•
I
r
r
I
F. R. Connor 1982
1979 Second edition 1982
may be retI"ieval . . "... or transmitted in any means, ..........'''' ......."" . . . . '..... 1I1ecJt1al1l1CalJ, p,n01:0CIDp~(lnJ~, nth.;coT''UI,nc:l'.;co without the l.PubHs:hers) Ltd. L .......... _
British
t.:altaU)2UID2 in Publication Data
F. R. Noise.-2nd ed. 1. Electronic circuits-Noise I. Title 621.3815'3 TK7867.5 ISBN 0-7131-3459-3
A"''''V& , .....,'"',
Scotland
Preface
In this new edition, certain parts of the text have been extensively revised. A new section on random variables is introduced in Chapter 2 and some basic ideas concerning matched filtering, decision theory, and estimation theory are presented in Chapter 3. A further treatment of circuit noise is made in Chapter 4 and a new section on low-noise amplifiers is included in Chapter 5. In Chapter 6, a comparative study of the signal-to-noise performance of various systems has been extended to cover digital systems and satellite systems. As an alternative approach, the energy-to-noise density ratio and its effect on the bit error rate is also included. A further feature of the book is the extended use of appendices to cover such topics as narrowband noise, decision theory, estimation theory, and the probability of error. It is intended for the reader seeking a deeper understanding of the text and is supplemented by a large number of useful references for further reading. The book also includes several worked examples and a set of typical problems with answers. The aim of the book is the same as in the first edition, with the difference that Higher National Certificates and Higher National Diplomas are being superseded by Higher Certificates and Higher Diplomas of the Technician Education Council. In conclusion, the author wishes to express his gratitude to those of his readers who so kindly sent in various corrections for the earlier edition. 1982
FRC
Preface to the first edition This is an introductory book on the important topic of Noise. Electrical noise is of considerable importance in communication systems and the book presents basic ideas in a coherent manner. Moreover, to assist in the understanding of these basic ideas, worked examples from past examination papers are provided to illustrate clearly the application of fundamental theory. The book begins with a survey of the various types of electrical noise found in communication systems and this is followed by a description of some mathematical ideas concerning random variables; Circuit noise, noise factor, and noise temperature are considered in the following chapters, and the book ends with a comparative study of some important communication systems.
IV
1
1
1 2 4
1.1 1 1.3
and
2
6 6 7 9 10 12 17
3
4 4.1
C·onlenlS
VI
4.4
5
tranMstor noise FET nOIse measurement
5.1 63
6 6.1
B C D E F H I J
L
m
B C
]
F
Vlll
Symbols
Pc PD PF PT R Req
R(r) R.Jt) Rxy{-r;) S
SIN StiNt SolN 0 S(w)
T
average carrier power detection probability false alarm probability transmitted power bit rate resistance equivalent thermal noise resistance autocorrelation function autocorrelation function of variable x cross-correlation function of variables x and y average signal power signal-to-noise ratio input signal-to-noise ratio output signal~to-noise ratio power spectral density absolute temperature periodic time antenna noise temperature effective noise temperature receiver noise temperature system temperature energy highest modulating frequency noise power of standard source , noise power of antenna
y
Y-factor
rx
transistor forward current gain transistor d.c. forward current gain Dirac delta function wavelength frequency of events correlation coefficient standard deviation time interval angular frequency
cX o
£5 (t)
A v p (J
L OJ
= ------------
I
III
I
2
1.1
3
o~O'-1----------~~-----------1~O------------1~O~O f(GHz)---
1.2
4
5
il
statistics. 6
if rnB is
occurrences
=
n- co
n
Bwe
statistics
p
n events
+
p
B mAB n
-------n
p
\..'IIJUJl.iIIJl.IJlk
the
two
P
v\..I1I.IUI.I.\..II.&..J
P
IA)P
or are
p A
Three coins are tossed all heads or all tails?
'nr1jQ ...... jQni'"1jQj"'lt
=1'
=p B is
random. What is the
Drclbaloili1tv of all tails P sum of P
events
Pr<)b~LbIjllty
for all heads and
i, the total !
uu,aul,u",
for
for
i.e.
A box contains six red beads and three blue beads. Two beads are drawn out In succession. If the first bead is what is the that both beads are of dIfferent colour? dniwlln2 out a red bead is six times out of nine or the conditional Drc)b~lbillitv been i.e. we obtain
statistics 9 Drc.oaltn1l1tv of dra.WlrUl out the two beads is The theorem as
and is
= or
10
08
030
025
X
8: 0-20
it
a.. 015 010
06 0-2
005
(b)
(a)
2.1
=
Noise
a
x
2.2
or
etc. are
+ ... + or
p
statistics
Two random variables x and y have standard deviations Determine the variance of their sum.
(J x
Solution
If z = x + y, the
'1118'1"'18,." ....450
y)
Now
o
and (y
Hence
y=o +
=
and
11
12
Noise
P(A) = 0 P(B) 75
2
3
2.3
4
statistics
m
13
Noise
0·25 0·20'-
vT::~3
015~
0·10!0051-
2
0
•
I
5
6
I 7
I
8
m
2.4
=1
10
05
o
x ...............
(b)
(a)
2.5
m= u u
e
m
Noise
= or
=
= or
= is
2.6
(J
= l-e
where
the variance.
we
1'\11 {",\,-p.,("n,,,",-
du Since
Hence
or
2ku
have
18
Noise Xl (t)
t __
Fig. 2.7 wherea~, In
time averaging, a partIcular sample function only is considered, e.g. (I),and averaging IS performed over a lime interval of this sample. Two of the most u~eful quantltIes are the average or expected value E[X] and the autocorrelation function R,(r) (whIch IS discussed In the next chapter). Xl
Stationarity A random process 111 which the average valuc of thc random variable X IS the same usmg ensemble averaging at any time t = r lor l = r2 is called a stationary process a~ its statistics are independent of the particular time position chosen. For such a process, the average or expected value IS given by E[X]
=
f~:XP(X)dX
where pIx) IS the probability density function of the random variable X in the Interval 0 < x < x + dx. Similarly, the autocorrelation function R,(r) over the
- d
E
E
R
'!)
r
y
()
o o
o
have zero mean
IS
and if is a
and
'lI<:lor1#:lnr"DoCO
r"f"U"'Icoi"'r:1ni"
since since or
x
p= Hence
+
0"2+
and
Show
Solution sin V2
Hence
with
0"1
=
=0
there is
or
v t v,2
Now
p=
Hence
p
0 VI V ,2
0
0'10',2
0'10',2
0
=:
or to
=:
:E: )( Vo
~
TI2
0
t-
T
-Vo
-, I
Vo
I I
I I I I
TI2
0
t-
T
I
I I
I
I
..J
Vo
3.2
+
24
(1 -
with
R
r)dt
sin
i
sin {
dt
r)
dt
dt function over R
and
.3.3
R
zero. Hence
autocorrelation function of the random the time instants zero ,...,.,........... 9"11,.......
Derive shown in distribution. + Vf-
tcol,con-Y"fJI't"'l.h
~
~
0
t--
Vi---
'---
'"--
3.4 For m ,..h''.:Ir...... coC' which more
Assume that the time interval T, cO]t1ve~nH~ntly written
To
-r)
± of variable
in Section
Here of chalngc~s
Since the
or
any interval of
as
3.5 and is shown in
3.5. SInce -t)
more
Flo",,,,,,,,,,c;a.U.J
26
Comnlent We observe that the curve in 3.5 is narrow at low values the narrower is is the number of in the interval T = t, n·u"I .....~lt • ..,.n the presence treaUlen(:v clomponellts. This will be confirmed later when the spe:ctral obtained for this
Two random are where A and B are constants and cross-correlation function
:=
O.
Solution
-'t")dt 1" ....
1
+
T ::::::
dt
sin
AB
sin 8)
T
-cos
x
AB
or
Each of the
1nt,{!!Ian1"I!.lIIC!
SIn dt
sin (J) 1 t sin
8
+sin 8 cos
cos (J) 1 t sin
-cos8 sin
sin
cos
sin 8 SIn
cos
cos
dt
is of standard form and is
is
28
Noise
2.
2
",""''::'''<'or
if
With the of the Wiener-Khintchine 'the:'r.r,p'l"\""t determine the of the random 3.4. What the the result?
T l I r . ' I I . . " P ·. . .
function of
Slgn:lhcan(~e
was ........c.·.,1r~'."'I .. , obtained as
The autocorrelation function of the random t-.:o.","or')1"'\h we dr
e
or
e -2vl_1
dr
+
e - Jwr
]
e -2\,1·1
+
dr
e
the form
=2
-2v
+
or and the maximum
is
It
(0-
3.6
IS
R
>
Wo
amount of
30
Noise
Solution From the Wiener··-Khintchine theorem we have
f~: S(f)eJWTdJ
R(T.) =
or
R(O)
= f~: S(f)dJ
which is the average output power of the network. Here
.
Sowo
Sowo
0
R(O) = - - e = - -
.
