. A specific data model such as a sum of sinusoids or white noise passed through a narrowband filter is best analyzed by a spectrum analysis technique compatible with that data model. Robinson (1982) emphasizes this Marple (1987) point in his historical review of spectrum estimation. stresses the importance of using an appropriate model fitting analysis and gives a very well organized discussion of classical and modern spectral estimates using digital techniques. 3.2.1 Fast Fburier transform techniques The Doppler power spectrum may be estimated from the Discrete Fourier Transform (DFT) of the complex signal. The DFT decomposes the observed data into a sum of sinusoids having amplitude and phase that will exactly It is easy to show that these N discrete reproduce the observed data. components are adequate to reconstruct the entire continuous spectrum so long as the complex data samples {zk} are taken at a rate equal to or greater than the bandwidth of the signal. The advantage of measuring the full Doppler spectrum is that spectral impurities such as ground clutter,
18
bi-modal
spectra
or artifacts
can be
suppressed by
intuitive
(if non-
optimal) algorithms.
The so called "periodogram", a frequently used estimator in weather radar as well as many other fields, is an N point spectrum estimator in which the standard deviation of each spectral value equals its mean value. one averages several
Usually
spectra from a divided time series or smooths over
several points in the periodogram to improve the accuracy.
The periodogram
is defined as the squared magnitude of the transformed data sequence {Zk} (Blackman and Tukey, 1958; Cooley and Tukey, 1965; Oppenheim and Schafer, 1975), N-1 P(f) = N- 1 Z hkzk exp [-j27fk]1 2 k=-O 1
where
the """ denotes
an estimate.
(3.6)
The hk term is the "window" which
modifies the waveform being transformed.
In general,
window functions have a maximum value centered on the time
series and are tapered near zero at the ends.
This tapering
spectrum
energy
smearing,
a
"leakage"
of
spectral
reduces the
introduced
discontinuity imposed by sampling when the end points are joined.
by
Windowing
also effectively reduces the number of points in the time series. simplest
window
is
hk =
1
(or no windowing).
the
The
For this window, the
periodogram of a single point target has the first side lobe only 13 dB down from the peak.
This is not a problem for estimating the mean and variance
of the designed signal, but if strong clutter is present, then the sidelobe
power from the clutter that leaks throughout the Nyquist interval can mask weaker weather echoes. windows. description
Harris
Table 2 shows characteristics
(1978)
of window
and Marple
functions.
(1987) both
In general,
of several
give
the
19
extraordinary
lower the
offered by a window, the broader its main lobe response. degrades the spectral moment estimates.
an
common
sidelobes
This broadening
Table 2 Characteristics of time series data windows (Marple, 1987).
Window Name Rectangle Triangle Hann Hamming Gaussian Equiripple
Highest Sidelobe -13.3 -26.5 -31.5 -43 -42 -50
Sidelobe Decay Rate
dB dB dB dB dB dB
-6 -12 -18 -6 -6 0
The windowed periodogram P(f) Nyquist
interval.
dB/octave dB/octave dB/octave dB/octave dB/octave dB/octave
Equivalent Bandwidth (Bins) 1.00 1.33 1.50 1.36 1.39 1.39
1/2 Power Bandwidth (Bins) 0.89 1.28 1.44 1.30 1.33 1.33
can be evaluated at any frequency f in the
The Fast Fourier Transform
(FFT) is simply a highly
efficient technique for evaluating the DFT at N equally spaced discrete frequencies
(Welch, 1967).
Although the FFT algorithm is attributed to
Cooley and Tukey (1965), a recent historical investigation into the history of the Fast Fourier Transform by Heideman, et al.
(1984) attributes an
algorithm very similar to the FFT for computation of the coefficients of a finite Fourier series to Gauss, the German mathematician.
Apparently the
first implementation of the FFT on a weather radar was in December 1970 at the CHILL radar (Mueller and Silha, 1978).
3.2.2
Maximum entrpy techniques
The aforementioned Fourier transform techniques have been understood since the time of Fourier and Gauss and are well documented by Jenkins and Watts (1968).
Only recently have techniques based on covariance estimates and
probabilistic concepts been explored.
Kay and Marple
(1981) and Childers
(1978) have termed these parametric techniques "modern spectrum analysis". Marple (1987) points out that maximum entropy, maximum likelihood and other techniques are "modern" in the sense that short data
sequences produce
spectral resolutions better than the inverse duration of the data sequence, which is characteristic of classical spectrum estimators. digital
Furthermore, fast
algorithms have been developed which allow computing hardware to
perform the computations
in the required time frames. 20
This interest in
alternative
spectrum estimators
can be explained by categorizing expected
performance improvements as increased resolution or increased detectability. Both Jaynes (1982) and Makhoul (1986) attempt to clarify some confusion and misleading notions related to the maximum entropy techniques.
Maximum
entropy
(ME) spectrum
analysis
estimates
parametric techniques to define the spectrum.
the
spectrum
using
The parameters are typically
derived from the data samples or some estimated autocorrelation sequence. The
ME
technique
was
developed
by J.P.
Burg
(1967, 1968,
1975)
as
a
geophysical prospecting technique for high resolution measurement of sonic wave reflections and velocities. spectrum
model
can
approximate
increasing its order L.
Makhoul any
(1975) shows that the all pole ME spectrum
arbitrarily
closely
by
He shows that the ME spectra minimizes the log
ratio of the estimated spectrum to the true spectrum integrated over the Nyquist
interval.
The MST radar community
(Klostermeyer,
1986)
and the
lidar community (Keeler and Lee, 1978) have used the maximum entropy method for characterizing atmospheric targets. Sweezy (1978) and Mahapatra and Zrnic
(1983) have computed maximum entropy spectrum estimates on simulated
weather radar data and compared them with Fourier transform and pulse pair estimators.
Haykin, et al. (1982) describe how maximum entropy techniques
can be applied to Doppler processing of radar "clutter"
including weather
and birds for aviation hazard identification.
Atmospheric echoes, whether from precipitation, aerosols, or turbulence, can be modeled by "autoregressive" (AR) techniques as narrow band filtered noise.
These AR and the standard Fourier technique appear to represent the
essential
spectral
features
well
although
little
quantitative work
available for comparison in the atmospheric echo application. (1971) and Ulrych and Bishop
is
Van den Bos
(1975) show that maximum entropy spectrum
analysis is equivalent to least squares fitting of a discrete time all pole model to the observed data.
As noise is added to the observations the
autoregressive moving average
(ARMA) model
1980; Marple, 1987).
21
is more
appropriate
(Cadzow,
The justification for studying maximum entropy spectra is its ability to estimate complete spectra from the first few lags of the autocorrelation function rather than from all the autocorrelation lags that are required by the Fourier transform technique (Radoski, et al., 1975). Since only the first few autocorrelation values are known with any confidence, this property may be critically important when the sampled data sets are very short. Baggeroer (1976) computes confidence limits for ME spectra which are applicable to atmospheric echoes. The "order" of the maximum entropy spectra defines the number of lags, or equivalently the number of poles in the filter through which white noise is passed in modeling the data.
A larger order allows non-Gaussian spectral
detail to be more accurately represented, e.g. a weak atmospheric echo in the presence of a much stronger ground clutter. However, a larger order requires a longer data sample to obtain accurate estimates. The basic technique uses the sampled input data to compute R(0), R(1),...
R(L) for the Lth order estimator. Additional lags are realized by requiring that the entropy (in an information theoretic sense) of the probability density function having the extended autocorrelation function be maximized. This extended autocorrelation function allows computation of coefficients for a whitening or linear prediction filter.
The ME spectrum is computed
from these filter coefficients which are defined by the matrix equation A = R-1 P
(3.7)
where A is the filter coefficient vector, R is the autocorrelation matrix The and P is the autocorrelation vector (Ulrych and Bishop, 1975). coefficient estimates can be rapidly computed using the Levinson algorithm (Makhoul, 1975; Anderson, 1978). This filter removes the predictable components from the input data and the optimum filter of order L minimizes the prediction error. The Lth order ME spectrum estimate can then be computed
22
^2 (L)
SME (f)
(3.8) L
I1 -e
am exp[-j2rfm] 12
m=l where am are the elements of A and ao2(L) (1967)
gives
the
"forward-backward"
is
final prediction error.
technique
of
estimating
the
Burg linear
prediction coefficients directly from the data which frequently permits more detail to be shown in Kesler
(1976)
Friedlander
give
(1982)
the spectrum. the
complex
Smylie,
and Makhoul
form
(1977)
of
et al. the
describe
ME
(1973)
and Haykin and
spectrum
estimator.
lattice structures for ME
spectrum estimates which are computationally more efficient and identical to Burg's method.
Papoulis
(1981) attempts to interrelate the various aspects
of maximum entropy and spectrum estimation in his mathematical review paper. Marple
(1987)
presents a
more readable
exposition.
Cadzow
(1980,
1982)
extends the ME concept to rational models.
Keeler and Lee (1978) and Mahapatra and Zrnic
(1983) have shown that the
pulse pair frequency estimator is identically the mean (or the peak, in this special case) of the first order maximum entropy spectrum.
The atmospheric
remote sensing community has been using the simplest form of ME for almost two decades! radars,
Its relevance to accurate parameter estimation for weather
ST/MST profilers
and
lidar
signals
is an
active research
area
(Haykin, 1982).
3.2.3
Maximum likelihood techniques
Maximum likelihood (ML)
estimation is
a statistical concept that gives the
most likely outcome or minimum variance estimate of an experiment based on a set
of
known probabilities.
"efficient", variance.
i.e.
there
ML estimates
of
is no other unbiased
spectral
parameters
estimator having
a
are
lower
It is well suited for estimating parameters of a spectrum whose
shape is known or assumed when neither a priori knowledge nor a valid cost
function associated with moment estimator error is known (Van Trees, 1968). Zrnic (1979a) uses ML techniques to derive the minimum variance (Cramer-Rao) 23
bounds of spectral moment estimators for application to atmospheric radar data. He compares present estimators to these bounds and interprets Levin's (1965) results in a modern framework.
Moreover, he shows that the pulse
pair estimator is ML for a Markov process. In general closed form solutions for ML estimates of spectrum moments are
quite complicated and difficult to compute. The optimum (ML) processor depends on the underlying signal statistics which in turn depend on the spectrum
shape
and SNR.
Shirakawa
and Zrnic
(1983) evaluate
the ML
estimator for sinusoids in noise and find a slight improvement over the pulse pair estimator at low SNR's.
Novak and Lindgren (1982) derive the
exact ML mean velocity estimator for Gaussian shaped spectra using more than
one autocorrelation lag. Their technique is similar to Lee and Lee's (1980) poly pulse pair velocity estimator.
Miller and Rochwarger (1972) show that
for independent pairs, the pulse pair estimator of mean frequency is ML for
an arbitrarily shaped spectrum so long as the normalized width is small. Sato and Woodman
(1982) use a least square
fit algorithm to estimate
spectral parameters, including noise and clutter parameters, by assuming
prior knowledge of the spectral shapes. Woodman (1985) shows that this technique is a ML estimator of the spectral characteristics. It is gratifying that the simple pulse pair estimators approach the minimum variance bound over a wide range of SNR's.
If the spectrum shape is completely unknown, the ML spectrum gives the most probable estimate which concentrates the spectral energy at the input signal
frequencies while minimizing other spectral energy in a statistically optimum sense (Capon, 1969; Lacoss, 1971). The statistical rationale for using ML estimation is that the ML spectrum estimate provides a minimum variance, unbiased estimate of the power at a given frequency. Burg (1972) has shown that in the mean the Lth order ML spectrum is just the following combination of ME spectra up to order L:
[SML(f) -
1
L = L1 Z [SME,m(f)m=l
24
1
(3.9)
Thus, the mean ML spectrum is a smoothed version of mean ME spectra.
It has
many of the same properties as ME spectra but the details are obscured by combining all order ME spectra.
There have been theoretical studies of ML
spectra but little application to atmospheric data.
Klostermeyer (1986) has
computed ML spectra for VHF radar data.
3.2.4
Classical spectral moment computation
The spectrum moments can be directly related to the reflectivity, velocity, and dispersion parameters desired
for further analysis.
Computing these
moments has historically been performed using classical moment calculations based
on techniques
from probability theory when considering the power
spectrum as a density function of frequency or velocity components of the desired
signal
(Denenberg, 1971,
"sampling theorem"
1976).
imposes certain
For
sampled data systems
requirements on moment
the
and transform
computations that cannot be ignored -- namely replication in the frequency domain and circular convolution (Oppenheim and Schafer, 1975).
Let the power spectrum of the received signal be denoted by S(f).
Then the
classical spectral moments are given by
Mn =
The zeroth moment (MO)
x
fn S(f)df .
(3.10)
is the area under S(f) and represents total signal
clutter, and noise power.
Of course, we are usually interested only in the
signal power, so the clutter and noise powers must be estimated and removed. Noise
power
is
generally
easy
to
remove,
but
clutter
removal
causes
difficulties to the parameter estimation process.
The classical normalized first moment represents mean velocity and is given by the linear weighting of S(f) over the Nyquist interval
fc =
f S(f)df / Mo
V = (X/2) fc
(3.11a) (3.1Ib)
25
Note that white noise biases the velocity towards zero and for a pure noise spectrum the mean velocity is identically zero.
Various techniques have
been described for mitigating this bias, most of them requiring manipulation of the power spectra. the
noise
spectral
Thresholding the spectrum points with some value near density
is
but
common,
some
sensitivity
is
lost
(Hildebrand and Sekhon, 1974; Sirmans and Bumgarner, 1975a; Klostermeyer, 1986).
The "spectral balancing technique" rotates S(f) until the signal spectrum is near
zero
velocity.
so that
the signal
and the
noise share
the same
zero mean
The amount of rotation represents the mean velocity of the signal
component and removes errors due to aliased spectra.
The same effect is
obtained by computing the offset first moment
4
(3.12)
(f-fc) S(f)df = 0
where fc is varied to obtain equality.
The
normalized
second central moment represents the velocity dispersion
within the pulse resolution volume.
Shear, turbulence and precipitation
motion (fallspeed oscillations, etc.) contribute to a distribution of radial velocities
(Nathanson and
Reilly,
1968).
A
contribution
from antenna
scanning during the finite dwell time may also be significant 1969).
(Nathanson,
The velocity dispersion (width) is the square root of the second
central moment of the spectrum estimate:
af2 = I
(f-fc)2 S(f)df / M0 W = (>/2) of .
(3.13a) (3.13b)
Spectrum estimation algorithms are fairly time consuming to invoke, and once the frequency domain is entered, there is still substantial computation to accurately extract the meteorological moments.
The main reason for entering
the frequency domain lies in the ability to more easily filter spectral 26
artifacts unimodal
or
identify multi-modal
and generally free
from
spectra.
In cases where spectra
artifacts,
more efficient time domain
are
processing is typically used.
3.3
TIME DIXMAIN SPECTRAL MO'ENT ESTIMATIC
The basis for time domain moment estimation is the transform relationship of the autocorrelation function of the complex signal to the power spectrum.
An estimate of the autocorrelation can be easily calculated from the complex input time series {Zk),
1
R(m) = (N-m)
N-m-1 2 Zk* Zk+m k=0
(3.14)
where m is the lag between the two data series. usually
For uncontaminated spectra,
only two or three lags are necessary to obtain the moments of
interest.
This represents a substantial savings in computation over the
spectrum domain approach.
The general relationship between the complex
autocorrelation function and the nth classical spectral moment is
Mn =
where
R[n] (0)
is
the
evaluated at lag = 0
nth
R[n](0)/(j2w)n
derivative
(Papoulis, 1962;
of
(3.15)
the
Bracewell,
autocorrelation 1965).
function
The first three
spectral moments are used to estimate the reflectivity, radial velocity, and velocity dispersion or width respectively (Miller, 1970; Miller, 1972).
3.3.1
Gecmetric interpretatins
The complex autocorrelation
function, which is the basis for time domain
moment estimation, is often depicted as its real and imaginary components, but
an
alternative
3
dimensional
representation
allows
a
better
understanding of the covariance, or pulse pair, mean frequency estimator. Consider the complex R(m) to be a 3D helix that is wide at the center and
tapered toward zero radius at the ends having a Gaussian shaped envelope.
27
Figure
shows a drawing of this continuous autocorrelation helix.
3.2
A
sampled autocorrelation helix will consist of points on this helix spaced at the PRT.
Note that zero lag, R(0), is
at the center and has no imaginary
The radius at lag 0 represents the signal power and the real
component.
delta function
at lag 0
represents the noise power.
The width of the
Gaussian envelope of the helix represents the inverse velocity spectrum width
The rotation rate of the helix defines the mean
or dispersion.
velocity of the signal.
For a given spacing of autocorrelation function
samples the angular rotation between a pair of samples is a measure of mean velocity. velocity
the angle of the complex estimate R(1) gives the mean
Thus,
of the received signal
expressed as a fraction of the Nyquist
interval which is the "pulse pair estimator" used almost universally for mean velocity in weather radar and lidar processors.
A
useful
geometric
spectral moments and the autocorrelation function can be found in and Siggia (1983).
classical
interpretation of the relationship between
Passarelli
This interpretation illustrates many of the properties
of pulse pair estimators.
3.3.2
"Pulse pair" estimators
The advent of the so-called pulse pair, double pulse, or complex covariance technique (Rummler, 1972;
1968a; Woodman and Hagfors, 1969; Miller and Rochwarger,
and Groginsky, 1973;
Berger
Woodman
and Guillen,
1974)
for mean
velocity estimation was revolutionary since the algorithm arose at about the
same time that it could be implemented in hardware for a significant number of range bins. of hardware However,
Lhermitte (1972) and Groginsky (1972) reported the first use
signal
processors
covariance processing
and weather for velocity
radars using this technique. measurements
apparently was
first used in March of 1968 for ionospheric velocity measurements and Hagfors, 1969).
(Woodman
Woodman and Guillen (1974) also reported covariance
based velocity measurements in the mesosphere at the Jicamarca MST radar in 1970.
This algorithm development in the MST community was independent of
Rummler's work.
The pulse pair algorithm led to an exciting growth in the
use of Doppler radar by the scientific community (Groginsky, Ihennitte,
1972; Sirmans,
1975; Ihermitte and Serafin, 1984). 28
et al.,
1972;
Real axis x
R(O) Pn PS
R(I) Imaginary axis
lag m
v1 J
Fig 3.2
Three dimensional representation of the complex function as a helix. Radius of helix Rs(O) is total signal power, Ps; rotation rate of helix is velocity, V; width of envelope! is inversely velocity spectrum width, W. Delta function Rn(0) power.
29
autocorrelation proportional to proportional to proportional to represents noise
Other time domain algorithms such as the "vector phase change" (Hyde and Perry, 1958) and the "scalar phase change" (Sirmans and Doviak, 1973) are closely related inferior.
to the pulse pair
Sirmans and Bumgarner
estimator, but (1975b)
their performance
capare these
and
is
other mean
frequency estimators.
It is well known that the first few lags of the autocorrelation function are sufficient to deduce spectrum parameters of interest. 1984),
Bracewell
(1965),
Woodman and Guillen
Papoulis
(1965 and
(1974), and Passarelli and
Siggia (1983) show that the autocorrelation function can be represented by a Taylor series
expansion
in terms of the central moments of the Doppler
spectrum with the low order moments being the leading terms.
In other
words, the first few lags of the autocorrelation function contain the moment information of interest.
For an arbitrary spectrum, these expansions have
the form
R(mits) = A(mrTs) exp[-j0(mrs)] .
(3.16)
The even function A(mrTs) is determined primarily by the even central moments (e.g.,
power, variance
and kurtosis),
while the odd
function 0(mTs)
is
determined primarily by the mean velocity and the odd central moments (e.g., skewness).
Estimators
can be generated
for any moment, provided that a
number of autocorrelation lags are measured. the magnitude for lag zero.
Therefore,
sufficient
White noise power Pn biases
the total received power must be
corrected for noise,
Pr = R(0) - Pn
(3.17)
The pulse pair mean velocity estimator is not biased by white noise and is obtained by taking the argument of the first
V = (
autocorrelation lag,
1 /2) (2Ts)tan[Im R(T)/Re R(Ts)] .
30
(3.18)
The pulse pair spectrum width is given by W = (X/2) (2fTs)-
1
[1 - p(Ts) (1 + SNR-1)]
(3.19)
where p(Ts) =|R(Ts) |/R(O) is the normalized first lag and the noise power must be determined independently. 3.3.3
Circular spectral rmment computation for sampled data Sampled data systems utilize the complex plane and z-transform theory to formally express the relationships between the time and frequency domains (Oppenheim and Schafer, 1975). For example, the DFT of the autocorrelation function is formally the z-transform of the sampled autocorrelation function
evaluated on the unit circle in the z plane, i.e. |z|=1 or z = exp[-j27f]:
S(f) =
N-1 z R(mTs) z-m m=O Iz=exp [-j 27rf]
The unit circle on the complex z plane is important
(3.20)
in understanding
concepts of sampled or discrete data systems, specifically concepts of
digital signal processing. Figure 3.3 shows the z plane and the frequencies associated with various points on the unit circle. Zero frequency, where ground clutter usually appears, corresponds to z=l and the Nyquist frequency
(where velocity spectra alias into the next Nyquist velocity interval) corresponds to z=-l. Thus, the z plane representation of spectral space allows an immediate and simple geometric interpretation of velocity aliasing
and the velocity ambiguity arising from sampling too slowly. Analysis and synthesis of digital filters requires heavy application of z transform theory, thus easily allowing visualizing the effect of various types of ground clutter filters, for example. It is natural to compute spectral moments on the unit circle rather than along the frequency line in the Nyquist velocity interval. The zeroth moment or total receiver power, is still that area under the spectrum
31
Imag
f = (2Ts)-'
Real I0
f = -(4Ts )-I
Fig 3.3
Z plane
Periodogram power spectrum plotted on unit circle in the z-plane. Note velocity aliasing point, the Nyquist velocity, at z=-l.
32
whether on a line or on a circle.
However, higher order moments can be
different for the two cases (Passarelli, et al., 1984).
A simple geometric derivation shows that the first circular moment estimate, fc, of the estimated spectrum, S(f),
is the normalized frequency at which
the center of mass on the circle is located, S (27m/N) sin(27m/N) fc = (27)-
1
tan-1
(3.21) Z S(27m/N) cos(2nm/N)
where the summations run over 0 to N-l.
Trigonometric manipulation converts
this equation to N-1 Z S(27m/N) sin[27(n/N - fc)] = 0 n=0 Thus, fc is the sinusoidal weighted mean of S(f)
.
(3.22)
(Zrnic, 1979a).
Further,
we see that the numerator and denominator of (3.21) are the imaginary and real parts of R(mrTs) and that the circular first moment is identically the pulse pair frequency or velocity estimator.
Two points are clear from this discussion: 1) white noise does not bias the pulse
pair
particular
frequency
estimate
frequencies
on
the
because
the
circle,
and
noise 2)
does
not
symmetric
weight
spectra
any have
identical first moments using either the classical (linear weighting) or the circular
(sinusoidal weighting) computations.
Asymmetric spectra produce
different first moment estimators but there are no compelling reasons to
prefer linear weighting over the more common sinusoidal weighting (the pulse pair estimator).
Indeed,
for sampled data
systems the circular moment
computation is more natural than classical moment computation.
3.3.4
Pbly pulse pair techniques
If we accept the premise that knowing lags of the autocorrelation function past the first allows a processor to extract additional information about the
received
signal,
then one should expect
to reduce the variance of
velocity estimates by using, not only R(1), but R(2), R(3), etc. 33
The
Classical:
2(f - f,,i) S(f)= o
m
2Ts
freq -
2Ts
Y
Circular: Zsin[27r (f
fci)]S(f) 0 -t~~~~~i,,]scr, ·o~~~~~~~~~~~~
fcir
m
+
2T5
Fig 3.4
Comparison of classical
and circular
(pulse pair)
first moment
estimators. Classical estimate is determined by linear weighting of spectrum estimate and circular estimate, by sinusoidal weighting.
34
variance reduction can be realized only if the received signal is coherent over the additional lags. Lee (1978) proposed the "poly pulse pair" algorithm for lidar signal processing. Velocity estimates can be found from a weighted average of the estimate given at each lag, where the smaller lags
are given higher weighting since the correlations are higher. Poly pulse pair velocity estimates (using a few lags) produce lower variance estimates than the pulse pair estimates when the spectrum width of the signal is only a few percent of the sampling frequency (Lee and Lee, 1980). Strauch, et al. (1977) evaluated poly pulse pair for 3 cm radar processing. They concluded that for typical velocity spectrum widths and PRF's (sample rates) used with X-band Doppler radar, the coherence time was frequently too
short to give a significant improvement in the velocity estimates. However, for infrared lidar the coherence times and sample rates permit a significant improvement in reflectivity, velocity, and width estimates (Bilbro, et al., 1984).
Furthermore,
Rastogi and Woodman
(1974) and Srivastiva, et al.
(1979) use multiple lag estimates of the correlation function to estimate
moments of a Gaussian shaped spectrum.
Several independent estimates of the
autocorrelation function can be found and a Gaussian shaped curve fitted to
these samples.
Sato and Woodman (1982) have used this nonlinear curve
fitting technique to estimate signal, clutter, and noise parameters at the Arecibo ST radar. 3.4
UNCERTAINTIES IN SPECThUM
UMENT ESTIfM4AI
Any estimator has an associated uncertainty. In atmospheric radar signal processing the velocity spectrum moments are being estimated with some uncertainty that depends on the processing interval, the coherence time or velocity width, and the SNR. Zrnic has published extensively on weather radar spectrum estimator uncertainties
and his results are succinctly
described in Doviak and Zrnic (1984). A summary is given here. 3.4.1 Reflectivity Marshall and Hitschfeld (1953) describe the probability density function of the distributed weather target.
The received signal is a complex Gaussian
process which has a Rayleigh amplitude distribution and an exponential power
35
distribution. Thus, the mean received signal power is Ps with variance Ps2 and the coherence time is determined by the spectrum width of the signal. The number of independent signal samples in a given integration time Td seconds is approximately M I = 2/WT d
(Doviak and Zric, 1984) where W is the
spectrum width (standard deviation) in Hertz. The number of independent noise samples is just M = Td/Ts, the total number of samples in the dwell time. Therefore, the variance of the mean power estimate is approximately var(Pr) = Ps2 /MI + Pn2 /M . Doviak and Zrnic
(3.23)
(1984) show that if the number of independent signal
samples is smaller than about 20 and a log receiver is used, the bias in the
estimated received power depends on M I and its variance is not exactly proportional to 1/MI. A square law receiver does not encounter these problems.