4
4
watts
If V rrns is the output voltage in a 1 Q load then
volts
or
3.6
White noise
Random noise signals generated by various sources are known to have a uniform power distribution over a very wide range of frequencies up to about 10 13 Hz, which is in the ultra-violet region, after which it falls ofT as predicted by quantum theory. Such noise is defined as white noise by analogy with white light which has a uniform power distribution over the band of optical frequencies. The nearest examples are thermal noise in a resistor and shot noise in valves or transistors which also have a Gaussian amplitude distribution and are known as Gaussian white noise. If No is the noise power spectral density per Hz for positive frequencies only then, assuming both positive and negative frequencies (for mathematical purposes), the noise power spectral density is N o/2, as shown in Fig. 3.7(a). Hence, we obtain for the autocorrelation function
1
R(r)
=~
I
""II.;
or
1
('+ oc'
S(w)e lWt dw =
-<:I)
R(T)
=
~ ""II
f+
oc
N e lWr 0
-0:'
2
N
dw =
1'+ J eJ"H dm 2II -"'0
~2
1
oc
N ~o(r)
2
where 6(T) is the Dirac delta function shown in Fig. 3.7(b). Since R(T.) has a value at T = 0 only, there is no correlation between any two samples of white noise separated by an interval, > 0 and they are therefore statistically independent. From Fig. 3.7(a) it will be observed that the average power, which is given by
Pay =
+- f+ "" .... 1f
- oc
S(w)dw
Correlation techniques
31
R(.)
I
No/2 S (Ul) .....
____
L _ _ _ __
0
............... -Lv
+Ul(b)
(a)
~vl
R(.)
1 ("I o
~B
B
f-
T-
(d)
(c)
Fig. 3.7
becomes infinite and cannot be physically realised in any practical circuits. As most communication circuits are band-limited, it is more practical to consider the results of passing white noise through a filter with some defined bandwidth. The output noise is then called band-limited white noise or coloured noise. 3.7
Band-limited white noise
Ifwhite noise is passed through a low-pass ideal filter with a bandwidth ± B Hz, the output noise can be obtained by means of the transfer function H(w) of the filter. Hence, we have SI)(W) Sj(w)
= IH(w)12
where S, (w) and So(w) are the input and output power spectral densities and IH(w)1 = 1, with 1 r+ cc No No f+B 1 "+cc. Pav = 2 So(W) dw =? I -2 dw = df = NoB watts n -00 _n,; _ -oo 2_B
J
and is illustrated in Fig. 3.7(c). The autocorrelation function R (r) of the filtered white noise is R(r)
1 =;;-.t..1t
1
=-
2n
f+
00
-co
So(w)eiW!dw
= -1
f+
00
2n:",_oo
Sj(w) IH(w)1 2 e JWt dw
J"+ ~eJW!dw=~! N. N i+B N [eJ21
_
(f)
2
2 .!
-B
2
J2m
-8
Noise
or
R
x x
iovv-uass filter shown in
3.8 is white noise with a power power and average
1IlI .... ""...,"I.Ir.l.1.
Determine the noise power.
3.8 Solution If is the network transfer function we have
or and
are the
and
power
1IlI ........'...,"I.'u,&
1
or The average
noise power is
densities respec:tlvely then
the substitution u = wRC with du = RC dw we obtain ex; du -1
4nRC
4nRC
=--
4nRC
or
A sine..wave carrier ....... with Gaussian white noise is IF and detected a linear detector. that follows or a distribution oel>enOll1l2 ",.0. ............
If the
then the carrier
Vcsin
band..limited noise from the IF
= {Vc+
+
where the
cos and
y
or where r The
that and
where The Since
is the variance. of the x and y
r y)dx
Also
=
y)
34
Noise
or since dxdy = rdrd(p. Hence re - [,1 -
2rf', cos¢+ v~J;2al
p (I', ¢) = - - - - - - - : : - - - - -
To obtain the amplitude distribution p (r) we must integrate this expression over all values of ¢ from 0 to 2n. Hence p{r) = { " p(r, ¢)d¢ =
J
"2
n
0" "'.LV re - [" r -~"" COS 'Y'
271.a 2 I"e - (,' + ~. ~ )/2a' f21r
1 J,,2 ,
~a
d¢
o
= .
,
27[a~
e d ', cos 1>/,,2 d4)
0
To evaluate thIs II1tegral we use the standard integral 10 (z) where
which is a modified Bessel function of zero order. Hence
The quantity Vc /a 2 can be related to_tb~input carrier and noise powers C, and N, respectively since V~/2 = C 1 and a 2 = n 2 (t) = N j • Hence
rV,
r.j2C, ,-
(J
.IN,
r z=_· )2((',11'1,)
or
(J
p(/')=---(J2
or
p(r).~
re - r2 '20'2 - - , .... (J.
(since V~/2(J2 ~ 0)
which is a Rayleigh distribution with the peak value at r = a.
e=
])
at
t'
r
3.9
3.8
Noise
3.10
can
R
+
+
37 are
n or
]
f
n
38
Noise
digital signals, it is pointless having an output anywhere between 0 and 1. Hence, on the basis of the received waveform, it is necessary to decide which of the two states the signal is in. In this decision approach, the filter at the receiver must be matched to the received waveform to achieve the maximum output signal-to-noise ratio. Such a filter is designed specifically to maximise the output signal-to-noise ratio and is called a matched jilter. The matched filter is the optimum filter for detecting signals received with additive white noise, i.e. noise with a uniform power density spectrum over a wide frequency band. It is shown in Appendix F that the transfer function of such a filter is given by
H (01)
=
kS* (w)e -JW!d
where k is an arbitrary gain constant, S*(o1) is the complex conjugate of 5(w), the spectrum of the received signal s (t), and td is the time-delay of the filter. Furthermore, the impulse response of such a filter is given by h(t) = kS(td -t)
where s(t) is the input signal to which the filter is matched. The time-delay td is required to make the filter physically realisable and the filter has its maximum output at some time t = to. The optimum decision is then made at time t = to to determine the nature of the output corresponding to a 0 or 1, and the output depends only on the original signal energy and is independent of its waveform. A practical implementation of the matched filter for a rectangular pulse is shown in Fig. 3.11.
Input Signal
Pre-filter
Comb filter
_
i
Output pulse
I Fig. 3.11
In many practical cases, it is not feasible to provide a suitable matched filter as it may be physically unrealisable and so somewhat simpler filters are used with some loss in signal-to-noise ratios. Table 3.11ists some typical examples of these and it is observed that the maximum loss is only about 1 to 2 dB. In Appendix F it is shown that the matched filter is mathematically equivalent to a correlation detector. In practical applications it may therefore be more convenient to implement the matched filter by means of a correlation detector. In the correlation detector shown in Fig. 3.12(a), the transmitted signal waveform s(t) is stored at the receiver and correlated (multiplied) with the received signal plus noise. After integration over the time interval to, the detector then decides whether a 0 or 1 has been received.
Table 3.1
s ' - - - - - - - v(t)
s(t)
t --+-R
3.12
40
Noise
which converts the non~white noise to true white noise and modifies the input signal slightly. The modified signal is now mixed with white noise and can be optimally detected as was shown earlier. The modification to the input signal leads to some intersymbol interference in the case of digital signals which can be minimised if the bandwidth of the pre-whitening filter is large compared to 1/1' where Tis the duration of the digital signal.
3.10
Decision theory 4
In the design of optimum receivers, an important problem is the detection of the received signal in a background of noise. This may involve simply the detection of a given signal or its absence or it may involve the detection of a zero or one, as in digital data communications. In either case, the receiver must make the best possible decision on the basis of various criteria. The statistical nature of the decision making process involves hypothesis testing and is known as decision theory or detection theory. Receivers designed to minimise the average cost of making a decision use, as a basis, Bayes' decision rule. When a sufficient knowledge of costs and a priori probabilities is available, the decision process can be optimised according to an expected cost criterion. It is shown in Appendix G that such decision procedures which are based on expected cost minimisation involve testing a likelihood ratio which is the ratio of the a posteriori probabilities of the observations against a threshold which depends on the a priori probabilities and costs. It is given by the expression L[ (t)J~PdC21-Cll) Y ~~p2(e12-e22)
where the Lh.s. is denoted by the threshold value L t , P 1 and P 2 are the a priori probabilities associated with the hypotheses Hl and H2 respectively, and ell, e 12, e 21, and C 22 are the conditional costs of making decisions. The Bayes' criterion is therefore characterised by the average cost or risk involved in making a decision. In many cases, it is more useful to express the risk in terms of the probabilities of detection PD and false alarm PF , provided the a priori probabilities and costs are available. If the costs and a priori probabilities are not available, a useful decision strategy is the Neyman-Pearson criterion which maximises P D while holding P F at some acceptable value. This type of criterion can also be reduced to a likelihood ratio, where the threshold is determined by the allowed false alarm probability. It is shown in Appendix G that its value is given by dPD L[y(t)] = dPF
where (dPD/dPr) is the slope of the receiver operating characteristic at any given point.