Marshall and Hitschfeld (1953), as well as a recent review by
Ulaby, et al.
(1982),
show that the ratio of the mean power to the
fluctuating power associated with a single sample of a Rayleigh quantity is 5.6 dB.
Therefore, for M I independent samples the signal power estimates
are known within 5.6/Mj dB. I
Averaging independent samples obtained in
range can further reduce the variance. 3.4.2
Velocity
Woodman and Hagfors
(1969)
used statistical analysis of Gaussian random
variables to estimate the uncertainty of pulse pair velocities. Berger and Groginsky (1973) applied perturbation analysis to derive the variance of the independent and contiguous pulse pair frequency estimators. later extended
Zrnic (1977b)
their results to spaced but correlated pulse pairs.
Two
conditions, both of which are usually satisfied for a large number of
samples (M), are necessary for the analysis to be accurate: M >>
M >>
A
/47
(SNR- 1
W Ts
(3.24a)
+ 1)2 /
p2 (Ts)
(3.24b)
where W is the velocity spectrum width and p(Ts) = R(Ts)/R(0) is the autocorrelation function at lag T s (the PRT) normalized to unity. At high 36
SNR and for large enough M contiguous
pairs
typical
that both conditions are satisfied,
of
and for
radar Doppler processing, and for Gaussian
shaped spectra, the variance of the velocity estimate is
var(V) = f W/
8/7 M Ts .
(3.25)
Table 3 summarizes the velocity uncertainties at high SNR for three cases: 1) contiguous samples, variance bound. width in ms (Wn).
-1
2)
independent sample pairs,
Expressions are given both in
and 3) the minimum
terms of the actual spectrum
(W) and the width normalized to the Nyquist velocity interval
Figure
3.5
shows
the
standard
deviation
of velocity
estimates
normalized to the Nyquist velocity interval and to the square root of the number of samples M as a function of the normalized spectrum width. is
a
parameter
for
the
two
sets
of
curves
--
those
for
The SNR
the typical
contiguous pairs and for less typical spaced pairs of pulses (Campbell and Strauch,
1976;
Doviak and Zrnic,
1984).
Note that reasonably accurate
velocity estimates can be obtained for a given M doawn to SNR ~ 0 dB so long as the Gaussian standard deviation velocity width is less than about 0.2 of the Nyquist velocity interval 2Vmax.
Woodman
(1985) discusses
errors
for multiple lag velocity estimators
which the lags are statistically dependent.
in
By weighting the correlation
estimates in an optimum fashion, he concludes that for high SNR only a few (2 or 3) lags are necessary.
3.4.3
Velocity spectrum width
Benham, et al. (1972) and Berger and Groginsky (1973) applied a perturbation analysis to the spectrum width estimator and Zrnic their results to arbitrarily spaced pulse pairs.
(1977b) later extended
Their primary result for
high SNR, contiguous pairs, and narrow, Gaussian shaped spectra is that the variance of the velocity width is
var(W) = 3 XW / 64/T M Ts
37
.
(3.26)
THBLE 3
Expressions for variance of velocity estimators at high SNR. Assumes Gaussian spectra in white noise, low normalized velocity width (Wn=W/2Vmax) and large M. Expressions apply to both pulse pair and Fourier transform estimators.
Var(V) using W
Contiguous samples (typical case)
x
W
167r MTs 2
8]/7w MT
wn
X2
Independent pairs 2M
Minirum variance bound
Var(V) using Wn
8Mr5 2
48 TS 2 2
-MX
M
3\
2
Wn4
W4
X2
2
MTs
38
Wn2
C\M <>
Q
V(
'° 0
1.
z
0
u Q 0
0.5
Q 0
0.1
0.2
0.3
0.4
NORMALIZED SPECTRUM WIDTH oavn Fig 3.5
Spectrum Velocity error as function of spectrum width and SNR. M is =2WTs/X. vn=W/2Vn interval, Nyquist width is normalized to 1 velocity Nyquist to is normalized error and number of sample pairs interval, 2va = 2Vmax . Small circles represent simulation values (Doviak and Zrnic, 1984).
39
Table 4 summarizes the width uncertainties at high SNR for three cases: 1) contiguous samples, 2) independent sample pairs, and 3) the minimum variance
bound.
Expressions are given both in terms of the actual spectrum width in ms-1
(W) and normalized to the Nyquist velocity interval (Wn). Figure 3.6 shows the normalized standard deviations of the width estimates as a function of normalized
spectrum width for a range of SNR's. The width estimator is relatively good if the normalized width is between 0.02 and 0.20 of the Nyquist interval and the SNR > 5 dB.
40
TABLE 4 Expressions for variance of width estimators at high SNR. Assumes Gaussian spectra in white noise, low normalized velocity width (Wn=W/2Vmax) and large M.
Expressions apply to both pulse pair and Fourier transform estimators.
Var(W) using W
Contiguous samples (typical case)
Independent pairs
3 X2
3 w 64W,/ 642z MT
128J7r MTS 2
s
w2
x 2
8MT2 8MTs2 81]?
2M
Minimum
variance bound
2880 T 4 4
Var(W) using wn
45 X2
W6
M X4
MTS2
41
2 Wn2
Wn
c
C I
0
z 0.5 a
cn
Fig 3.6
n
I
0
.1 0.2 0.3 NORMALIZED SPECTRUM WIDTH, Ovn
0.4
Spectrum Width error as a function of spectrum width and SNR. width is normalized to Nyquist interval, vn=W/2Vmax=2WTs/X- M is
number of sample interval, 2Vma x .
pairs and error is normalized to Nyquist Small circles represent simulation values
(Doviak and Zrnic, 1984).
42
4.
SIGNAL PROCESSING TO ETTMINATE BIAS AND ARIT'ACIS
The primary goal of an effective signal processing scheme is to provide accurate, echoes.
unbiased
estimates
of
the
characteristics
of
meteorological
This means that in addition to moment estimation, the signal
processing algorithms must also eliminate the degrading effects of ground clutter targets,
range
aliasing and velocity aliasing.
Indeed, this
challenging aspect of signal processing has received considerable attention in the recent literature. 4.1
DOPPLER TECHNIQUES FM GROUND CLITER SUPPRESSION
Ground clutter poses a significant problem for both coherent and incoherent radar applications.
Clutter biases the reflectivity, mean velocity and
velocity spectrum width estimates.
It significantly reduces the effective
area of coverage at close range where the azimuth resolution is best. Even weak clutter can frequently mask clear air echoes. Fortunately, signal processing can greatly reduce the effects of clutter. (1981), Zrnic, et al.
Zrnic and Hamidi
(1982), and Evans (1983) address various aspects of
Doppler clutter cancellation. Clutter cancellation coherent systems.
is possible
for both coherent
(Doppler) and non-
Non-coherent techniques rely on the Rayleigh distribution
of the amplitude fluctuations of weather echo to differentiate between clutter and weather (Geotis and Silver, 1976; Tatehira and Shimizu, 1978; Aoyagi, 1983). The performance of this approach uses the correlation of successive samples which depends on the Doppler spectrum width (Sirmans and Dooley, 1980). Clutter cancellation on most modern systems is performed via Doppler techniques.
Coherent ground based systems rely on clutter being
nearly stationary and use high-pass digital filters to eliminate targets in a narrow bandwidth near zero velocity. requirements
and
clutter
filter
Groginski and Glover (1980) give
specifications
particular to weather radar systems.
43
and
design
concepts
4.1.1
Antenna and analog signal considerations
The first line of defense against clutter is an antenna with low sidelobes
and a good radar site. Main lobe clutter is very difficult to suppress because clutter targets are usually much stronger than weather targets. However, since sidelobes are usually down at least 20 dB (one way) from the peak power, signal processing is effective in suppressing resulting clutter
power without problems caused by a saturated receiver. Shorter wavelengths generally offer better signal-to-clutter ratios than longer wavelengths given the same targets. This is because the power returned from Rayleigh scatterers goes inversely as the 4th power of the radar wavelength, while large clutter targets will behave more like specular
reflectors having a lesser wavelength dependence (Barton and Ward, 1984). Superior clutter cancellation performance depends critically on the linear
dynamic range of the transmitter/receiver system. This dynamic range is governed primarily by the system phase noise and the linear dynamic range of the receiver itself. The phase stability of the oscillators used in the radar will determine the degree of clutter cancellation that is possible. The effect of phase noise is to spill power from a coherent target into white noise. In the case of a strong clutter target and a weak weather target, even a relatively small amount of phase noise can obscure a weather target under the phase noise floor. For Gaussian distributed phase noise and a coherent clutter target, the maximum clutter-to-phase noise power ratio (CNR) that can be achieved for small phase errors is straight forward to compute (Skolnik, 1980) as CNR= exp(-/2 }/(l-exp{-p 2 })
« P-2
for P<<1
(4.1)
where p is the pulse-to-pulse rms phase error in radians of the complex (baseband) signal. The maximum CNR that can be tolerated is equal to the clutter-to-signal ratio (CSR) that corresponds to a signal-to-phase noise power (SNR) of about 0 dB.
For example, a klystron transmitter can achieve
better than 0.1 degree rms phase error which corresponds to 55 dB CNR. 44
A
signal at 55 dB CSR would have an SNR of 0 dB. If 55 dB of main lobe clutter power could be cancelled, and only the phase noise power or clutter residual
remained, there would be an adequate SNR for Doppler processing. Some coherent-on-receive magnetron systems may achieve only 5 degrees of phase stability (21 dB CNR) depending on the quality of the phase lock loop that synchronizes the receiver to the transmitted pulse. Therefore, it is frequently not cost-effective to design a signal processor capable of more
than 20-25 dB of clutter cancellation for many magnetron systems. designed magnetron system can achieve much better phase stability.
A well
In many systems it is the dynamic range of the linear receiver that poses the fundamental limit on the ability to separate weather signals from strong clutter signals. If the linear receiver has a dynamic range of 50 dB, then this will be the order of the maximum clutter-to-signal ratio that can be handled.
High performance clutter cancellation that is commensurate with
the phase stability of a klystron typically requires a "fast AGC" gain control and, essentially, floating point digital data conversion. Other AGC techniques are less effective and may degrade the existing inherent quality of a stable system. But because they introduce less noise than a typical magnetron transmitter, they can be used in magnetron systems without sacrificing overall system performance. The simplest form of clutter cancellation by Doppler signal processing is to simply ignore strongly reflecting narrow width targets that have velocities
near zero.
On a color velocity display, for example, those bins can be
assigned the background color. More sophisticated processors use either time
domain digital filtering or frequency domain filtering.
Which approach is
used depends on the general philosophy of signal processing that is employed
for spectrum moment estimation. 4.1.2
Frequency domain filtering.
Frequency domain processing was discussed earlier. narrow spike (<1
ms-1)
Clutter is typically a
centered about zero frequency or DC (direct current).
Weather echoes are usually broader, so that it is possible to remove the clutter and then interpolate the weather signal across the gap. The first 45
step in frequency domain filtering is to enter the frequency domain via some spectrum estimation technique. This is usually done via an FFT. The choice of the time-domain window is critical since the window sidelobes should be matched to the dynamic range characteristics of the transmitter/receiver system. For example, a 57 dB Blackman window (Harris, 1978) might be used in a klystron system but it would not be justified for a magnetron system that has a phase noise limited CNR of 25 dB. Removal of clutter in the frequency domain is easily performed by the human eye, and it
is
not difficult to develop algorithms that achieve ;30 dB
suppression. Passarelli, et al. (1981) discuss several algorithms for frequency domain clutter cancellation and point out the adaptive nature of the general technique, i.e., both the notch width and depth of the filter can be adjusted to remove only the clutter that is present, with minimal distortion of overlapped weather or noise. On the other hand, time domain filters usually, but not necessarily, have a fixed notch width and stop band attenuation. 4.1.3 Time domain filtering Time domain digital filtering has been an active research area for over 20 years (Kaiser, 1966; Gold and Rader,
1969; Oppenhiem and Schafer,
Rabiner and Gold, 1975; Tretter, 1976; Roberts and Mullis, 1987).
1975; Precise
control of the digital transfer function allows filter characteristics not obtainable with analog filters. Digital filters fall into two general categories, finite impulse response (FIR) filters and infinite impulse response (IIR) filters. Both of these are used in current weather radars wherein the I and Q values are filtered separately. An example of a simple IIR filter is an exponential average of the I and Q values to determine and remove the DC offset. An example of a simple FIR filter is to calculate the DC offset over a fixed number of pulses and then subtract this value from the pulses. In practice, more general FIR and IIR filtering techniques are used that attenuate not only the DC, but also the low frequency components around DC to achieve clutter suppression of more than 40 dB. Figure 4.1 shows a typical high-pass filter.
Filter design is fairly mechanical and
the parameters that are adjusted are the stopband attenuation, the stopband 46
0
I I I c·Vs-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~___ ~~~~~~~~~~~~I
I
-10 Z z
I
V
/
I
I
I
TV__-
-20
I
0
-30
I
I -50
I\ 0
I I
!
!
2
3
I
I, I
I
4
5
-
I
6
VELOCITY (m s') Fig 4.la Clutter filter frequency response for a 3 pole infinite impulse For ground clutter response (IIR) high pass elliptic filter. width of 0.6 ms-1 and scan rate of 5 rpm this filter gives about 40 dB suppression. V s = stop band. Vp = pass:band cutoff, Vmax= +
16 ms -1 (Hamidi and Zrnic, 1981).