Correlation techniques
3.11
41
Estimation tbeory 16
It was shown earlier that decision theory can be used to detect the presence of a signal. Similarly, estimation theory can be used to make an estimate of some
unknown parameter of a signal such as its frequency or phase. Since the signal is received with additive noise, only an estimate of the signal parameter is possible. Two useful estimators are Bayes' estimate, which endeavours to minimise a cost function, and the maximum-likelihood estimate, which tends to maximise a likelihood function. As the latter does not require any a priori information, it is often employed and will be considered here. Maximum-likelihood estimation (MLE) Since the received signal is a function of time and contains additive noise, it can be represented by z(t) = s(t, e) + n (t) where s(t, 0) is the transmitted signal, e is the unknown parameter to be estimated, and n (t) is assumed to be Gaussian white noise. By observing the signal continuously or by sampling it, it is possible to make an estimate of e which is denoted by ti. lt is shown in Appendix H that, for a sampled signal, the likelihood function ofz(t) is p(zIO)=Aexp
{ -NoIf
T 0
[z(t)-S(t,e)]2 d
tJ1
where A is an unknown constant. To obtain the maximum-likelihood function, it is convenient to differentiate In p (zIO) with respect to e and then equate it to zero. This yields a 2A as(t,O) -[lnp(zIO)]=[z(t)-s(t,O)] ~'e dt ao No 0 0
IT
and so the maximum-likelihood estimate of 0 is a solution of the equation
I
~
T
o [z(t)-s(t,O)]
as(t,
ao
ti)
dt=O
where 0 has been replaced by the estimate ti. The estimate ti depends on the signal received and on the number of observations made. Hence, it can vary as a random variable with a mean and variance. The variance of an estimator, under certain conditions, cannot be less than the lower bound known as the Cramer-Rao bound. It is shown in Appendix H that, if the expectation ofti, Le. E [tiIB], is equal to the true value e, the variance of ti is given by 2
u·
o
1
fJ
~ -=~------~==
E[ {aln~~zIO)
42 Noise
Example 3.10 A signal of known amplitude and frequency has the form s(t, 0) Make an estimate of the phase angle cPo
o ::;; t ::;; T.
= A sin (wot + t/»
Solution We have
s(t, 0)
= A sin (wot + 4»
os(t 8) T= A cos (Wot + cP) and the phase estimate
f:
[z(t) - A sm (wot
IT
or
o
;p IS a solutIOn of the equation + $)] A cos (wot + $)dt
~
= 0
A fT -sin2(w o t+t/»dt A
z(t)cos(wot+cP)dt =
0
2
z(t)
(a)
y
zIt)
tan-· 1 ylxl---__ ,p
(b)
Fig. 3.13
where
The ...... t,:>rT ....... 1
the
if
where
is an
or
1. Hence
T
o
= tan
n"'Il111".r.I1,o ... "
and matched
In
4 Circuit noise
The two most important types of noise associated with electronic components, such as valves, transistors, and resistors, are thermal noise and shot noise. The physical basis of each will be considered and, as both of them give rise to noise power in the system, they can be regarded as producing one combined noise effect. 4.1
Thermal noise
A metallic conductor or resistor contains a number of free electrons. Due to thermal agitation, these free electrons are moving about continuously in the conductor causing collisions with the atoms and a continuous exchange of energy takes place. This accounts for the resistance property of the conductor and, though there is no current in the conductor on open-circuit, the random motion of electrons in the conductor produces voltage fluctuations across the conductor which accounts for a mean-square noise voltage v~ at its terminals. The thermal noise effect was investigated experimentally by Johnson 17 and theoretically by Nyquist. 18 Experimental results showed that the thermal noise voltage depends upon temperature and its mean-square value v~ is given by v~
= 4kTBR
where k is Boltzmann's constant, T is the absolute temperature, B is the bandwidth of the system, and R is the resistance of the conductor. For example, if R = 1 kQ, B = 5 MHz, and T= 290K then with k = 1·38 X 10- 23 11K we obtain vf = 80 X 10- 12 or Vrms = [vtJ 1 / 2 ::::: 9/lV. Nyquist's investigation of the effect was based on thermodynamical reasoning and similar results were obtained. He showed that the thermal noise power Pn associated with any resistor is given by Pn
= kTB
watts
where k, T, and B have their previous meaning. The derivation is given in Appendix I and is based on the assumption of available noise power. This implies matched conditions, as is usually the case in most communication channels, since it is necessary to transfer the maximum
Circuit noise
45
signal power through the system. However, in practice, the concepts of noise voltage, noise power, or noise power spectral density can be equally well employed in the study of noise problems. If the noise voltage spectral density is S. (f), it can be shown that* Sv(f) = 2kTR
a result which depends on T and R but is independent of frequency up to about 10 13 Hz. This implies that thermal noise covers a broad band of frequencies and has a uniform response. Hence, it is often called 'Johnson noise' or white noise due to an analogy with white light which has a uniform power distribution over the band of optical frequencies. Equivalent circuit It is convenient in practice to represent thermal noise in a resistor as due to a thermal noise source v~ in series with a noiseless resistor R, which is based on
Tbevenin's theorem. Alternatively, a current source it in shunt with a conductance G may be used and this is based on Norton's theorem. This is illustrated in Fig. 4.1.
R
v~·-)
TL-_ _o Fig. 4.1
Under matched conditions, the load is also R (assumed noiseless) and the maximum noise power available from the source is obtained as follows. In Fig. 4.2, iUis the current in the circuit then i = ~/2R and the maximum
R
R
Fig. 4.2
*
See Appendix C
we
or a
Two noise
in series and 'l'"A«."~A"tnl·AI" tenmIIlals in each case.
Solution
Series 'l'"A«." . «."t.,. . 'I'"«."
4.3 If vn is the rms noise .. '. . .
11'11" ............
at the
........,"" .........."... then
where Hence
+
paI·au~~1.
If the
~~.~n.~~Atherms
noise vn = = T, as is gerlentUY the case in pnlctllce. then
or In .......................'. . . . . _ if
resistor of value
and the two resistors behave as a Parallel resistors
4.4 The noise the noise these two where
vn
at the are rms
=
terminals due to terminals due to 1.'.2 the total rms
and at the
terminals is
we have
with
and
or In
nQ11"f'11"'nl~I"
when
practlc:e. then
+ or
Vn
Noise resIstor of va]ue
and the two resistors behave as a
o
t-
4.5
noise
4.6
IS
50
Noise
+
Fig. 4.7
with an equivalent resistor Req. For a triode, the hypothetical resistor is inserted in series with the grid and it is given by RCq ::::::
2·5/g m
where gm is the mutual conductance of the triode. The equivalent circuit for combined thermal and shot noise is shown in Fig. 4.8. +HT
+HT
+HT
Rg
Fig. 4.8
The equivalent rms 'thermal' noise voltage VI
where
=
vf =
VI
of Fig. 4.8 is given by
Jv~ + v~ 4kTBRS!
v~ = 4kTBRcq
Hence
Example 4.2 State two sources of noise encountered in high-gain amplifier circuits and briefly explain their origins.
Circuit r"" ..... "'f.'... ........... ""
51
of 4 kQ over
with
the
4.9
10 or Also
with
= 4kTB
or or kQ noise factor
in Section 5.1 F
52
Noise
where (S,IN,)'is defined at the source and (So/No) may be referred to the input side at points P and Q, with the amplifier noise represented by Req. Hence (2vY
F
(Vsl2
4DOOkii3, (iooo+:i3jO)I~fB
=
4 4000 x 3330
F = 3·33 or 5·2dB
or
4.3
I
=
Partition noise
In multigrid valves, such as tetrodes and pentodes, the division of current to one or other electrode is sut-ject to random fluctuations also. This gives rise to a further noise effect which is basically similar to shot noise. It is known as partition noise and can be evaluated using statistical ideas similar to that for calculating shot noise. This effect can be accounted for by increasing the value of the- equivalent noise resistance Req obtained previously for the case of a triode. The value of Req for a pentode is given by
Req =
(Ia
Ia ) [2'-+-25 20Is] + Is
grn
gm
where 1a is the anode current, I, is the screen current, and gm is the mutual conductance of the valve. Typical values for RCq are between 1000 0 to 10 kO. Due to partition noise, multigrid valves are more noisy than triodes and should be avoided in the early stages of an amplifier if noise is of primary concern, as in low-noise amplifiers for space communications. The noise generated in the early stages is amplified in subsequent stages and will give a large noise output. In the case of multigrid mixers, the conversion conductance gc is used instead of gm for obtaining the value of Req. Since gc is much smaller than gm' values of ReQ around 100 kO are possible and so multigrid mixers are quite noisy.
4.4
Bipolar transistor noise 20 • 21
Noise in junction transistors shows some similarity to that in valves and the three types of valve noise, namely thermal noise, shot noise, and partition noise, are found to exist in transistors. This is basically because random fluctuations in the movement of the charge carriers (electrons and holes) cause variations in the various transistor currents. However, their exact nature and evaluation is more complicated and still not clearly understood. For the bipolar transistor, it is found that thermal noise is associated with the
b
4.10
54
Noise
trans-
III
I
5.1
F
power power
the
the '-J~"'-''''''''
• ""' .. A''''''....,'.....
power
IS
IS
measurement
an F
= 1.
3. a .......,,-........,.LJ
4.
Discuss the sources of noise in an ampll:ner and the manner in which
limit
am,pU1tler deSilgIled to from a 75 n source contains SU:U"UlU ratio 1 the first valve enl11V~=llelnt noise resistance of the valve to be draw the arr'an,geIne][}t and hence calculate the source emf that would be reCIUllred ratio at the in bandwidth of 200 kHz. The value of 4·14 W. nU!n-l!aln
LJ ..............:LILJ.:I ..
Solution The main sources of noise overall effect of the noise this ratio below a certain minimum is limited. the
described in Sections 4.1 to 4.3. The ratio of the If of value and the
SH!nal-IO-IHHse
Problem The circuit is shown in circuit is
5.2
Noise IS the SHlnal-H)-n.OI~;e ratio
and
s
4 x 1200 k7 B
= 4·14 X 10- 21
200 x
16
noise of the ampUJher and
F=
Also
1200+ 700 1200
IS the
=1·58 1)
F=
e;
Hence
we have
300
8·28 x 10
or
=1.
'LU ...., ...
e:
=--I
the ..., . . .
16
1·58 x 8·28
or
e; = 0-392
and
es
10
12
= 0-63
ratio and a network. netVtrork with ItS nOise properties two of nlcan-~quare and the other of meandn\·en from a source an interna1 resIstance to hc uncorrelated. an for the nOIse factor sourcc and that It IS a mlnlmum when expressIons for 2 and Ifthc nctwork consists shunt-connected or a ~cnc~-connectcd rC\I~tor J~. Define the terms nOl~e between them In 5 ~ho\\'~ a
~1~nal-U) .. ntOI!\e
. n l l .......... ,... .....
n&>'I"U''''r'lll'"'ltl'tc"
r&>£:"I£:''I'''''''
N
5.3 Solution
answer to
first part
IS
gIven In Section 5.1.
circuit
is the
---+
+ 4kTB
5.5 Shunt
R
at
Series resistor
=0 Hence
4kTBR
.5.6
1+
or
F
1+
where J1
or
if
1+ J1
Noise measurement
Common-base
5.7
=1
J
+
J
(1-
F=
+----------------------~--------
=1+ IS
source
62
Noise ,-----0
JFET
+ VDD
IGFET
Fig. 5.8
where Rn is an equivalent noise resistance, g,n is the input conductance, and gm is the forward transconductance.
5.3
Cascaded networks
When two or more active networks such as amplifiers are connected in cascade to give greater amplification, the overall noise factor F of the arrangement is important. Consider two networks in cascade with noise factors of FI and F2 and power gains of Gland G 2 respectively. Let F be the overall noise factor of the combIned networks with a relevant bandwidth B. If a noise signal with power kTB is fed into the first network we have output noise of first network
=
FI G} kTB
output noise of ideal network
=
GjkT B
nOlse generated
1I1
first network = F 1G 1H B-G 1k7 B
= (F I -l)H BG I The noise generated in the Ilrst network therefore appears to be due to a hypothetical input noise component which is equal to [(F1 - l)k TBGdG l ] = (FI - l)k TB and is shown dotted in Fig. 5.9. Similarly, the equivalent noise generator associated with the second network is (F2 -l)kTB. Hence, the first two networks can be regarded as 'ideal' for the purposes of this analysis if the noise contributions from the two input signals are taken into account and the total noise output from the second network = kTBG 1 G 2 + (F I - l)kTBG J G 2 + (F z -1)kTBG J = FlkTBG} Gz + (F2 -1)k7BG 2 ·
measurement 1)
5.9
F
F=
or
+
F
2.
a
Lean
to
64
Noise
~) RF
o+HT
~
choke ')
c
?s R
Fig. 5.10
Active network
NOIse diode
Attenuator f-------i Power meter
F1 .G 1
Fig. 5.11
equivalent circuit in Fig. 5.10 is a shot noise current generator delivering a shot noise mean-square current Let the input resistance to the active network which is used be Rgo The measurement consists in noting the output power Po with the diode current zero initially and wilh zero attenuation in circuit. The diode current is then adjusted to a value I a which gives the same output reading with 3 dB attenuation in circuit, i.e. the output power has been doubled. To obtain a suitable meter reading for Po, the network gain is set at the same convenient value for both readings. The use of 3 dB attenuation maintains the reading at Po in both cases and avoids any scale non-linearity. Hence, we have
i; .
noise power available from resistor Rg
.
.
e2 := _ 1 _
4R g
e;
=
nOIse power output of dIOde = _0- = 4R g
Hence, when Ia = 0 we have
Po = kTBGF
(since F = No/kTBG)
kTB
eIaRgB
-----=~-
2
Noise measurement
65
When the diode current is set at the value 18 we have
Po+Po = kTBGF + elaR g BG/2 or
2kTBGF
Hence
= kTBGF + el a Rg BG/2
kTBGF = el a R g BG/2
with Typically, e/2kT ~ 20, hence F::::: 20I a R g
and it can easily be evaluated since the values of 18 and Rg are known directly in the measurement. Comment
At higher frequencies in the microwave region, the diode noise power is insufficient for measuring large noise factors and so a gas discharge tube is used. It is placed at a small angle across the waveguide to produce an impedance match. At present, solid state noise sources are also available. (See Section 5.9.) 5.5
Noise temperature
The thermal noise power P" available from a resistor in a bandwidth B is kTB where T is the absolute temperature of the resistor. Hence, an alternative concept associated with noise power is the effective noise temperature Te which is given by Te = Pn /kB for a resistor at temperature T K. The idea of effective noise temperature can be extended to other noisy sources which are not necessarily associated with a physical temperature as is the case with a resistor. For example, a non-thermal device, e.g. an antenna which picks up noise power due to the random radiation it receives from various directions, may also be associated with an effective noise temperature Ta • If the noise power received by the antenna is Po in a bandwidth B then Ta
= P,jkB
The value ofTa depends on the direction in which the antenna points and its radiation pattern. Different parts of the sky are associated with sources of random radiation usually called galactic noise, solar noise, etc. Hence, the sky effectively has a 'noise temperature' and it varies with frequency as shovm in Fig. 5.12. The concept of effective noise temperature may also be applied to an active network, such as an amplifier. It is found to be more useful and meaningful for low-noise amplifiers, such as masers or parametric amplifiers. In the case of
5.12
G
.5.13
F
1+
measurement 1-0...,r'\ ....... ,~ ... ..,,1- •• T·O
here value
the K
F
F
where
IS
a
A B C
6dB dB 20dB
the
HHilJ-It':vel
2-0 ..... r' ...... the arnlpllheJrS must 1"'11 ..... lt:>,C'
the . . . . . ,,.+."'.•• 1,,....
B,C
F
+---
0·25 +0-047
F= For
B
+--1)
+--+--4 x 100
4
+
+0·025
F = 2·475 then
B,
are, the
aJ.J.A ........ .., ..........
the no1s iness'. are inter-related 4
3. 4.
the
.".V ...·H·.".""'.r-, .....
Both effect.
In
""-.. '", ..... '""' systems.
Noise measurement
69
case of an antenna connected directly to a receiver, Ts is defined by 1~
= 1~ +1'r
where Ta is the antenna noise temperature associated with its radiation resistance in thermal equilibrium with the environment and Tr is the effective input noise temperature of the receiver due to its internal noise sources. However, for convenience, a transmission line often connects the antenna to a receiver and gives rise to thermal losses, while a preamplifier may be used ahead of the receiver to improve system performance. In this case, it is convenient to refer noise temperatures of the 'receiver chain' to a point just behind the antenna, as shown in Fig. 5.14. The system noise temperature T" is then given by Ts
=
1'a + (L -1)To + LTp + LTr/G
where L is the loss factor of the transmission line, To is the ambient temperature, Tp is the noise temperature of the preamplifier with power gain G, and Tr is the effective noise temperature of the receiver.
>r
•
L,To
l"s
•
Tp,G
I
I
I
Tr
I
Fig. 5.14
Example 5.4 An antenna with a noise temperature of 57 K is connected by means of a cable to a preamplifier and receiver. The cable loss is 1 dB and the preamplifier has a 20 dB power gain and a noise temperature of90 K. If the effective temperature of the receiver is 290 K, what is the system noise temperature? Assume an ambient temperature of 290 K. Solution We have
Ts
=
Ta + (L -l)To + LTp + LTr/G
with
L = 1 dB = 1·26
and
G = 20dB
=
100
Hence
Ts = 57 + (1'26 -1'0)290 + (1'26 x 290) + (1'26 x 290)/100 or
Ts
=
57 + 75·4+ 113-4+ 3·65
=
249·5K
Noise factor L. nOIse
the
is t43rnn,o.ro:Jtl1'·43
of the receiver
28
>
1
> d> c w
C)
>
C)
d> c w
- - - ' ' - - - - - - ' - - - - W, Th ree-Ievel maser
maser
5.15
71
.5.16
72
I
t _______ Output
5.17
measurement Input
Input Idler circuit
5.18
74
Noise
performance and is fabricated by the planar process. The thin conducting GaAs epitaxial layer is deposited on a semi-insulating GaAs substrate and the Schottky barrier gate is formed by the deposition of a metal on the epitaxial layer. The construction of a typical device is shown in Fig. 5.19. Schottky barrier
s
\
\
G
D
\ s
D
Semi-insulating GaAs
-1~ Gate length
Fig. 5.19
At present, the most important application of GaAs FETs is in low-noise amplifiers. GaAs FETs designed for low-noise applications have Schottky barrier gates with typical gate length of 0·5 ,urn. Noise figures range from 1 dB at 4GHz to about 2dB at 12GHz with a power gain of about 10-12 dB. A microwave low-noise amplifier may employ one or two GaAs FET stages followed by a bipolar amplifier. Alternatively, the first stage can be an ultra lownoise parametric amplifier followed by a low-noise FET stage. An overall noise figure of 3 dB can' be achieved over a 1 IvlHz band in the freqi,lency range 7·25-7·75 GIlz. FET amplifiers have other advantages, such as long shelf and working lives, small size, and low power dissipation. They are finding increasing use in the front end of various kinds of microwave receivers for both radar and satellite communications. In earth station applications, they are a very reliable and costeffective way of implementing a low-cost satellite ground station with an excellent figure of merit GIT.