47
Xk
Fig 4.lb
Implementation of 3rd order IIR clutter suppression filter; z -1 is K 1 - K4 are filter coefficients (Hamidi and Zrnic, 1 PRT delay.
1981).
48
width, the transition band width and the passband ripple which if too large, can bias the mean velocity. The IIR filter is computationally more efficient to implement than a FIR filter but, because of its transient response comparable is best run in a continuous mode with minimal characteristics, it perturbation such as those caused by slow AGC changes or PRF changes. Initialization of the filter can improve the transient response characteristics. Hamidi and Zrnic (1981) and Groginsky and Glover (1980) evaluate IIR filters for weather radar systems. FIR filters offer linear phase performance and are well suited for batch processing of pulses since they operate on a finite number of pulses. This makes them well-suited to slow AGC or multiple PRF techniques (i.e., where
the PRF is held constant while a batch of pulses is collected and then changed for the next batch). There are other types of clutter suppression algorithms that should be Anderson's (1981) test of the mean block level subtraction mentioned. The parametric technique offers 20 to 30 dB of clutter cancellation. clutter cancellation techniques described by Passarelli physical
models
of
clutter
and weather
along
with
(1981, 1983) use estimates
of
the
autocorrelation function at various lags to compute the clutter power and
then estimate various Doppler spectral moments. dB or more has been achieved.
Clutter suppression of 30
Sato and Woodman (1982) use a nonlinear
processing scheme to fit the observed clutter spectrum and extract the spectral moments when clutter is about 50 dB stronger than the signal.
When a separate calibrated log channel is used for reflectivity measurement,
an uncalibrated linear channel can be used to remove the clutter contribution from the log channel power estimate. The ratio of the signal r = Pr[S]/Pr[ S + 3c is the same in power to the signal plus clutter power both the linear and the log channels. Therefore, after computing r from the coherent (linear) channel data, the log channel signal power is
49
10 log Pr[S ] = 10 log Pr[S +
]
+ 10 log r.
(4.2)
When multiple PRT measurements are made for the purpose of extending the
unambiguous velocity interval, nearly all clutter filters have problems. Anderson (1987) describes an interpolation scheme for the dual PEF ASR-9 radar.
4.2
RANGE/VELOCTTY AMBIGUITY RESOLUDTCN
A fundamental tradeoff exists with constant PRF Doppler radar. A large unambiguous range (IRma)
requires a low PRF PRF = ,/2RPx ;
(4.3)
however, a large unambiguous velocity (Vmax) (and accurate spectral moment
calculations) requires a large PRF PRF = 4 Vmax /A
(4.4)
Another PRF tradeoff is that accurate measurement of the mean velocity requires a high PRF since the Doppler spectrum width must be narrow relative
to
the
Nyquist
measurements
interval (high coherency) whereas accurate intensity require a low PRF to acquire independent samples (low
coherency).
Signal processing offers several techniques for expanding the
unambiguous range and unambiguous velocity. These tradeoffs illustrate that the choice of PRF must be optimized for different applications. A performance benchmark for comparison purposes is an S-band (10 cm) radar
operating at 1 KHz PRF with an unambiguous velocity range of ±25 ms'1 and an This unambiguous range is too small for The unambiguous assuring that second trip echoes will not be present. velocity is also too small to ensure that aliasing will not occur, but large unambiguous range of 150 km.
enough that double aliasing (velocities greater than 75 ms -1 ) will be rare. At C-band, the unambiguous velocity is halved so that double aliasing will be fairly common and single aliasing will occur routinely. Reducing the PRF
50
to minimize second trip echoes, will make the velocity aliasing problem even
more serious at C-band. Coherent lidar and profiler systems do not exhibit range/velocity ambiguities. For Doppler lidar, the sampling rate during a single pulse can be made sufficiently high with no impact on the unambiguous range. For the case of a wind profiler operating at a high elevation angle, the long wavelength and the steep angle of incidence provide such a large unambiguous velocity that most profiler processing schemes utilize coherent averaging to reduce the effective sample rate while simultaneously preserving processor resources.
For microwave radar, range/velocity ambiguity is a serious problem in many applications techniques
(Doviak, et al.,
1978).
Fortunately,
there
are several
for mitigating these ambiguities and each technique has its
advantages and shortcomings. for specific applications.
Selection of a technique is usually optimized
4.2.1 Resolution of velocity ambiguities There are several techniques for handling range/velocity aliasing that are not truly signal processing techniques, but rather techniques that use physical modeling to correct aliased data. Frequently, continuity can be used to detect velocity folding. For example, one does not expect to see 25 ms -1 discontinuities in velocity from bin to bin (in range or azimuth), so
they are assumed to be caused by aliasing. The disadvantage of this approach is that one must have some region with a known velocity to correctly invoke continuity. Also, this technique requires that the echo coverage be fairly continuous and may need manual input to perform final editing (Bargen and Brown, 1980). Hennington (1981) uses another physical modeling approach by estimating the mean wind profile obtained from a sounding or other source to correct aliased velocities. The technique works well when the perturbation velocities are small compared to the Nyquist interval. A similar technique described by Ray and Ziegler (1977) uses the velocity distribution along a radial to dealias velocities. Merritt (1984) employs both continuity and a wind field model to dealias isolated areas. 51
Bergen
Boren, et al. (1986) describe an artificial intelligence approach. and Albers
(1988) have investigated 2 and 3 dimensional dealiasing for
NEXRAD algorithms.
There are several signal processing techniques for extending the unambiguous
range/velocity. 1.
The criteria useful in evaluating the techniques are:
The algorithm should not preclude the use of clutter cancellation
techniques. 2.
The final moment estimates should have a conparable accuracy and be
made in a comparable time (number of pulses) to standard velocity estimation techniques. 3.
The cost of implementing the technique should be comparable to standard
velocity/range processing. Batch PRT.
interlaced
One approach to velocity/range ambiguity resolution is to use
PRT
sampling
whereby
a
short
PRT
is
used
for
velocity
measurements, and a long PRT is used for reflectivity estimates (Hennington, 1981).
For example, several pulses at a short PTR
are first transmitted,
followed by a clearing period (no transmission) and then one or two pulses separated by a long PRr for the reflectivity estimate. The basic assumption is that the PRT for the reflectivity estimate is sufficiently long so that
there are no second trip reflectivity echoes. The short PRr velocity estimates will have two classes of range aliased echoes,- those that are overlaid with the first trip echoes and those that are not overlaid with the
first trip echoes. When there is no overlap, the velocity estimates can actually be assigned to the correct range. When first and higher trip echoes are overlaid and one dominates the others in power by 10 dB or more,
then the velocity of the strong echo can be correctly estimated. disadvantages of this batch technique are:
The
1.
Loss of velocity data where first and second trip echoes are overlaid
2.
and powers are nearly equal. The technique may preclude the use of effective clutter cancelling.
52
3.
The data acquisition time is increased because the long PRT pulses are
unusable for making velocity estimates. A similar approach is to have two radars share a common antenna which is
also known as a dual-frequency approach (Glover, et al., 1981). One radar can sample at a long, constant PRT and the other can sample at a short, constant PRT. Alternatively, two scans can be made at each elevation, a long PRT scan for reflectivity and a short PRI scan for velocity. These techniques are clearly more expensive but they allow excellent clutter cancellation. Multiple PRT and multiple PRF techniques can be used to dealias velocities. Here, "multiple PRT" shall mean that the PRT is changed on a pulse-to-pulse basis whereas "multiple PRF" shall mean that the PRF is fixed while a batch of samples is collected and then changed for the next batch of samples. general technique is described by Sirmans, et al. (1976).
The
Dazhang, et al.
(1984) and Zrnic and Mahapatra (1985) describe an actual implementation. Dual PRT technique.
In the dual PRT (or staggered PRT) method the two PRT's
usually are in ratios of either 3/2 or 4/3.
First, one calculates the
first lag complex autocorrelation for each PRT, averaging over a number of pulses. Then, the expanded velocity is calculated from =
(981 -2)/47r(T2-T
1)
.
(4.5)
The corresponding unambiguous velocity is Vma x = +
4/4(T2-T1
) .
(4.6)
According to this expression, a 3/2 PRT ratio yields an unambiguous velocity that is twice that corresponding to the short PRT, while for a PET ratio of
4/3, the expanded velocity range is 3 times. Why not expand further? Since the variance of the expanded range velocity estimate is based on the difference between the two fundamental estimates, its variance is roughly proportional to twice that of each fundamental estimate. 53
Fortunately, the
expanded velocity estimate need be used only to roughly dealias the two fundamental estimates. The velocity estimate can be improved by averaging the two velocity estimates to get the final estimator provided they have been correctly dealiased. This averaging technique provides an estimator that uses all available pairs of consecutive pulses, rather than half the available pairs. Since the dual PRT technique dealiases velocities by a large factor, one can operate the radar at a lower PRF and thus have a larger unambiguous range. Doviak and Zrnic (1984) point out that another advantage of the multiple PRT
technique is that second trip echoes will be incoherent or "whitened" and thus not bias the first trip velocity estimates. Dual PRF technique.
A disadvantage of the dual PRT technique is that
standard clutter filters are very difficult to implement.
This can be
overcome for some filtering schemes by using a dual PRF technique wherein a
sequence of pulses is collected at each of two PRF's and then each sequence is processed separately. The data processing is identical to the standard pulse pair processing except that the velocity from the previous sequence is
used along with the velocity from the current sequence to dealias the current velocity. The sampling statistics are similar to the pulse pair, except that for this technique to be viable the mean velocity change between
adjacent sequences must be small. Because the PRF is
fixed while each batch is collected, the dual PRF
technique can employ a batch processing clutter filter such as an FFT or an FIR filter.
An IIR filter can be used, but several pulses will be required
to clear the filter between PRF changes.
Because the basic dual PRF processing is essentially the same as standard pulse pair processing at a constant PRF, it is easier to implement on an existing system. Unfortunately, the dual PRT feature of "whitening" the second trip echoes is lost when dual PRF sampling is used.
54
4.2.2
Resoluticn of range ambiguities
Low PRF radars minimize overlaid echo but require sophisticated velocity dealiasing techniques. If we promote the occurrence of overlaid echoes by using a higher PRF to provide a large unambiguous velocity, then the range aliased echoes must be resolved. Most range dealiasing techniques use phase codes to distinguish between first and second trip echoes. The simplest is the "magnetron" technique for which each transmitted pulse has a random phase. A typical magnetron is This means that coherent-on-receive only for the current pulse. contributions from multiple trip echoes are not coherent so that they appear
Consequently, the mean velocity and as increased white noise power. spectrum width are unbiased by overlaid multiple trip echoes. A problem with this technique is that the reflectivity cannot be deduced unless various received noise sources can be evaluated quantitatively. Also, the additional white noise that is caused by multiple-trip echoes reduces the sensitivity to first trip echoes and degrades the accuracy of mean velocity
and width estimates. A similar technique can be developed using a fully coherent system such as a
klystron in conjunction with a phase shifter to change the phase of the transmitted pulse. This permits the transmission of pseudo-random phase sequences that have "white" properties (Chakrabarti and Tomlinson, 1976; Sawate and Dursley, 1980).
The I and Q values can be "recohered" relative
to the first trip or the second trip, etc., by using the appropriate phase
shifts so that Doppler spectra can be evaluated for each trip (Laird, 1981). This technique offers information for both the first and second trip returns, but does not solve the problem of reduced sensitivity for overlaid echoes. Siggia (1983) addresses this issue by filtering the first trip echo from the The second trip echo and vice versa, to reduce noise contamination. technique works well as long as the two Doppler spectra (1t and 2 nd trip) are not so broad that they occupy a large fraction of the Nyquist interval. Zrnic and Mahapatra (1985) have evaluated this technique. 55
Sachidananda and Zrnic (1986) describe a different technique where, instead of inserting phase shifts to "whiten" the 2nd trip echo, the phase shifts
are inserted to cause the second trip Doppler spectrum to be a split bimodal spectrum whose autocorrelation for lag 1 is zero.
This means that the
second trip echo does not bias the first trip velocity estimates. All of these clutter
"phase diversity"
filtering
techniques.
techniques However,
are well there
suited for standard
are
substantial
signal
processing computations to implement some of them.
4.3
POIARIZATION SWITCHING CO(SEQUENCES
Bringi and Henry (Chap. 19A) describe various polarization techniques which provide
valuable
velocity scattering
target
dealiasing and
more
information
but
difficult.
instrumental
make
clutter
Differential
effects
preclude
processing techniques (Schnabl, et al., 1986).
use
suppression
phase of
and
propagation,
simple
Doppler
However, it is possible in
principle to extract both the Doppler information and differential phase shift simultaneously (Sachidananda and Zrnic, 1989; Doviak and Zrnic, 1984). Keeler and Carbone (1986) describe a dual PRT scheme which allows processing two orthogonal polarization states separately prior to velocity dealiasing. The alternating horizontal and vertical polarized pulse sequence mitigates contamination caused by range aliasing since the overlaid second trip echo is depolarized (Doviak and Sirmans, 1973).
Processing velocity
techniques and
range
to
simultaneously
dealiasing,
and
provide
polarization
beginning to receive serious attention.
56
clutter
suppression,
processing
are
just
5.