Example 5.5 Sketch and discuss the variation of the sky noise temperature as a function of frequency. A receiver has a system noise factor of 10 dB and it is proposed to improve its sensitivity by adding a preamplifier of 3 dB noise factor and 10 dB power gain.
Noise measurement
75
Solution
minimum when horizon. behaves
Problem of aerial without
290
2900K ----=841K
1rn.r"\rr'l.''I7o.r'\'''!ol'>'nt
factor
2900
3-45
2668 K with pre:amlplrDer
+----
7.,
lrn'l"'Iorr'l.'l.l01r'\"'!ot:>lnt
factor
K 6·4dB
the latter
dB when the
76
Noise what
Assume that the bandwidth of 290 K. defined with reference
linear
IS
the main same and noise
K
1
5.20
Lis
with
When referred
Hence
we have
the mput
290
=1
280
or Also or
40dB
the 9
F=
and or
or
Noise measurement
5.8
77
Noise temperature measurement 31
The effective noise temperature Ta of an antenna is usually measured using the 'Y-factor method'. The principle of the method is to compare the noise power received by the antenna to the noise power generated by a standard noise source and, from the ratio cif these noise powers, Ta can be determined. The circuit arrangement used is shown in Fig. 5.21.
Low-noise preamplifier
rn~~wer
I
! meter , ! i
Matched load
Fig. 5.21
With the standard liquid-cooled load connected to the input, the output noise power Ns is noted on the meter. The antenna is now connected to the input and the output noise power Na is noted on the meter. If the ratio NjNa is denoted by Yand the receiver noise temperature is TR we have k(T, + T R)B Y - -Ns - -'---- Na - k(Ta + TR)B
where B is the relevant system bandwidth. Hence
Y= Ts+TR Ta+TR or
YTa + YTR = 1~ + T R
T = ~+TR(l- Y)
and
a
Y
To measure TR , the output noise power N h of a 'hot load', i.e. a load at room temperature, is compared with the noise power output N s of the standard cooled load. If the ratio of these noise powers is Y' then we have Y' with or
N = k(Th+ T)B R Ns k(Ts + TR)B
=_h
Y'Ts + Y'T R = Th + TR Y'T R - TR = Th - 'f'T,
78
Noise
Th - Y'Ts
T ------R - (Y'-1)
Hence
which can be determined since Y', T h , and Ts are known by direct measurement. The value ofTa can then be determined from the previous expression using the known values of 7'., TR , and Y. 5.9
Excess noise ratio (ENR)32
Microwave noise generators are usually of the solid-state or gas discharge types which use room temperature as the standard reference by simply de-energising the noise source and assuming room temperature as the standard value To. In this case, the effective receiver temperature T R is related to the noise factor F by T = R
with or
T - Y'T h
(Y' _ 1) F _
° = (F -1)1'°
_ (Th/1'o) - Y' 1- (Y'-l)
F
=
_(T-"h-,---IT--,o,-,-)-,---_1 (Y' -1)
where Th is the hot-load temperature of the energised source. The quantity in the nUlneratQr is a measure of the power output of the :Q.Qise source and is called th~_~xc,=-~s noise rqUo.(ENR) which is giveI1l:>i" -and
ENR = 10 log (ThiTo -1)dB F = ENR - 10 10g(Y' - 1) dB
Solid-state noise sources are semiconductor p--n diodes operating in the avalanche region. The randomness of the avalanche multiplication process produces fluctuations in the avalanche current which generate random noise over a wide frequency range. Typically, the diodes operate at voltages of around 20 to 30 volts and are driven from a constant current source from between 5 to 20 rnA. The choice between solid-state and gas discharge noise sources is based on frequency coverage. For laboratory work below 18 GHz requiring operation at many frequencies, solid-state units otTer economic advantages, in addition to small size, low mass, and low power consumption. Usually, solid-state noise sources are calibrated at several frequencies and, typically, may have an ENR of around 15 dB in the frequency range 1-12·4 GHz or, in certain cases, an ENR as high as 40 dB. They can be used to determine the noise figures of amplifiers, mixers, or receivers and also to check the performance of radar and communication systems.
34
80
is
~
'0
components 1------
>
o 6.1
f--
Systems
81
If S (f) is the noise power spectral density, then the power in a bandwidth (~f associated with the single 'impulse' is S (f) (jj = No (5j where No is the power spectral density in watts/Hz. If the corresponding single noise component is vn(r) we have V,,(t) = Vn sin 2nUIF+ jn)t where };F is the IF frequency and fn is the frequency of the particular noise frequency component with a peak value Vn and No (5} = V~/2. When of --+ 0, all the noise components cover the IF bandwidth 2B continuously and the total noise power N is N, =
S
and
J::
I
No 6} = 2NoB
m2 V 2
1/\';, =
m 2 V2
J
~j
2N oB =
m2 P
8No~ = 4No~
where Pc is the average carrier power. To obtain the output noise power oflhe detector, each noise component v,,(t) within a bandwidth ~rwill beat with the carrier and the resultant voltage is given approximately by Vn(t) = Vc [1 +(Vn/Vc)coswntJsinwct
if Vc no(t)
}>
Vn . When this is applied to the detector, the output noise voltage
= Vncoswnt. Hence, the average noise power (jNo in a bandwidth of is 6N o
=
V;/2
= No~f
(as obtained earlier)
and for the whole IF bandwidth this becomes
f
+B
No
=
No
df =
2N () B
-B
Hence
S m2 V 2 I m l V2 m2 P -;'- = _ _c / 2N 0 B = - - " = - - " No 2 4N oB 2NoB
I
which is equal to twice S,/N, or (So/NolAM = 2(SjNJAM
and amounts to a 3 dB improvement at the detector. The 3 dB improvement is due to arithmetic addition of power in the sidebands and quadratic addition of the independent noise in each sideband.
DSBSC system The only difference between an AM system and a DSBSC system is the carrier power which is present in the former. Hence, for equal average power in the
X2
N01.\e
ratlO5J
the
the two
must
same,
and
amounts to a 6
on . . . . . . . . . . .,. . . . . '. . . """ ...., it can
opf~ratlon
an
of the
en~/eU)ne
amlplltu(le-lmodu.latc~d
detectors for use in the wave under conditions of very low
C'uY"U''''hrn1nnllC'
83
or
v(t)
carner
The
and with the
and the
blocked
84
Noise
[ ...... "' ..... .,. ....~ •• ,...,.. """I .. '
and
from
to
due to
If since
is small. Hence cos does not contain the modulation ,...,,,,"'I"1""'A'11C" from 0 to
The
'\1<:lr''\l1r1,n
1,\1
n'V''O',ty,.'fh,
as last the modulation called the AM
Comment The results obtained above show in the threshold effect and the modulation is """ ..."'.... ""'' ' ' . . noise. In the case the ~n"~I£'~ del:ect:or. effect noise is when SI
an
of the C" .. ,9',,.. ...................... ,,'" detector. there no embedded the threshold be useful
IS
Systems
85
where K is a constant of proportionality. The average signal power is So where So =
Kl !J.f2
2
watts
To evaluate the effect of random noise, we observe that each noise frequency componentJ;, will beat with the carrier wave to produce amplitude modulation and angle modulation as illustrated in Fig. 6.3.
Fig. 6.3
If the noise component has a peak voltage Vn where Vc ~ Vn we obtain
vn(t) = (Vc + Vn cos wnt) -+ j Vn sin wnt
= Vc [{l + (Vnl Vc)cos wnt} + j (V,jVc) sin wnt] Vn (t)
or where
:::::: Vc sin (wet
(J =
tan
-1
+ 8)
(VnIVc)sinWnl (T' I ). :::::: Vn Vc smwnt 1 + (Vn/ Vc )cos wnt '
since Vc ~ Vn • The output noise voltage from the discriminator Vd(t) will be proportional to the frequency modulation produced by the noise signal Vn (t) which is related to the phase modulation it produces by 1 de
Vn
= K2n:- -dt = K-. fn cos Wilt . ~/ c
Vd(t)
The average output noise power bIVo in a bandwidth of is then given by K2 V 2j'2 ~
11.1
OlV 0
=
D
n
2
2Vc
Previously it was shown that N oof = V;/2 and Pc = V~ /2 is the average carrier power. Hence = K2(N0 (1)f2n o 2Pc
.sN
BHz
In
or Hence or
B
B
~f
.6.4
ratio
B
an ""' . . . . 1=.£........... ""
Hence In particular, if ~f = 75 kHz and B = 15 kHz we have m f = 5 and so the signal-to-noise improvement due to FM is 3 x 25 = 75 or 19 dB. The factor of 75 can be increased further by the use of pre-emphasis at the transmitter and deemphasis at the receiver. It can be shown'~ that this amounts to about 4 dB. giving an overall SIN improvement compared to AIv1 of 23 dB. 6.2
S/N ratnos
Typical graphs of input and output SIN ratios are shown in Fig. 6.5, assuming the same input noise bandwidth for the vanous systems. A t Jaw values of S,,' N" the AM systems appear to be the best, but for S,/ N, above the 10 dB threshold, FM is superior to the AM systems. To achieve large values of Sol No> however, PCM appears to offer the best advantage. As a comparison, the ideal system shown is still about 8 to 10 dB better than the FM or PCM system.