EPDXRAfTORY SIGNAL PXCESSIN
TECHXNIQUES
Implementations of modern signal processing algorithms on atmospheric radars have evolved slowly in the last several years. Modern digital signal processing algorithms have been difficult to implement for a variety of reasons, but the algorithms are well known (Kailath, 1974). Programmable processors with the speed to implement many of these algorithms and to explore their application to distributed targets, rather than point targets, is now possible. 5.1
PUISE OCfMPRESSICN
Pulse compression, or wideband waveform, schemes for improved radar range resolution were first theoretically described by Woodward's (1953) fundamental paper.
Klauder, et al. (1960) and Cook (1960) later described
the linear FM (chirp) pulse which has been widely used in military radars. Reid (1969) described a CW meteorological radar using pseudo-random coding. Barton (1975) has edited a collection of pulse compression papers which
details the chirp technique. Lewis, coded pulse compression waveforms.
et al.
(1986) emphasize poly-phase
Probably the first use of pulse compression for atmospheric distributed targets was on the Arecibo ionospheric radar (Farley, 1969; Gray and Farley, 1973).
The
STORMY
weather
group
at
McGill
University
implemented
a
compression scheme for reflectivity processing in the early 70's (Fetter, 1970; Austin, 1974).
Their use was to provide many independent samples of
intensity within a given range cell to improve the reflectivity estimate. They did not attempt any velocity measurements using their pseudo-random phase coded pulse. In the late 70's Krehbiel and Brook (1979) reported using a wideband noise waveform on the New Mexico Tech "Redball" radar to provide reflectivity estimates during the short dwell time of their fast scanning radar. Chadwick and Cooper (1972) and Keeler and Frush (1983a and 1983b) have described the principle of pulse compression Doppler measurements on microwave weather radars using distributed targets. Browning, et al. (1978) describe the 10 cm pulsed Doppler radar at Defford, 57
England which was modified to generate 4 Js, 5 MHz chirp pulses and measure Doppler shifts from ice crystals at 8 km range.
Chadwick and Strauch (1979)
demonstrated an FM-CW waveform on a 10 cm Doppler weather radar.
Woodman
(1980b) shows how a continuous wave phase coded waveform was used in the
bistatic mode at Arecibo. Recently information using this technique.
he
has
obtained
full
spectrum
Pulse compression is a well established waveform design technique in the
military and aviation radar communities and has been used in the ST/MST radar community Woodman,
(Crane,
1980;
Gonzales
and Woodman,
1985) and the lidar community (Oliver, 1979),
1984;
Sulzer and
but has not been
seriously investigated for microwave Doppler weather radar use.
The reasons
for this are: 1.
Range resolution and transmit power using standard high peak power
pulsed radars have been adequate to achieve the required scientific goals. 2.
Dwell times have been limited by mechanical scanning rates to tens of
milliseconds, thereby yielding the several independent samples of the Rayleigh fluctuations necessary to obtain accurate reflectivity estimates. 3.
Presence of range time sidelobes on pulse compression waveforms causes range smoothing and large bias errors in high reflectivity gradients.
5.1.1
Advantages of pulse omupression
The driving force for exploring pulse compression in weather radars is the
desire for ground based and airborne Doppler radars to rapidly sample the volume at a spatial resolution adequate for mesoscale or cloud physics analyses.
These systems fall into the short dwell time category.
Dwell
times of only a few milliseconds are insufficient for averaging independent
Rayleigh
fluctuations
to reduce
the variance
of parameter estimates.
Therefore, independence must be gained in some other way, in particular by
multiple frequency schemes or spatial averaging.
Marshall and Hitschfeld
(1953) pointed out that frequency separations greater than the inverse pulse
width give independent Rayleigh returns. 58
Pulse compression waveforms give
independent returns (to first order) at spatial resolution proportional to the inverse bandwidth (Nathanson, 1969). Either technique gives independent returns over short dwell times ( <5 ms) so that the antenna beam can be scanned at least an order of magnitude faster than typical weather radars (Keeler and Frush, 1983b). Strauch (1988) proposes a burst chirp waveform relevant to short dwell time weather radars. Another application of pulse compression waveforms is in solid state transmitter systems which typically are peak power limited to low values compared to klystron transmitters, but can sustain very long pulse widths and generate average powers comparable to the tube systems with greater reliability. Pulse compression techniques could be used with these high duty cycle systems to achieve range resolution corresponding to a much shorter pulse length. The NOAA network wind profilers will incorporate pulse compression for this purpose. 5.1.2
Disadvantages of pulse ccmpression There are tradeoffs associated with using pulse compression to achieve faster scan time.
The tradeoff involves reduced radar sensitivity with a
compressed pulse compared to a single frequency pulse of the same duration
and power.
While the full benefit of the average transmitted power is achieved, however the noise bandwidth must be increased to accommodate the pulse bandwidth.
Therefore, the SNR of the individual samples is degraded.
Keeler and Frush (1983a) describe how this tradeoff relates to the "timebandwidth product" (TB) of the compressed pulse. For the same average transmitted power the increase in independence is TB and the decrease in SNR is TB. For example, a chirp waveform 1 microsecond long sweeping 10 MHz of bandwidth has a TB = 10.
Range samples spaced by more than 15 m are
independent and have a SNR ten times lower than the uncompressed 1 microsecond pulsed waveform. Frequently, the independent range samples can be averaged to provide estimates having a reduced variance while allowing much faster scan rates. The primary disadvantage is a contribution to the backscatter from range time sidelobes. Because the receiver filter output is the cross-correlation 59
of the received waveform and the time reversed transmit waveform (a matched filter), range time sidelobes will cause data "blurring" in range space similar to that caused by antenna sidelobes in the transverse spatial dimension. Range time sidelobes (and antenna sidelobes) are especially troublesome in high reflectivity gradients.
Because atmospheric targets are
distributed in space, it is the integrated sidelobes that contribute to the distortion. They are analogous to the integrated antenna sidelobes which contribute interference from distributed targets at the same range. The contamination problem is particularly troublesome in downward looking radars from air or space platforms when one desires to estimate precipitation directly above the strongly reflecting earth surface. Careful waveform design and tapering based on digital waveform generation rather than analog devices may alleviate the range time sidelobe distortion (Farina, 1987). For echoes with sufficiently ST/MST radars
using
Schmidt, et al.,
long correlation times, as
is the case of
long wavelengths, complementary codes
(Golay, 1961;
1979; Woodman, 1980a; Gossard and Strauch, 1983; Wakasugi
and Fukao, 1985) completely cancel the range time sidelobes. robust schemes, like quasi-complementary codes
However, more
(Sulzer and Woodman, 1984)
show good results in practice when non-linearities in the system distort the
desired pulse shape.
The direct application of complementary codes is not
compatible with the shorter wavelength weather radar and lidar system.
The second disadvantage for pulse compression waveforms is the increase in
minimum range caused by transmitting a long pulse. Reception cannot begin until the entire transmit waveform is finished. Pulses longer than several microseconds
are unacceptable
for close ranges.
The NWS wind profiler
solution is
to extend the scan time using a short pulse mode for short ranges and use a long pulse mode for long ranges. Other techniques also exist. A third disadvantage relates to the availability of bandwidth.
Research
systems are not seriously constrained, but operational systems may require bandwidths which do not fit into the channelized frequency assignments.
60
5.1.3
Ambiguity function
The tradeoff in sensitivity for a larger number of independent samples gives
considerable flexibility in waveform design that the concept of the
so much flexibility in
"ambiguity function"
fact
was developed by Woodward
(1953) to study the effects on range and velocity ambiguities for a specific waveform.
For our purposes this ambiguity function is indispensable for
understanding the receiver response to targets at other ranges and other velocities from that to which the receiving filter is matched.
Weather
targets are distributed in range and velocity by their very nature and are especially sensitive to these undesirable responses.
The
ambiguity
function
defines
the
ability
of
a
waveform
to
resolve
different targets in range and velocity based on the power response of a filter matched to some specific range time and Doppler shift (Nathanson, 1969;
Skolnik,
1980;
Brookner,
1977).
Figure
5.1
shows the
ambiguity
diagram for a single FM chirp waveform in range (r) and velocity (¢) space. Note that targets having non-zero velocities at ranges different from the desired range function
(r=0)
evaluated
contribute significantly along
the
r
axis
to the
(i.e.,
autocorrelation function of the waveform
0=0)
filter is
output. identically
The the
(Frank, 1963; Cook and Bernfeld,
1967; Barton, 1975).
Atmospheric
radars
involve
estimation
of the
return power and velocity
rather than detection of such a target at some position in range-velocity space.
Our primary interest in the ambiguity diagram is to study the range
time sidelobes as a function of Doppler offset. plot
of
the
ambiguity
function
autocorrelation function. order only
103
Hz
along
the
It is easy to show that the range
axis
is
simply
the
Real weather targets having Doppler shifts of
compared to pulse bandwidths
of
107
Hz
allow us to
concentrate our attention to this narrow strip of the ambiguity function along the range axis. kept
small
velocity.
to
All the range time sidelobes in this strip must be
avoid contamination
Known waveform
of targets at the desired range and
design techniques may
allow tailoring of the
waveform to our "small velocity" case to keep sidelobes in this narrow
61
Ijx
(r, ) I Io1O6
/ I
Fig 5.1
Ambiguity diagram for single FM chirped pulse waveform with TB=10. 0 is velocity dimension. Targets r is range dimension. distributed in (r,¢) space contribute to the filter output proportional to the ambiguity function. For atmospheric targets, Doppler shift frequencies are typically very small relative to pulse bandwidth (Rihaczek, 1969).
62
ambiguity region acceptably small (Deley, 1970; Kretschmer and Lewis, 1983; Costas, 1984; and Lewis, et al., 1986).
5.1.4
CCpariso
with multiple frequency sdceme.
Krehbiel and Brook
(1968) and Keeler and Frush (1983a) show that a pulse
compression waveform with
time-bandwidth
product TB has
characteristics
similar to a multiple frequency radar using the same time and bandwidth factors.
Consecutive pulses may be
generated at different frequencies and
processed in separate receivers tuned to the different frequencies.
This
scheme yields the same number of independent samples for the same total pulse duration and total bandwidth.
The advantage of the multi-frequency
scheme, aside from the straightforward parallel receiver implementation, is reduced range time sidelobes.
5.2 At
AiAPTIVE FJIIERING AIXIcTHMS Stanford University
in
the early
1960's, Widrow and his
colleagues
(Widrow and Hoff, 1960) developed a class of filters that could "learn" their received signal environment and, in time, adapt their characteristics to optimally
filter
an
incoming
signal.
Initial
applications
were
in
pattern classification (Widrow, 1970), but use in adaptive antennas (Widrow, et al.,
1967)
and the closely related field of spectrum line enhancement
(Zeidler, et al., 1978) and noise (interference) cancelling (Widrow, et al., 1975a)
quickly
followed.
Griffiths
(1975) has described
instantaneous
frequency estimation techniques applicable to Doppler radars.
Atmospheric
radar applications (i.e., non-military) have been sparse mainly because the computational load associated with constantly changing filter coefficients could not be accommodated until recently.
Keeler and Griffiths
(1977) have
reported adaptive frequency estimation schemes applied to acoustic radars sensing boundary layer winds.
With the advent of fast programmable signal processors, we can expect to see a rash of new applications
in radar
for adaptive
filtering techniques.
Adaptive filter systems are characterized by both a time variable transfer function and the ability to self adjust, or be trained, to their environment for optimizing some measurement criterion 63
(Alexander, 1986b).
A
common
index for optimization is the minimum mean squared error (mmse) between the
processed output signal and a known desired output (or at least one which is correlated with the desired signal). Figure 5.2 depicts a 2 dimensional (2 weight) error surface. Widrow's (1970) popular Ieast Mean Square (IMS) algorithm estimates the gradient of the quadratic error surface and steps
the weights toward the minimum error value. Nearly identical adaptive techniques have been developed for antenna beam steering by Howells (1976), Gabriel (1976, 1980), Appelbaum (1976), Monzingo and Miller (1980), and Compton (1988). Adaptive antenna systems have the capabilities of tracking desired signals in space, maximizing the SNR, and nulling out undesired interfering signals. The optimization criterion is maximization of signal to interference plus noise ratio, which for many cases is identical to the IMS criterion. For radar applications the beam can be steered to the desired direction and the adaptation can simultaneously maximize the SNR by spectral shaping and spatially nulling any interfering sources.
Van Veen and Buckley (1988) give a tutorial review
of spatial beam forming techniques. 5.2.1 Adaptive filtering applications The structure for a performance feed back adaptive system is shown in Figure 5.3 where we note the input signal xk, the adaptive processor output yk, the
yet to be defined desired response dk, and the error signal, ek = dk - Yk. This error signal drives an adaptive algorithm which controls the transfer function of the adaptive processor, and its output yk. Various closed loop structures are possible as are a variety of adaptive algorithms. Widrow and Stearns (1985), Honig and Messerschmitt (1984), Alexander (1986b), and Haykin (1986) give excellent overviews of these structures and algorithms. Widrow, et al. (1976) describe the learning characteristics of IMS adaptive filters in both stationary environments when the filters converge to an optimal setting and non-stationary environments where the filter continues to adapt to the time variable input signal statistics. Adaptive filters have found application in data prediction schemes, system identification or modeling, parameter tracking, deconvolution and equalization, and interference (clutter) cancelling (Alexander, 1986a). Usually the 64
LU ()
2
.0
W2
WI WI
Fig 5.2
Prediction error surface for 2 weight adaptive filter. The LMS algorithm estimates the negative gradient of the quadratic error and steps toward the minimum mean square error (mse). The optimum weight vector is
W* =
(0.65,-2.10).
If the input statistics
change so that the error surface varies with time, the adaptive weights will track this change (Widrow and Stearns, 1985).
65
Inni it
Xk
Error k
Fig 5.3
is (dk) The desired response structure. Adaptive filter coefficients filter The adaptive the application. by determined (Wk) and/or the output signal (Yk) are the parameters used for spectrum moment estimation (Widrow and Stearns, 1985).
66
application determines the origin of the reference signal and the specific adaptive algorithm to be used.
Sibul
(1987) has edited a collection of
application papers for adaptive
filters.
Further applications in neural
networks and fault tolerant computing are being explored (Lippmann, 1987; Shriver, 1988).
As
an example
of an atmospheric
radar application, an
adaptive
linear
prediction filter will improve the SNR of the received signal so that the moment estimation will yield improved estimates.
In the frequency domain
the prediction filter acts as a narrow band pass filter having time variable center
frequency which passes the received signal while suppressing the
spectral
noise
components.
Tufts
(1977)
describe this enhancement procedure. signal, dk.
and Anderson,
et
al.
(1983)
The input signal xk is the desired
The previous input samples (xkl, xk_2,
..., xk-L} = XT are
filtered to predict, or estimate, the present sample xk.
The error signal
is the difference between Xk=dk and its estimate Yk, i.e. ekx-yk.
The
filter is adjusted using the IMS algorithm so that the mean squared error signal is minimized.
Sequentially then, the filter adjusts itself to
predict the input signal more accurately.
Some error will be present but
the predicted signal will have an improved SNR over the input itself.
In
this sense, we have an adaptive matched filter which can track the input signal as
its characteristics
(e.g., its Doppler shift and width)
change
with time (Tufts and Rao, 1977).
Probing
deeper
into
the
mathematics,
we
find
that
the
algorithm
estimating the negative gradient of a quadratic error surface dimensional
is
in the L
adaptive filter weight vector space and adjusting the filter
weight vector Wk to step towards the minimum mean squared prediction error with
every
defines
the
iteration. highly
This operation efficient
steepest
plus some gradient
supporting mathematics descent
IMS
adaptive
algorithm (Widrow, 1970; Widrow, et al., 1975b).
Wk+l = Wk + 2gek Xk,
67
(5.1)
where g is a precisely defined constant which determines the convergence rate and the excess noise generated by the adaptation process.
It is easy to show that the one step prediction structure leads to the Lth order maximum entropy Griffiths, 1975).
(ME) spectrum estimate
(Lang and McClellan, 1980;
Keeler and Lee (1978) have shown how the complex, first
order, one step prediction filter yields the pulse pair frequency estimator, which has been made adaptive.
Keeler
(1978) further reports a bias and
variance of an adaptive ME frequency estimator.
What
makes
difference
adaptive in
prediction
correlation
time
and
SNR
of the
enhancement
desired
sinusoid) and the unpredictable white noise.
possible
narrow band
is
the
signal
(or
Similarly, the long coherence
time of clutter input components may allow these interfering signals to be rejected using adaptive interference (noise) cancelling filters (Widrow, et al., 1975a).
For example, airborne Doppler clutter can be represented by a
strong, narrow spectral
return having a variable
Doppler shift
and sea
clutter may be sufficiently offset from zero Doppler that an adaptive scheme may provide adequate suppression in both cases.
5.2.2
Adaptive antenna applications
Adaptive beamforming was motivated by a desire to steer the main beam in a desired
direction
while
simultaneously
nulling
interfering
sources
and
maximizing the signal to interference plus noise ratio at the output of the adaptive beamformer (Haykin, 1985a; Compton, 1988).
Atmospheric radars are
troubled by interfering ground clutter returns and could benefit from using an adaptive antenna.
For example, an RHI scanning radar could dynamically
place a line of nulls along the dominant ground clutter return angles near 0° elevation.
Or a ground reflected multipath ray could be suppressed.
radio communications present slowly time varying
interfering sources
UHF for
wind profilers which could be suppressed by adaptive array techniques.
Forming nulls in the array antenna patterns in real time as the interferers become active or as the antenna elevation increases may be feasible in many cases.
Constraints on the adaptation speed and antenna scan rates may limit 68
performance of these proposed systems since stationarity over a finite time period is usually required.
Furthermore, narrowbeam systems require several
thousand array elements and a digital control system for a truly adaptive 3 dimensional
beam.
Cost is a limiting
factor in this regard
(Mailloux,
1982).
Array processing utilizes multi-channel processing algorithms to process the individual signals from each element to effect both spatial beamforming and temporal filtering. difficulties.
Vector and matrix based algorithms introduce special
Haykin (1985a) describes array signal processing algorithms
which have been applied to a variety of fields, e.g. seismology, radio astronomy, tomographic imaging, sonar, and radar. et al.
(1985) have proposed sequentially changing (at the pulse repetition
rate) the pattern of a phased array antenna. allows
Recently, Sachidananda,
contributions
to velocity
Subsequent Doppler processing
estimates entering through the antenna
sidelobes to be whitened and/or removed (Zrnic and Sachidananda, 1988).
5.3
UITI-CHANNEL PRDCESSING
As atmospheric remote sensors become more sophisticated and programmable processors
achieve
algorithms
will
channels
greater
become
computational
more
can be thought
common.
of as
a
power, multi-channel
The
vector
signals
from
processing
separate
input
time series and processed, or
filtered, collectively by using the correlated information in the channels to produce more accurate parameter estimates than if they were processed separately
(Marple, 1987).
The coefficients of these multi-channel filters
are found by solving a set of linear equations similar to the single channel equations used in linear prediction filtering and associated applications. Wiggins and Robinson "normal" equations.
(1965) give a recursive technique for solving these Strand (1977) and Morf, et al. (1978) describe multi-
channel maximum entropy spectrum estimation, which is a direct result of solving the normal equations.
In addition to radar array antenna data, dual polarization data is another example of a multi-channel complex input signal.
Horizontal and vertical
channels of a dual linear polarization radar can be processed to yield cross 69
parameters. Each input data point can be thought of as a 2x2 matrix, the polarization matrix, rather than a complex I and Q estimate. The set, or vector of these matrix inputs is then processed using complex matrix algorithms which are designed to optimally and jointly estimate target parameters. Processing both channels simultaneously yields additional information that could not be obtained if they were processed independently. Integrated sensor systems can benefit by multi-channel processing schemes. A multi-channel algorithm might make use of 10 minute wind profiler data and 1 minute radar or lidar data.
Wind profiles on multiple scales would be
produced with lower error than either system operating alone. Application of coherence functions to these multi-channel sensors provides an analytic tool for correlated data which improves the analysis. 5.4
A IPRICI INFR4ATICN
Information that is known in advance, a priori information, can be used to improve atmospheric parameter estimates. Most remote sensors treat each spatial resolution volume independently from all others. However, there are physical constraints in the atmosphere that limit the rates of change of certain parameters. These constraints are known in advance and can be used to constrain the processing algorithm to produce better estimates of velocity, for example, than if they were ignored. To be most effective this a priori knowledge should be used as early in the processing chain as practicable.
For example, if one knows (or is confident that the received
signal consists of a Gaussian shaped signal spectrum in white noise, then one should be able to use this prior information to generate a lower error
velocity estimate than if the information were ignored. Signal processing algorithms constrained by known a priori information typically yield simpler and faster algorithms that give lower variance estimates than unconstrained estimators. Frequently these estimators are maximum likelihood, i.e., minimum variance, and can be readily computed using modern processing hardware.
70
6.
SIGNAL PFXFESSCR IMPTrFMTATION
Signal processing encompasses analog and digital processing of both the transmitted and received radar signal. Because of timing requirements, most pulse-to-pulse control functions are also handled by the signal processing system. In this section we discuss the signal processing implementations that are found on modern radars and the tasks typically allocated to the signal processor. 6.1
SIGNAL PROCESSING CONTROL FUNCTICNS
Signal processors usually perform a variety of radar control functions and
serve
as the interface between the radar system and the radar data processing system (usually a host computer). These control tasks include: 1.
Pulse waveform selection
2.
Polarization switching
3.
Phase sequencing
4.
Pulse sequence generation
5.
Range gate trigger generation
6.
Linear channel gain control
7.
Calibration pulse injection
Radar control starts at the transmitter.
The signal processor usually
generates the PRF, although good practice dictates that the basic clock be derived from a reference oscillator that is shared between the processor and PRF control by the processor minimizes the possibility of range bin jitter caused by timing uncertainties in the A/D sampling and is the radar.
particularly important if a multiple PRF processing scheme is employed since
the processing must be synchronized with the PRF. Because of the need to preserve the duty cycle limit of the transmitter, it
is a safety feature and a convenience to have the signal processor also control the pulse width and bandwidth filter selection.
71
Since the signal processor is in control of the PRF, it is typically assigned the task of controlling all pulse-to-pulse functions such as phase control for pseudo-random phase processing and polarization switch control. This approach assures that the processing is properly synchronized with all aspects of the transmit-receive sequence. Built-in calibration test units that operate during normal data collection
are now found on some systems.
The idea is to inject a pulse of known power
and phase characteristics in the last few range bins for each transmitted pulse or during antenna repositioning intervals with the transmitter off. These bins are then processed identically to all other bins. The output values can be monitored in real time to verify that the system is functioning properly, and for system power calibration. In addition, the injected signal can be made coherent so that the Doppler processing can be
checked.
The advantage of this approach is that the entire receiver and
processing system can be verified without interrupting normal operations.
The remainder of this section is devoted to linear channel gain control techniques. Currently, the receiver systems for most applications use analog signal -processing techniques for deriving the linear channel I and Q
(in-phase and quadrature) and log channel outputs.
The log channel output
is typically used for quantitative power measurements because of its dynamic
range capabilities (90-100 dB). The linear channel measurements are used for extracting information related to the phase of the signal, i.e., mean, velocity, spectral width and clutter measurements, and can provide power estimates as well. The linear channel measurements operate over a more restricted dynamic range, typically z40-60 dB, that is usually shifted by means of an automatic gain control (AGC) loop over a range of ~100 dB. It is the linear channel gain control problem where digital signal processing
often makes its first appearance in the radar processing chain. Linear receiver gain control is typically performed via one of the following
methods: 1.
IF limiting 72
2.
Sensitivity time control (STC)
3.
Slow AGC
4.
Fast AGC
5.
Multiple receivers
In the first case, a "soft" limiter is inserted at IF before phase detection
(Nathanson, 1969;
Zeoli, 1971;
Frush, 1981 ).
The advantage of this
technique is that it is extremely simple to implement and permits the linear receiver to operate over a fairly wide dynamic range with good mean velocity
retrieval.
However, if the Doppler spectrum is bimodal, such as for ground
clutter mixed with a weather spectrum, this technique tends to "capture" the
This behavior makes it stronger signal and suppress the weaker one. unsuitable for systems that require clutter cancellation. For the STC case, the linear channel gain is increased with range in an attenpt to represent the average characteristics of weather and clutter. Since there is no feedback based on actual power measurements, it is easy to implement.
However, it is a near certainty that strong clutter targets will
cause saturation of the linear receiver at close range unless an IF limiting
approach is used as well. detectable at full gain
Likewise, weak clear air echoes that would be at close range, will be attenuated beyond
detectability. For the slow AGC, the log receiver measurements from the previous ray are used to optimize the linear receiver gain for the targets that are actually
present at each range. The samples for an integration period are collected If the log receiver is used for while the gain is held constant. quantitative power measurements, the actual gain does not need to be known
with great precision (within 3 dB is usually satisfactory). Also, since the gain is held constant, the phase shifts that are introduced by the gain control are constant from pulse-to-pulse so that these do not have to be corrected. The primary drawback is that the ability to distinguish between the clutter and weather components of the signal may be limited by the Furthermore, strong fundamental dynamic range of the linear receiver. reflectivity gradients will cause erroneous gain settings. 73
The fast AGC, or instantaneous AGC (IAGC) approach, for which the gain of
the linear receiver is adjusted for each range and each pulse, is used where there is a high degree of phase purity in the transmitted pulse
(e.g.,
klystron systems). The power measurement for either the previous pulse, or the current pulse (in which case a delay line is required) is used to set the receiver gain. This is the most complicated form of AGC to implement since it requires a very accurate calibration of both the amplitude and phase response of the receiver as a function of gain and the input power.
Mueller and Silha (1978) employ a real-time calibration and correction scheme so that the output phase of the linear receiver requires no correction. Properly implemented this approach provides wide dynamic range linear response for high-performance clutter cancellation and more accurate
estimates of the power than a log channel. Another approach is to employ multiple receivers, each optimized for a fixed range of input power with the advantage that all samples can be digitized and the optimal receiver can be decided with a digital algorithm. Moreover, switching transients and calibration procedures are minimized. 6.2
SIGNAL A/D CONVERSICN AND CAIBRATICN
Figure 6.1 shows a block diagram of a typical digital, time domain Doppler signal processor. The digital signal processor provides the interface to the radar I, Q and log signals, and connects to a host computer that provides the communications.
user
interface,
data
processing,
display
and
After analog phase detection, the I, Q and log values are digitized.
data
In the
case of a fast AGC, a digital AGC value may also serve as an "exponent" for
a floating point representation. The precision that is required for digitizing the I and Q values depends primarily on the underlying precision of the linear receiver and the dynamic range limitations imposed by ground
clutter induced phase noise. In computing dynamic range, an additional bit amounts to 6 dB more power measurement capability. However, because the receiver noise level requires about two bits to coherently integrate weak 74
g ication
Fig 6.1
Block diagram of a typical signal processor.