LpCM AM
-----;~-i----:;7~1 O.dlglt code)
30dB
..Y.
per"l (5·dlglt code)
o
(S;iN,) dB
Fig. 6.5
The most familiar pulse systems used are pulse amplitude modulatJOfl (PAM). pulse position modulation (PPM), and pulse code modulation (peM). " Sec F R Connor. ModliiarlOli. Edward Arnold (19R2)
88
Noise
PAM system Pulse amplitude modulation is normally employed in the early stages of PPM or PCM systems since it is easy to multiplex PAM pulses. It is generally not used as the final system, however, since the SIN ratio obtainable is not as good as those of the other pulse systems. It can be shown that PAM gives results very similar to those obtained previously for AM. Since noise directly affects the amplitude of the pulses, it appears as direct amplitude modulation in the system, as with the modulating signal. Hence, we have
PPM system In pulse position modulation, the modulating voltage causes a time displacement of the pulse. To evaluate the SIN ratio, assume that the maximum time displacement due to the modulation is to and so the peak signal volts is equal to K Co where K is a constant of proportionality. Hence, the average signal power at the output of the detector is
So = K2t5/2 The effect of noise in the signal is to alter the time displacement which leads to an error e, as shown in Fig. 6.6. The rms noise volts causing the error e produces an rms time displacement lit such that e Vc
=
I1t tr
where Vc is the peak pulse volts and which carries the modulation.
tr
is the rise-time of the leading pulse edge
v
..../ "
/With noise
t-
Fig. 6.6
The output rms noise volts is thus K I1t and the average output noise power at the detector is
Hence .....o r•• ' .. 't'"orl
IS
or
Quantlsed Analogue signal
E~============~
Sampling
Quantised signal
6.7
to
mean
90
Noise
The error c: can take on all possible values between -/1 V/2 and + /1 V/2 and may be considered as due to added noise in the signal. Hence, the mean-square value of the error gives the mean-square value of 'quantisation noise'. To calculate it, assume that over a long period of time all levels have an equal probability of occurrence and so the occurrence of any level is the same. Hence, we obtain
N
or
L1 V 2
o
= -12-
watts
(for a 1 Q load)
To calculate the signal power for q levels spaced L1 V volts apart we have
v=
(q -1).6. V
volts
Assuming further that bJpolar pulses are used (since less power is consumed), the pulse heIghts are ± 6 r /2. ± 311 Ii /2 ..... ± (q - 1)11 V/2. For equal probabIllty of occurrence of all levels 111 a long message. we obtain the average Signal power as
or
51 -"
61
2
0
= - 2q -- [! - +
_,
j -
+
_0
+. .. +
y
(I} -
1) 2 J
This may be written as
NOlA
from whlc]) we obtaIn
+ 2- + ... + (If -
0 '
1-
_,
.
1~+2~+.,.+
,q(q-l)(2(/-1) 1)- = ...._ 6
(l -q _2')" ..... . \ 2 /
Hence
L1 j So = -'1")
2
,61
'2
q(ij - l)(q - 2)
4x6
[1/((/ - 1)(2ij - 1) - (j(q - l)(Cf - 2)J
Lit
=
._-
12
(lj-l)I'{2(j- IJ-Uf-2)]
'..
,
Systems
91
AV 2
=U(q-l)(q+l) AV 2 =_(q2_1)
12 or
AV2 So ~ 12 q2
(for q
~
1)
Hence, the signal-to-noise ratio due to quantisation noise becomes (S o
AV2
/N
)PCM ~ _ _ (q2 0
12
-
/AV 2
-1) - ! 12
~(q2_1)
(for q
or
~
1)
The result depends on the square of the number oflevels used and so a large number of levels is required for a large SIN ratio. As an example, if q = 128, (So/No) ~ (128)2 ~ 42dB. This requires theuseofa 7-digit code since 27 = 128 levels.
According to the Hartley-Shannon law of information, the communication capacity C of a system with a bandwidth B and a signal-to-noise ratio SIN is given by C = B log] (1 + SIN) bits/s This rate of information transmission may be regarded as the ideal if it is assumed that the error rate is less than 1 in 10 5 bits/so Comparing this with a binary PCM system, we observe that, for a sampling frequency of 2 HI where HI is the highest modulating frequency and q quantised levels are used, the amount of information H transmitted is given by H
=
log2 q bits
For a sampling frequency of 2 W, at least one pulse is sent in each sampling period and so the total number of pulses sent per second is n = 2 W. Hence
C = H' = nH = 2Wlog 2 qbits/s or
C
=
B 10g2 q2 bits/s
since B = 2 HI is the system bandwidth. Previously it was shown that
AV2 So = 12 (q2 -1)
C
B
C=B
ratio
was obtained """..."'........."......
ratio of where q is the number of . . . ",....... f-'C"oA For
nB or
n
group or
1.. ,
Ideal
5 PCM
40
PCM
N
-
:::c 23
:3
8 peM
Error
o 6.8
10·
A
rot
=0 ~
two
p
';:).lJ:.".l.lU.l';:)
and
one
well
the
bit ratio
1,
is the same.
we
-E/No
Noise Table 6.1
1
'2
SER 10- 3
6.9
Systems
97
budget, the critical parameter to be considered is the effective isotropic radiated power (EIRP) at the satellite which is important in achieving maximum power output from its transmitter. It is given by the expression EIRP
PTG T
=----------------
LFLS or where
EIRP = PT+GT-LF-L s dB P T = transmitter power LF = feeder and diplexer loss Ls = free space loss G T = transmitting antenna gain
Given below are typical values of the relevant parameters at an operating frequency of 6 GHz for a geostationary satellite orbiting at a distance of about 36000 km above the earth. The losses include the large space loss (201 dB) and the miscellaneous losses due to transmitter ageing (1 dB), antenna pointing error (2 dB), and rain attenuation margin (2 dB). Transmitter power Transmitting antenna gain EIRP Free space loss (4nd 21A2) Satellite antenna gain Miscellaneous losses Received carrier power Satellite noise power Carrier-to-noise power density ratio Carrier-to-noise power ratio
23dBW 60dB 83dBW -201 dB 28dB -5dB -95dBW -126dBW 97dBHz 31 dB
The critical downlink parameter is the figure of merit G IT since it directly determines the carrier-to-noise power density ratio (C / No), which is the ultimate criterion at the receiver's demodulator for analogue signals, or the bit energy-to-noise power density (E I No) for digitaJ signals. Hence, it is easily shown that C No or
C -N - 0
and
EIRP X GR kTsLsLm M
= EIRP+GR-kTs-Ls-Lm-M dBHz
98
Noise
C = carrier power at the demodulator No = noise power spectral density N = noise power EIRP = effective isotropic radiated power G R = receiving antenna gain k = Boltzmann's constant T, = system noise temperature Ls = free space loss Lm = miscellaneous losses 114 = margin for multiple carriers J-V = relevant bandwidth
where
Typical figures for a ground station receiving a signal at 4 GHz from a geostationary satellite are given below. It is convenient here to consider the parameter (C I No) for analogue signals since it is independent of the system bandwidth used. For digital signals, the ratio (EI No) is easily determined from C
Cr
1
ER
-=-x-=No No r No
E C -=--dB No NoR
or where
T
is the bit duration, E is the energy per bit, and R is the bit rate.
Satellite EIRP 18 dBW Free space loss ·····197 dB Receiving antenna gain 60dB System noise temperature 70 K GIT ratio 41-5dB/K Miscellaneous losses - 5 dB -124dBW Received carrier power Noise power density -210dBW/Hz Carrier-to-noise power density ratio 86 dB Hz 20dB Carrier-to-noise power ratio Bit rate (digital systems)* 8·5 x 10 6 bits/s Energy-to-noise power density ratio 17 dB BER 1 x 10"4 Here, the system noise temperature includes the antenna noise temperature and that of the subsequent receiver chain, the losses include the large space loss (197 dB) and the miscellaneous losses due to atmospheric attenuation (2 dB), antenna aperture efficiency (1 dB), and mispointing and polarisation loss (2 dB). ,. 132 channels
communication
Qlagra.ms the arr'an,gerneIlt of a microwave satellite an active earth in geo-
.........'...., .. I.... ...., ......... ,"'...
par'amete:rs and
s
6.1 chain and . . a,....,.,.l',.",,,,, modulators drive The ",.n,n~rol'Ufj"'r arnpl1lher .. """" ......... &r, C''lIf-,o.lh1~a the trallsrrUSSlon back earth. received with a bandwidth up to MHz band at 4 GHz and so low-noise wideband are ""' ...... ,o.n1l· ....
station which are isolated IF of 70 MHz
eqlUl]:lm<~nt
corlslsts
t-""'lInC'lI"""lt-t-a ....