75
signals
and one bit denotes the sign of bipolar data, the usable instantaneous dynamic range is limited to ~54 dB for 12 bit samples. This range provides a margin for an AGC that may not optimize its use of the receiver dynamic range and offers reasonable clutter rejection. For the log channel, the quantization of the digitized signal determines, to some extent, the accuracy of the final power estimates. However, it is usually the inherent large fluctuation of «30 dB for Rayleigh signals (Nathanson, 1969) that imposes the more fundamental limit. The A/D converter values should not saturate.
I and Q saturation causes
harmonic generation in the frequency domain. Furthermore, image spectrum generation about DC in the spectrum is frequently caused by imbalance in the amplitude and/or phase of the I and Q signals (Hansen, 1985). Time domain averaging is an important step
in processing ST/MST radar
signals to reduce the noise (Strauch, et al., 1984).
The averaging not only
increases the SNR by N, but also increases the dynamic range by 10 log N. The discussions above illustrate the need for time series and power spectrum
displays
to optimize radar performance.
Just as
important,
the host
processor must be equipped with software to provide the interactive displays that are required for accurate system adjustment and verification.
6.3
REEIETlVlTY PROCESSING
The precise measurement of the received power is an important objective for most weather radar systems, and for noncoherent systems, this is the primary measurement. In the pre-Doppler era, there was interest in the so-called "power-fluctuation spectrum" and spectrum width estimates (Rutkowski and Fleisher, 1955; Atlas, 1964;). Most radar systems, whether Doppler or noncoherent, employ a wide dynamic range log receiver that operates at IF. These systems merely average the log values which results in an asymptotic
(with the number of independent samples) 2.51 dB bias in the estimation of the average power for Rayleigh distributed targets (Doviak and Zrnic, 1984). There are other types of receiver responses, such as the linear and square law receivers, and the log receiver has the largest standard deviation for power estimates (Zrnic, 1975a). However, in view of calibration errors and 76
the uncertainties in relating power measurements to rainfall rate, the log receiver performance is adequate for many applications. When differential reflectivity measurements are required, one attempts to measure small differences in power so that the square law receiver is preferred (Bringi, et al., 1983; Chandrasekar, et al., 1988). Two common techniques that are used for power averaging are the exponential average (Zrnic, 1977a) and the uniformly weighted average. Exponential averaging is calculated using Pk = Pk-1 *(1-C) + Pk *C
(6.1)
where Pk is the current estimate of average power based on the new sample pk and the previous estimate Pk-1 . C is a weighting constant between 0 and 1. When C is close to 1, the current pulse is more strongly weighted. This technique is extremely simple to implement in real time and provides a new estimate for each pulse. Since real time digital processing capabilities have improved, and analog CRT displays are rapidly being replaced by color displays, this technique has been largely replaced by a simple uniformly weighted average over a fixed number of pulses. Averaging of independent samples is required to obtain accurate reflectivity
estimates.
Since. independence is governed by the coherence time this
imposes a fundamental constraint on the scan rate for data collection. For example, at 3 rpm and 500 Hz PRF, one can average only 27 pulses per degree of antenna rotation. Depending on the wavelength and the spectrum width of the scatterers, not all of these pulses will be independent.
A technique
for increasing the number of independent pulses is to average in range using
a range bin spacing that is greater than the pulse width. This requires somewhat more processing power, but results in more accurate reflectivity estimates. Also, averaging can be adjusted as a function of range so that the resulting dimension.
average range
interval
77
is comparable to the beamwidth
The conversion from dBm to dBZ is done via the radar equation which involves
the radar constant and range normalization. The term "STC" is sometimes inappropriately used to refer to the digital range normalization that is performed in the processor.
This term is a reference to the analog
technique that was used in the past to represent the radar reflectivity on CRT display. Digital range normalization merely adjusts the output values appropriately without causing the loss of sensitivity at short range.
6.4
'THRESHOEIDfING
R DATA QUALITY
The goal of thresholding is to have the signal processor flag data that
may be corrupted by bias and artifact. Clarity of presentation of the spectral moments is important to a user trying to interpret a display. For subsequent data processing and product generation
(e.g., CAPPI's, cross-
sections, rainfall accumulations), noise, bias and other artifacts increase the computational demand on the data processor and degrade the final product. Finally, thresholding followed by run length encoding for data compression can greatly reduce the communications bandwidth requirements for
transmitting radar data and products and reduce the archive resources that are required to store them. There are numerous thresholding criteria and variables that are employed in
modern radars: 1.
Incoherent signal-to-noise power
2.
Coherent signal-to-noise power
3.
Doppler spectrum width
4.
Clutter-to-signal power
5.
Zero velocity
6.
Geometric criteria
7.
Statistical criteria
The incoherent signal-to-noise power is calculated by comparing the received
power at a range bin with the system noise power (S+N/N).
This criterion is
most commonly used to threshold the wide dynamic range power measurements (e.g., from a log receiver). The coherent signal-to-noise power is the area 78
under the signal portion of the power spectrum divided by the total noise power (S/N). It can be calculated directly from the spectrum, or using the measured autocorrelations. Similarly, the spectrum width itself can be used as an indicator of the accuracy of the Doppler mean velocity and spectrum width. Both a low coherent signal-to-noise ratio and a large spectrum width contribute to a large variance in the velocity and width estimators. Ideally, thresholding should be made at a constant variance level, e.g., velocity is accepted if it's expected error is less than 1 m/s. Unfortunately the relationship that governs the effect of SNR and width on
the variance of the velocity estimator is not a simple one (Zrnic, 1977b), hence it is usually not implemented as a real time thresholding criterion. Instead, the typical approach is to use either the coherent SNR and/or the width separately and adjust the threshold until the displays are reasonably
free of speckles. A popular measure of the quality of velocity and width estimates, which accounts for the effects of both the coherent SNR and the spectral width is
the normalized first lagged autocorrelation magnitude IR(1)|/R(O). easily computed,
conveniently
bounded
between
0 and
1
It is
and thresholds
unreliable estimates reasonably well.
The measured
clutter-to-signal ratio
(CSR) is often calculated for the
purpose of correcting the log receiver power for the effects of clutter. When the actual CSR exceeds the dynamic range capabilities of the receiver
or the ability of the clutter filter to accurately remove clutter, then the data should be discarded.
The calculated CSR can then be used as the
thresholding criterion. Another method of thresholding range bins that are affected by clutter is to
simply not display bins that have a mean velocity within a narrow band about zero velocity. This technique is effective for Doppler radars that have no clutter filter, or Doppler radars of limited linear dynamic range available for
cancelling
clutter.
Both
the velocity 79
and reflectivity
can be
thresholded using this criterion. Unfortunately, any weather that falls into the threshold velocity band is also rejected. Simple geometric considerations can be used for thresholding data that are
not physically reasonable. A very simple threshold is to eliminate all data that are above a fixed height where weather echoes are assured not to occur, e.g., 20 km. Another threshold that is easily implemented in a processor is
a "speckle remover" that eliminates all isolated range bins that have no nearest neighbors in range or azimuth. Use of a speckle remover eliminates aircraft and point clutter targets. It also allows other thresholds to be set to lower values for greater sensitivity since only double speckles will
be passed. Finally, statistical criteria involve considerations of local continuity and
rejection of data that are a few standard deviations away from local mean values.
Strauch,
et
al.
(1984) utilize
a very
effective
"consensus
averaging" technique (Fischler and Bolles, 1981) to delete wild points or outliers for time domain integration of wind profiler processing. One or two dimensional median filtering techniques also allows deletion of individual or isolated groups of anomalous data. The application of thresholding requires caution. One common problem develops when a linear channel index is used to threshold both the velocity and the reflectivity. If this is done it is not uncommon to observe "black holes" of rejected reflectivity echo (so called if the display background is black). These often occur in regions of large shear or turbulence such as thunderstorm cores (Hjelmfelt, et al., 1981) where there is ample reflectivity present. This points out that different threshold combinations, and perhaps threshold levels, should be used for the different spectral moments. For example, an acceptable threshold for velocity will generally not be appropriate for spectrum width since spectrum width requires a stronger signal for proper estimation.
80
7. 7.1
IN SIGNAL PROCESSING
ITREND
REALIZATIC[N FACT
Several key components comprise a realizable signal processing system-chips, memory, and a large bandwidth output device. This digital technology has found wide applications in modern radars (Rabinowitz, et al., 1985).
7.1.1
Digital signal processr chips
In the last 5 years integrated circuit chips specially optimized for digital
signal
(DSP)
processor and
memory,
the
operations logic
supporting
multiply-accumulate, on-chip
such as have
developed
exceeding hardwired processors of several years ago.
computational
power
These DSP chips are installed
on
commercially available high speed busses, such as VME and Multibus II.
As
available
from
a
variety
of
and
manufacturers
can
be
integrated circuit developments in memory continue, on-chip memory will expand
to
allow
caching
and
make
DSP
algorithm's
more
efficient.
Interconnectability using multiple fast busses and fast communication ports still allow full implementations of many DSP algorithms. The commercial availability of families of DSP chips and busses provides documentation, technical support, and probable upgrades for faster and compatible processing speed. Current 32 bit DSP chips are based on silicon technology (TTL and CMOS)
and
can achieve clock rates of tens of MHz and execution rates of a few Million Instructions Per Second (MIPS). The next generation of microprocessor and DSP chips will be fabricated from gallium arsenide (GaAs) and will allow several processors to be attached to a single chip component. Clock rates for these advanced devices will be a few hundred MHz with instruction rates
exceeding 100 MIPS. This technology is growing rapidly. However, within the next several years the number of components per chip will be limited by fabrication processes and shortly thereafter by physical constraints within the chip itself (Aliphas and Feldman, 1987).
81
An important factor that will allow rapid expansion of radar processing power is the trend of D6P chip manufacturers to develop higher performance
chips that are compatible with previous versions. Thus, a relatively simple redesign of the processor board using the same basic architecture, combined with reprogrammed algorithms, offers greatly enhanced processing power at low cost. The ready availability of the processing power obviates a move towards more
real time processing.
For example, as multi-parameter radars and faster
scanning radars evolve, more processing power will be necessary to compute
the quality-checked, auto-edited data that is so valuable to real time observations. The real time processing can perform all the "signal processing" plus an increasing amount of the "data processing" tasks.
7.1.2 Storage media External devices for mass storage have long been dominated by magnetic tape.
The half inch tape is the standard, but various other tape-based media and technologies are being explored. These include special high density tapes such as NCAR's obsolete TBM (terra-bit memory), magnetic tape cartridges, video cassettes, and the digital audio tape (DAT) devices using helical scan technology. All of these tape storage media suffer from serial access delays and are undesirable for on line, fast access storage. However, they are extremely well suited for "write-once" archiving applications such as radar data acquisition. Storage capacities of two or more gigabytes can be achieved today. Higher capacity and faster transfer rates will continue to evolve. Winchester disks using "vertical recording" techniques allow high density and fast access and fit many applications which require fast, random
access storage. The thrust in storage media development now seems to be in optical recording
techniques.
Compact disk (CD) technology, being a consumer product, has become relatively inexpensive. The data capacity of optical media is approaching several Gbytes on a 5.25" CD and data transfer rates of several
Mbytes/sec are possible. millisecond range.
Random access times are being reduced to the
82
7.1.3
Display technology
Real time color radar displays have become an important component of remote
sensor technology since their first implementation by Gray, et al. (1975).
Intensity modulated PPI and RHI scopes show high resolution reflectivity displays, but digital color displays show all the directly measured variables (e.g., velocity) as well as derived variables such as differential reflectivity, phase, and depolarization quantities. Plotting data from multiple sensors in real time, zooming into specific areas of interest, generating time lapsed images, and defining special overlays provides a measure of flexibility not available only a few years ago. Special purpose programmable graphics processors allow these new, yet fairly simple, image processing capabilities. The next generation of graphics processors will accommodate 3 dimensional real time image generation, color images with
transparency, easily manipulated images to change the viewing angle, and programmability in high level languages to allow a high degree of user interaction. The display is the investigator's or the user's contact to the environment being studied or watched. Particular emphasis should be placed on this aspect of the remote sensor to extract its maximum utility.
7.1.4 omrmercial radar processors Radar processors have historically been developed by the organization responsible for the entire remote sensor system. Recently, however, digital signal processors have become commercially available as special purpose computers for Doppler lidars (Bilbro, et al., 1984), and weather radars (Siggia, 1981; Chandra, et al., 1986, and Schroth, et al., 1988). The specialized processing algorithms being developed and applied to atmospheric
remote sensors can be efficiently integrated into many types of remote sensors and customized to the specific application by different software. System
engineering
improvements
in
of
signal
hardware
processors
technology
and
is
changing
architectures
because
(Allen,
of
the
1985).
However, the biggest change is occurring because of changes in the system
engineering methodology.