100
Noise 15 K followed demodulators are 1"'1'"'\1n''I1t:~1nt11'"'\'I''\n the and RX use conventional oa:set)an.a
I
1
4
5
7
102
Noise
8
Derive an expression for the overall noise factor of a combination of three two-port networks, connected in cascade, expressed in terms of the individual noise factors and gains of the individual networks A receiving system consists of a preamplifier connected through a length of cable to a main receiver. The noise factor of the preamplifier is 6 dB, while the corresponding values for the cable section and receiver are 8 and 13 dB. Given the attenuation of the cable is 8 dB, calculate the minimum gain required in the preamplifier if the overall noise factor of the system is not to exceed 9 dB. (U.L.) 9 (a) Describe briefly what is meant by the noise factor of a radio receiver. (b) Two amplifiers, connected in series, are matched in impedance and bandwidth. The first has a gain of 16 dB and a noise factor of 6 dB, the second has a gain of 10 dB and a noise factor of 3 dB. Calculate (i) the noise factor of the combination and (ii) the noise factor if the order of the amplifiers is reversed. (CG,L.I.) 10 A parabolic antenna has a noise temperature of 60 K and it is connected by a length of waveguide to a parametric preamplifier. The waveguide has a loss of 1 dB while the preamplifier has an effective noise temperature of 77 K and a gain of 20 dB. If a receiver is used with the preamplifier and its noise factor is 10 dB, what is the effective noise temperature of the system? 11 A communication receiver system has an input stage with a noise temperature of 100 K and a loss of 5 dB. This is followed by three IF amplifier modules, each with a gain of 10 dB, a bandwidth of 6 MHz, and a noise figure of 1·3 dB. With the system matched throughout, calculate (a) the system noise figure, (b) the equivalent noise temperature of the system when connected to an aerial with a noise temperature of 50 K, (c) the smallest usable input signal power if the communication recognition system operates correctly for an output signal-to-noise ratio > 1, (d) the effect on (a), (b), and (c) of adding a preamplifier with a noise temperature of 100 K and gain of 20 dB. The receiver system is at a (eE.l.) temperature of 290 K. 12 A signal s(t) is a triangular pulse of the form s(t) = Kt s(t)
13
=
0
O~t~T
at all other values of time
where K is a constant. Determine the output of a filter matched to this signal. If Gaussian white noise of zero mean value and noise spectral density No (positive frequencies only) is added to the signal, what is the maximum signal-to-noise ratio at the output of the matched filter? In a pulse radar system, the observed signal is received in the presence of Gaussian noise of zero mean and unit variance. Assuming the received signal is of 2 volt amplitude, determine for a Neyman-Pearson receiver
Problems
103
the probability of detection when the false alarm probability is set at 0·2. A 30 channel PCM system with uniform quantisation and a 7-bit binary code has an output bit rate of 1·5 Mbits!s. Determine (a) the maximum information band\vidth over which satisfactory operation is possible, (b) the output signal-to-quantising noise ratio for an input sinusoidal signal at a frequency of 3 kHz and maximum design amplitude. 15 Calculate the signal-to-noise ratio for a sinusoidal signal quantised into M levels given that the total mean-square quantising noise voltage (T2 = 0·083(I\V)2 where 11v is the step size. What assumption has been made about the quantisation? Hence, estimate the number of digits per character required in a PCM system carrying the above signal if the quality has to be satisfactory for the telephone system. Discuss whether speech processed in a similar manner has the same quality. Explain how the signal-to-noise ratio for speech can be improved by analogue or digital signal processing. (C.E.I.) 16 A coherent binary data system uses on-otT pulses varying in amplitude from 0 to V volts. The probability of a 0 or 1 in the presence of Gaussian noise is the same. For a peak signal power to average noise power ratio of 13 dB, calculate the probability of error. Also, for a probability of error of 10 - 5, determine the signal-to-noise amplitude threshold required. 17 An FSK communication channel transmits binary information at a bit rate of 100 kbits!s in the presence of Gaussian noise with a spectral density of 10- 19 W 1Hz. If the signal is transmitted with a peak voltage level of 1 volt, determine for a probability of error of 10 - 4 the path loss of the channel for bit by bit detection with (a) incoherent FSI(, (b) coherent FSK. 18 Explain why communications satellites fulfil an important role in worldwide communications. Mention, particularly, aspects of the role that cannot be readily fulfilled by alternative systems. A satellite in a geostationary orbit at 35800 km has a 4 GHz downlink transmitter which feeds 25 watts into an antenna with 20 dB gain. The ground station receiving system has a total noise figure of 1·5 dB. Calculate the antenna gain necessary at the ground station to maintain a 30 dB input signal-to-noise ratio over a 12 MHz signal band. (eE.L) 19 The parameters of a satellite-to-ship link are as follows 14
Satellite RF transmitter power per channel Satellite aerial gain relative to isotropic Free space path loss at 1·5 GHz Propagation margin at 5° elevation for 99 % of the time Ship aerial gain relative to isotropic RF input power to ship receiver
+PsdBW + 17 dB -189 dB +5dB +GrdB
-152dBW
20
4 5
8 9 11
McGraw-Hill Communication
STEINBERG, J
2
and Communication
Inter-
4 5
7 8
Modulation
9
1 12
JOHNSON,
cation
13 14
r"h't:ln1n.t:lIl
Generalised harmonic Korrelationstheorie 1934.
WIENER. N
KHINTCHINE, A
15 16
and RAJASEKARAN, John Thermal n"".1'" .... 1'" .............
flP1JllOQlUJns.
,,1" • B
... \J.J" ....." ...
18
.J
...nf"t:lf" • ...,'n
t:lIlt:lIt"t1"'1r"1i"u
in C01001uct:or:s, ...,"',,,,,.ro,. . ,
of electric
and
20 21
Electronic Advances
t'r4ryCE~ealn~7S
24 25
Electrical
26
Ultra low-noise
28
lortrnnlr
and
29 30
33 34
35 36
arnlpllhelrs in communication 1971. The
Sa(~eUl1[e
Communication t'rllcedeOljrzas Institute
ROWE, H E
of GaAs FET low-noise
POSNER, R
arrlpllne:rs~
31
1I"'\f"lI1.. <'lI'I"1""""t· ........
Advances in Low-Noise
........ ,u...... " ......... ..,.,
Institute
Microwave
KREUTEL, wand PACHOLDER, A 0 Measurement satellite communications R and aplpU(~at],on sources. M ullard
of a
1969.
Communication
CARLSON, A
and F
tprnf"loJo.rgtl11-p
McGraw-Hill
SCHILLING, D L
Introduction
z. Communication OLIVER, B M. and PIERCE, J R The Ph110:S0t)hV November 1948. SHANNON, the presence of PEEBLES, P
t'rc')Ct~ealnG'S
Institute
Radio bnazn!eel's 39
LAWTON,J.G
'"'-'v' ....... iJ ......... t..7'U.....
Convention on 40 MARTIN, J Communication Satellite 41 TURNER, L W Electronic
Prentice-Hall
Book. Newnes-Butterworth
Venn
rI.:;;>,nr""'rrl
A.1
s
(a)
s
(b)
A.2
s
not
110
Noise
theorem.
was
Error . . ,," ............ '1 ..........
To
Power
x we 1 T IS
x
Appendices Table A.1 x
erf x
x
erf x
0 005 010 0'15 020 025 030 035 040 045 050 055 060 065 070 075 080 085 090 095 1 00 1 05
0 00563 01124 01679 02227 02763 () 3286 03793 04283 04754 05205 05663 06038 o 642() 06778 07111 074;;.1 On06 07969 08208 08427 08624 08802 08961 09103 09229 09340 09437 () 9522 09596
1 50 1 55 1 60 1 65 1 70 1 75 1 80 1 85 1 90 1 95 200 205 210 2 15 220 22E, 2 30 2. 35 240 245 250 255 260 265 270 275 280 285 290 295 300
09661 09716 09763 09803 09837 09866 09890 09911 09927 09941 09953 09962 09970 09976 09981 () 9985 09988 o 9991 09993 09994 09995 09996 09997 09998 099986 099989 099992 099994 099995 099996 099997
110
1 15 1 20 1 25 1 30 1 35 1 40 1 45
xiI)
X{;)
,!
.V'\
-f~O
3T
-"2
T
2"
t~
fb)
(a)
Fig, A,3
, Hence, rearranging the order of integration then yields
r
+ T I L2. 1 +W [ X (t)dt = -7F(w)dw 7J2 ~rrT • - '"
f
-I
TIL-
-TIL
X (t)ej<"'dt
]
111
112 or
Noise
1 ['+Tl2
-
T ~ ---Tl2
f+
1
X2(t)dt = ----------
2nT -
00
00
X(t)eJw1dt
a : ; - - - 00
since X (t) is zero over the intervals Also, we have ,,+
f+
F(w)dw
< t < - Tj2 and + Tj2 < t < +
00
00
J_
X(t)eJw'dt = F* (w)
00
where F* (w) is the conjugate of F(w) such that F(w) F* (w) = 1F(w) 1__ T -
00.
f+1i 2 X2{l)dl • = -1
f+08IF(W)12
2n
-7/2
-
12. Hence
dw
T
00
As more 'samples' of x(t) are removed over time intervals of T and added to X(t), the signal X(t) will eventually resemble the periodic signal x(t) and so in the limit when T ---} 00 we obtain 1 J~ + J /2 1 lim ::-:: X2(t) dt = lim ::-:: T ---+ ~ 7 _ Ti2 7 -> oc 7
f
+ l/2
x 2(t) dt
_ T2
1 : + ex) F (w) 12 i "dw 2n J - 00 7 1
= lim T ->
oc;
The quantity in the centre is the average power of the periodic signal x(t) and so we obtain
P
.
dV
= Jlln -
J'
1
7-+0C;2n
+
08
1
F(w)
T
-00
12-dw = -1
:
+
00
I
S(w)dw
2nj_00
from the defmition of S(w). Hence, by inspection, we obtain S(w) = lim
1F(w) 12
l-+OC;
V oltage spectral density The voltage spectral density
Svtfl
7
is defined by
f::SJf)dj=V~ where v ~ is the mean-square noise voltage. In the case of thermal noise, the noise spectrum is constant over a finite bandwidth B. Hence
f
+B
-B
or
SvCf)
Sy(j)df=4k7BR
f
+B
-B
dI =
4kTBR
1
Hence or
we
or
e
14 ~"I'·I\'U/h~nil1
noise
f
--IJIoo-
f--..... (b)
A.4
+
115
n or
R
Hence
/
A.5
116
Noise
lDDlendlx
F: a
+
s
or
and
IH
N
Appendices
117
Furthermore, if E is the signal energy we have E =
f
+OO -
1
f+oo
2n
-
S2(t) dt = -
00
IS(wW dw
00
and substituting this expression into (S/N)max yields
a result which depends on the signal energy but is independent of its waveform. If the impulse response of the matched filter is h(t), the Fourier transform of an impulse <5{t) is unity and we have H( )
F[h(t)] F[b(t)]
=
w
where H(w)
= S*(w)e- Jwto.
h(t)
F[h(t)]
Taking the inverse transform then yields
1 h(t)=-
or
=
f+
2n
-
= -1
f+
2n
-
00
S*(w)e-JWIOxeJwtdw
00
00
S*(w)eJw(t-ro)dw
00
and the r.h.s. is simply the inverse transform expression for the signal s* (t - to). Hence, it follows that h(t) = s* (t - to)
=
s(to - t)
and the impulse response of the matched filter is the input signal delayed and time-reversed. For h (t) to be real, or for the filter to be physically realisable, we must have t ~ to, i.e. all the signal energy in s(t) must be received before the decision is made at t = to. Correlation detector If the input signal to the matched filter is s(t), the output signal so(t) is given by the convolution integral as so(t) =
f+OO ..
00
s(T)h(t - T) dT
where r is an arbitrary variable, h(t) = s(to - t) is the impulse response of the matched filter, and h (t - T) = S(to - t + T). Hence
Noise
o
G:
. A.6
costs
Hence
C IP
P
+
or
L
Noise
+00
p p
p may
A.7
a
Appendices
121
when it is present at the receiver, and so choosing H 2 correctly_ Similarly, we have P F =P(H 2 Ix 1 )=
r+ ~
cc
pdylxddy
},
which is the probability that x 2 (t) was not transmitted, yet deciding it is present at the receiver, and so choosing H 2 incorrectly_ To apply this criterion, it is usual to decide on a maximum acceptable false alarm probability PF and then to maximise the detection probability PD - The resulting values of P D and P F when plotted yield the receiver operating characteristics shown in Fig. A.8. Parameter d is the signal-to-noise voltage ratio at the output of the receiver.
I
10~
075
t 05
o
025
05
075
10
PF -
Fig. A.S
For any given false alarm probability P F , a corresponding threshold value YI is determined to yield the maximum value for the detection probability P D - If y(t) > YI' hypothesis H2 is chosen and, if y(t) < Y1' hypothesis HI is chosen. To show the relationship with Bayes' criterion, we obtain the derivatives of
d
or
x
(z
I
(k
or 1
we
or
In
2
where
=0
a
=0
1
1
we
{
=1
1
as is
as a minimum
o
........----- I (a)
-------t
(b)
A.9
or
m
or a
dW
or
N=
watts
dW=--e
t
126
Noise
where f is the frequency of oscillation and h is Planck's constant. For frequencies up to about 1013 Hz, L1 W :::::: kT which is the classical value. Appendix J: Shot noise The shot noise rms current Is in a diode is due to the random emission of electrons from the cathode. Each electron arriving at the anode carries a discrete electronic charge e which gives rise to a current pulse i(t) in the anode during the transit time T, as shown in Fig. A.l0(a). The actual shape of the current pulse is immaterial if the time-average interval chosen is such that T ~ T. Each pulse can be regarded as a Dirac delta function (5(t) and approximated by a short rectangular pulse, as illustrated in Fig. A.I0(b). Hence, we have
l+:
b{t) dt
=
e
l.e. the area of the rectangular pulse is such that e/T x
~ 0
~ r-
T
-1
T
= e.
t~
la)
Fig. A.10
If F (w)
IS
the Fourier transform of b(t) then
F (OJ) =
f+ -
and
'L
.
b (t) e - Jrot d t = e
00
sin OJ, /2 . OJT/2
IF (wW = e2 [~-i-~;;¥~T
where IF (OJW is the energy spectral density and is shown in Fig. A.10(c). From Fig. A.lO(c) we observe that if the transit time T is very small (about 10 - 9 s) then liT:::::: 10 9 Hz and the spectral density over a bandwidth L1f = B is fairly constant, especially at lower frequencies. Hence, the total energy W in a bandwidth B is given by
W= or
00 f+oo 12 d f = 2 J1'+_""IF(w) IF(wWdf 0
Appendices
127
If n electrons arrive at the anode in time T where T is sufficiently large, the average shot noise power in a 1 Q load becomes I; = n ~VIT = n2e 2 BIT
and substituting for the average anode current 1a = ne IT yields
I; = 2el"B ~
Is =y'2el a B
or Appendix K: Noise factors
Grounded-cathode circuit
Fig. A.11
Grounded-cathode amplifier The equivalent circuit is shown in Fig. A.lI where Rs is the source resistance, Rg is the grid-leak resistance, and Req is the equivalent shot noise resistance of the valve. If their rms noise voltages are 1.:" vg, and Veq respectively, the total noise voltage of the amplifier is Vo and for an ideal amplifier it is V1 where VI is due to the source resistance only. Hence
v;
mean-square noise voltage of amplifier F=·································· ........................······........ _ ...................................................... = mean-square noise voltage due to source vf
4kTB[Req + RsRg/{Rs + Rg)] [vsR~/(Rs + Rg)] 2 Since
v; = 4kTBR
s,
we obtain
F =
or
[Req (Rs + Rg) + RsRgJ (R, + Rg) RsR;
--'---=---"----"-'----::---"-=----'-~
+i
F= 1+RjRg
q [1+RjR g]2
s
128
Noise
Grounded-grid amplifier The eq uivalent circuit is shown in Fig. A.Il. Since the effective amplification factor of the grounded-grid amplifier is (/1 + 1), the rms noise voltages Vs and Vg appear as (/1 + 1)u, and (/1 + 1)v~ in the equivalent circuit. The grid noise voltage veq , however, is common to both grid and anode circuits and so its value is (p + l)vg - ug = pUg in the equivalent circuit. Hence, we obtain mean-square noise voltage of amplifier mean-square noise voltage due to source
F=--------------------------------
/124kTBReQ + (p + If 4kTBR sR g/(R, + Rg) (/1 + 1)24k7BR, [Rg/(R, + Rg) J2 [/1/{fl + 1) J2 RCq + R,Rg/(R, + Rg) ················i(tRJO~·:·+··R~")ji---
or
F = 1 + Rj Rg + (_/1_)2 Req [1 \/1 + 1 R,
+ R,,/R gJ2
Common-base transistor
Common base circuit
Common·emltter circuit
Fig. A.12
The equivalent T-circuit is shown in Fig. A.12 where R, is the source resistance and rc , rb, and rc are the emitter, base, and collector resistances respectively. The rms t101Se voltages v, and Vb are due to thermal noise in the source and base resistances respectively, whIle ve and 1\ are due to shot noise and partition noise Il1 the emitter and collector regions respectively. Hence, we have for a bandwidth B and absolute temperature T
v: = 4kTBR, v~ = 4kTBrb
vc2 =
[2,.2 ,e
= 2kTBr c
since f: = 2efoB and re = kT/elt if Ie is the d.c. emitter current Also, the
is
(Ial =
+
F = -------------source
or
+ source
or
F
+
Noise varIOUS
(1 F =
1+
]
re
+-+
Shot
(kHz)--
A.13
if
Vg
A.14
current
F=
----+ F
1
t A.15
For ance, gi
t
to zero, we
error IS ..........
2E
,L ........................ " " .....
]
two
the
was
If/=
or
-J it
] /2
I
VUJ.JlUVA"-'
noise power,
55
Band-limited
coloured communication conditional nrl"\t'VJIt'\111T'U correlation
91
113, 126
91
14 UU ....,LIV.U.
9, 17
108
Low-noise . . . . . . 'I"\h1ho,...
lUCl.l\.lU.lUlJU-llA'-'lUI\...'VU
74
'-'.:>lJIIIClllVJLI.
41
Random
7
Ul.:>IL.llU'ULlVll.
15
10
112
White Wiener-Khintchine 110
1-1->,..,.,..._,... .........
13