Open software standards for operating systems 83
(e.g., POSIX), for computer language (e.g., ANSII standards, Ada, etc.), run-time environments
(e.g., X-OPEN) are being developed and applied.
bus standards,
VME)
(e.g.,
are being clarified,
board and peripheral manufacturers.
and Data
updated and adhered to by
Open software standards and workable
data bus standards facilitate cost-effective development and manufacture of special signal processing boards that integrate and can be upgraded to the latest
DSP chip sets.
7.2
IREN1&S IN I
The
new generation
sensors
EGRA4TBILr[TY OF DSP of digital
is programmable.
processors
in
which
signal processors
for atmospheric
remote
This is a marked contrast to early hardwired
the
algorithms
could
be modified
only
with
great
difficulty and most often resulting in the loss of the original capability. Programmable
processors
allow
algorithm
modifications,
processing
experiments, diagnostic testing, and system testing while still retaining the capability of returning to a pre-existing mode of operation.
Modern
digital filtering and waveform processing using advanced algorithms is now
possible without the constraints imposed by physical limitations of hardware devices.
Schmidt,
programmable
signal
et
al.
(1979)
processors
and Woodman,
for VHF
et
al.
Doppler wind
(1980) describe
profilers.
These
present day DSP systems are directly programmable in modern languages, such as "C".
Advanced processing algorithms using matrix methods, such as singular value decomposition, orthogonalization, multichannel optimization techniques, and non-linear
processing
algorithms
using
adaptive
and
data
compression
techniques (Haykin, 1985a; Kay, 1987; Marple, 1987) can be coded and tested on
line
in
real
implementation. be modified.
As
destroying
the
original
algorithm
Optimization may become an easier task.
easier
algorithms,
without
Standard algorithms can be as easily replaced as they can
the DSP chips
becomes
time,
for
support
to
higher
achieve.
example
in
level languages,
Reproducability a
radar
network,
of is
algorithm portability clone
processors
feasible.
and
However,
programmable hardware leads to a new set of development and maintenance 84
A higher level of training and maintenance equipment is
problems. for
trouble-shooting
a
radar
malfunctioning
required level
Board
processor.
Programmability
maintenance may require a more expensive spare inventory. brings new headaches as well as many new features.
Another area of rapid development important to distributed signal processor architectures
is
the
application
of
multiprocessor
operating
systems.
Distributed computing power on a common high speed bus requires an operating system capable of controlling data transfers and bus arbitration and memory Presently these operating systems are targeted towards more
management.
purpose
general
processor chips
(e.g., the Motorola
68030),
but
application will find them on distributed DSP processors as well.
future
Software
development is a key issue in generating efficient realizations of the DSP algorithms.
UNIX is presently becoming accepted as the common operating
system of choice for many applications programs and for development of real time software, which then typically run under a UNIX compatible real time operating system (e.g., VxWorks, PDOS).
7.3
SHIOR
TERM EXPECCTATICNS
During the next 5 years we may expect a revolution in atmospheric digital signal processor technology.
However, this technology will tend to leave
the atmospheric science community behind unless we prepare ourselves to take advantage of the evolving hardware and software advances. the pulse pair processor for over a decade.
We have lived by
Other techniques have been
explored that in same instances provide better parameter estimates but have not been feasible to implement in the past.
This constraint is rapidly
disappearing. 7.3.1
Rarge/velocity ambiguities
Within the next 2 or implement
new
dealiasing.
pulsing
3 years we may expect several research groups to and
processing
schemes
for
and
range
velocity
These schemes, driven by the FAA's Terminal Doppler Weather
Radar (TDWR) procurement, as well as the Nexrad implementations, will allow ground
clutter
overlaid
echo
suppression suppression
simultaneously algorithms. 85
with There
velocity dealiasing will
be
exploration
and of
polarization processing improvements combined with resolving range and velocity ambiguities and clutter suppression. 7.3.2 Ground clutter filtering Effective clutter filtering will be readily implemented on conventional Doppler radars. However, efforts to integrate clutter suppression with other processing improvements will likely encounter several technical obstacles involving analog components (e.g., polarization switches, IF amplifiers, and transmitter instabilities). Fundamental limitations related to the narrow clutter spectra may well limit clutter suppression for radars using dwell times shorter than the clutter correlation time. Yet to be explored nonlinear filtering techniques may allow effective suppression even under these conditions. 7.3.3 Waveforms for fast scanning radars A major limitation of existing Doppler meteorological radars is their inability to scan a solid angle in space fast enough to measure a rapidly evolving atmospheric event with adequate temporal resolution. A dwell time of a few milliseconds is desired. The proper long term solution requires an electronically scanned phased array antenna -
a very expensive item.
The
mechanical solution of simply scanning faster and using short dwell times is
insufficient to preserve the parameter measurement accuracy. Scan rates greater than about 100 degrees per second for a 1° beamwidth cause spectrum spreading due to antenna motion that rapidly degrades the measurement accuracy. A reasonable alternative is to rapidly scan mechanically at a rate such that the spectrum spread is not dominated by the scan induced component and to use a wideband waveform (pulse compression or multiple frequency) that allows a reasonably large number of independent parameter estimates to be made in the short dwell time imposed by the coherence time
of the return signal. Some research groups are testing short dwell time waveforms (Keeler and Frush, 1983b; Strauch, 1988) on both airborne and ground-based weather radars.
86
7.3.4
Data cmpression
Data compression algorithms are an important aspect of signal processing. Data compression can be divided into two classes -- "truncation" for any range gates at altitudes
greater than the tropopause
encoding"
or
for strings
Typically
parameter
"compaction"
estimates
not
and
"run length
of data having the same value.
passing
arbitrarily set to zero and run length encoded.
some
threshold
test
are
Data truncation will become
more common as programmable processors are installed.
7.3.5
Artificial intelligence
ased feature extraction
Future computing will be directed at enhancing man's analytical and inferential skills, rather than routine physical or mental activities. Symbolic programming techniques combined with knowledge engineering and artificial intelligence techniques show potential for rapid advance; the same is true for meteorological image processing and automated recognition and extraction of atmospheric features. Two dimensional signal and image processing algorithms will be implemented using programming architectures, reducing development time and extracting more meteorological information from remote sensor data sets. 7.3.6 Real time 3D weather image processing Relatively new computing hardware allows ready implementations of various symbolic object processing systems that can be applied to problems in atmospheric science. Coupled with fast graphics processors we can expect real time 3D images produced with the latest image rendering techniques which allow reconstructed radar data fields overlaid with in-situ measurements
from
airborne
and ground
based meteorological
stations.
Graphics computers with large video memories allow time lapsing of high resolution 3D images and arbitrary cross sections to be displayed using a variety of techniques currently being developed. Transparency of data elements near the viewer allows observation of the storm interior.
7.4
IDNG TERM EXPECIATICNS
Several years from now we can expect revolutionary changes in the way signal processing will increase our ability to understand atmospheric dynamics in 87
real time. Combining new hardware forms and more efficient software development techniques with evolving communications technology and the tumbling cost of computing power will allow remote sensor systems to present
readily assimilated graphical formats.
These systems will provide an
interactive user interface taking forms that are only dreamed about today.
For example, tactile feedback technology will allow a meteorologist to manually pick up a "thunderstorm" and manipulate it to better examine the evolving towers and outflows. 7.4.1
Advance hardware The present development of GaAs (gallium arsenide) computing elements may replace silicon dominated chips if the promised five fold speed increases and higher reliability in thermal and radiation extremes are realized.
7.4.2
Optical interconnects and processing
Fiber optical communication is capable of extremely high bandwidth. Data rates and parallel processing using optical techniques can accommodate processing algorithms having throughput many orders of magnitude higher than
serial and most existing parallel digital signal processing schemes. optic back planes for computers are available now. 7.4.3
Fiber
Ctumunicatians
Processing of atmospheric radar signals has many concepts in common with communications processing and the same technologies can be incorporated. By logically combining the processing functions with the communications link, both locally and over long distance, new capabilities will be possible. 7.4.4
Electronically scanned array antennas
Military budgets have financed the development of highly efficient, very low
sidelobe,
multiple
beam, two
dimensional
electronically
scanned
array
antennas. The computing power necessary to control the beams is available but the communications to each array element, the phase shifters capable of handling high peak powers for radar systems, and the sheer number of elements required (several thousand) are very costly. These step scan antennas will allow more rapid volume coverage while retaining parameter 88
accuracy and will reduce the deleterious effects of antenna sidelobes.
The
very
for
high
cost
of
this
performance
increase
must
be
justified
atmospheric radar applications.
7.4.5 Self
Adaptive systems learning,
time variable processing
optimization that is
not possible today.
systems will
allow a
degree
of
Neural networking concepts utilize
interconnected arrays of processing elements which share the processing and communications maximized.
load
so
that
the
The algorithms used in
overall
computational
efficiency
is
these adaptive systems can be defined by
a training sequence or can be self learning during the processing time. Research
is concentrating on integrating distributed processing concepts
with expected hardware.
89
8. 8.1
CoNCrI3SIONS
ASSESSMENT OF CUR PAST
Radar signal processing engineers, in the meteorological radar community at
least, have taken a somewhat narrow view of signal processing in the past. A large effort has been dedicated to using the pulse pair algorithm for
estimating the first two or three spectral moments, largely because the existing processing power has been rather limited to these simple algorithms
and because for an important class of signals the pulse pair algorithm is optimum. Advances have been made in the ST/MST radar community in pulse compression, coherent averaging, and non-linear least squares parameter fitting techniques, and in the lidar community in multiple lag processing. Other techniques have been ignored or rejected simply because the scientific
need for these advances did not exist, or if it did, the risk of undertaking such a development was not warranted. The operational radar community and many researchers have been unable to explore weak echoes because of inadequate sensitivity. There are better ways
of improving radar sensitivity than brute force techniques of more power and larger antennas. Advanced signal processing techniques must be explored more thoroughly to achieve these sensitivity gains. Modern spectrum analysis methods for modeling distributed target echoes in strong clutter and multi-channel processing techniques to extract better information from collections of remote sensors is an area ripe for extensive research.
The digital boundaries of the signal processor are being extended in both directions. Digital IF quadrature mixers are presently available which will accept IF and local oscillator analog signals and put out digitized I and Q samples.
Digital matched filters operating at IF rather than baseband (DC)
will became a reality. The radar engineering community is ready to integrate these new components where warranted.
91
8.2
RECEMMENDATIONS
R CU
FOR RE
Aside from continuing to actively explore many of the modern signal processing techniques, there are two general recommendations we would encourage for utilizing modern signal processing algorithms. First, many universities have active digital signal processing groups in the
Electrical Engineering departments and many industries have vast experience in radar signal processing techniques. Our research community should strive to interact more strongly with these two on an international scale. The university cooperative education programs should be explored and encouraged. University exchange programs involving signal processing experts as well as
meteorologists should be encouraged. Industrial contacts with radar manufacturers and systems producers, such as NEXRAD and TDWR should be maintained so as to exchange signal processing expertise as well as meteorological expertise. Second, the meteorological radar community should maintain the lead in sponsoring signal processing sessions at AMS radar conferences and sponsor participation in other signal processing related meetings. Members of the ST/MST radar and coherent lidar communities should be encouraged to attend
these sessions (and vice versa) since our target models, our propagation medium, our processing problems, and our techniques are nearly identical. As noted before members of these communities have successfully explored modern algorithms and predated weather radars use of the pulse pair and poly pulse pair velocity estimators as well as use of pulse compression and complementary coding schemes. Finally, as R.W. Lee of the Signal Processing panel stated once, we can now build processors with "megaflops to burn". We can use them very easily by implementing new processing algorithms, for example, using a priori knowledge to improve estimates. Computing special diagnostic outputs which have no bearing on the data collected, but simply allow the operator to adjust processing parameters, is an effective use of processing power.
92
8.3
ACCEPiANCE OF NEW TECHNIQUES
New techniques are not usually accepted easily by any scientific community. Twenty years ago, Doppler processing using the now standard pulse pair estimator was not readily accepted.
Why should any new signal processing
algorithms using only statistical concepts improve the accuracy of moment estimates?
Skepticism is healthy in science.
Accepting a new technique
requires four critical conditions:
1.
An important application, a problem which needs to be solved.
2.
An intuitive, familiar basis for understanding the concepts involved in
the new technique, which includes a convenient interface for exploring the innards of the new technique. 3.
A field demonstration to convince the community that the new technique is indeed an improvement over the former.
4.
Real, live funding for development and demonstration.
8.4
ACKNoICMrEDGMENT
The authors wish to thank the panel members for their verbal and written contributions to this report. helpful with comments
D. Zrnic and R. Serafin have been especially
on various drafts.
V. Chandrasekar, J. Evans, G.
Gray, J. Klostermeyer, F. Pratte, R. Strauch, R. Wiesenberg,
and R. Woodman
and have provided helpful written comments that have been incorporated into this signal processing review.
J. Devine provided expert assistance with
integrating the text, the figures, and the references.
93
ACIHNYM isr
A/D
-
analog to digital
AGC
-
automatic gain control
CNR
-
clutter to noise ratio
CSR
-
clutter to signal ratio
DFT
-
discrete Fourier transform
DSP
-
digital signal processor
FFT
-
fast Fourier transform
FIR
-
finite impulse response
FM-CW -
frequency modulated continuous wave
IIR
-
infinite impulse response
IF
-
intermediate frequency
I/Q
-
in-phase / quadrature
IMS
-
least mean square
ME
-
maximum entropy
ML
-
maximum likelihood
PRF
-
pulse repetition frequency
PRT
-
pulse repetition times
SNR
-
signal to noise ratio
95
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