Spread SpectrumT SpectrumTechniques echniques and and Technology
Spread spectrum modulation was originally developed for military to be used in the battle ground and in the hostile territories where the enemy always tries to intrude into the communication system of the friendly forces to steal information and to jam the systems. The main objectives of spread spectrum modulation were :To avoid being be ing detect d etected. ed. To prevent prevent eavesdroppi eavesdro pping ng.. To prevent the jamming of signals. signals.
“Spread spectrum is a means of transmission in which the signal occupies a bandwidth in excess of the minimum necessary to send the information: the band spread is accomplished by means of a code which is i s independent of the data, and synchronized reception with the code at the receive is used for de-spreading and subsequent subse quent data recov rec overy ery..” Or A transmission technique in which a pseudo-noise code, independent of the information data, is employed as a modulation waveform to “spread” the signal energy over a bandwidth much greater than the signal information bandwidth. At the receiver the signal is “despread” using a synchronized replica of the pseudonoise code.
Advantages of Spread Spectrum
Anti-jamming (A/J)
Anti-interference (A/I)
Low Probability of Intercept (LPI)
Code Division Multiple Access (CDMA)
Message Privacy
High Resolution Ranging and Timing
Spread Spectrum
A signal that occupies a bandwidth of B, is spread out to occupy a bandwidth of Bss All signals are spread to occupy the same bandwidth Bss Signals are spread with different codes so that they can be separated at the receivers. Signals can be spread in the frequency domain or in the time domain.
WHAT IS SPREAD SPECTRUM
A type of modulation in which the modulated signal bandwidth is much greater than the message signal bandwidth. The spreading of the message signal spectrum is done by a spreading code called Pseudo Noise Code (PN Code) which is independent of the message signal.
Bandwidth of the spread spectrum signal has to be greater than the information bandwidth. The spreading sequence has to be independent from the information. Thus, no possibility to calculate the information if the sequence is known and vice versa.
Need to Know:
Spectrum Spreading Techniques
Processing Gain
System Comparison
PN-Codes
Gold Codes
Other Codes
Other way of classifying the spread spectrum system is
Averaging system: In this system the interference reduction takes place because the interference can be averaged over a large time interval. Avoidance system: In this system the reduction of interference occurs because the signal is made to avoid the interference over a large fraction of time.
SS technique is a wideband modulation technique.
BW expansion factor is too large.
One major drawback:
It can not combat white noise like other wide band modulation techniques. because bandwidth expansion is achieved by something that is independent of the message, rather than being uniquely related to the message.
Types of Spread Spectrum
Direct Sequence Spread Spectrum (DS/SS)
Frequency Hopping Spread Spectrum (FH/SS)
Time Hopping Spread Spectrum (TH/SS)
Hybrids
For BPSK modulation the building blocks of a DSSS system are:
Spreading:
In the transmitter, the binary data d t (for BPSK, I and Q for QPSK) is ‘directly’ multiplied with the PN sequence pn t, which is independent of the binary data, to produce the transmitted baseband signal tx b: tx b = dt . pnt
The effect of multiplication of d t with a PN sequence is to spread the baseband bandwidth Rs of dt to a baseband bandwidth of Rc.
Despreading:
The spread spectrum signal cannot be detected by a conventional narrowband receiver. In the receiver, the received baseband signal rx b is multiplied with the PN sequence pnr.
If pnr = pnt and synchronized to the PN sequence in the received data, than the recovered binary data is produced on d r. If pnr ≠ pnt , than there is no despreading action. The signal dr has a spread spectrum. A receiver not knowing the PN sequence of the transmitter cannot reproduce the transmitted data.
To simplify the modulation and demodulation, the spread spectrum system is considered for baseband BPSK communication (without filtering) over an ideal channel.
Modulation
Spread spectrum systems are spreading the information signal dt which has a BWinfo, over a much larger bandwidth BW SS:
The SS-signal spectrum is white noise-like. The amplitude and thus the power in the SS-signal tx b is the same as in the original information signal dt. Due to the increased bandwidth of the SS signal the power spectral density must be lower.
The bandwidth expansion factor, being the ratio of the chip rate Rc and the data symbol rate R s, is usually selected to be an integer in practical SS systems:
pnr = pnt
To demodulate, the received signal is multiplied by pn r, this is the same PN sequence as pnt synchronized to the PN sequence in the received signal rx b. This operation is called (spectrum) despreading, since the effect is to undo the spreading operation at the transmitter. The multiplier output in the receiver is then (pnr = synchronized pnt) : dr = rx b . pnr = (dt . pnt). pnt
The alternation is destroyed when the PN sequence pnt is multiplied with itself (perfectly synchronized), because: pnt . pnt = +1 for all t
Thus: autocorrelation Ra (t=0) = average (pn t . pn )t = +1
The data signal is reproduced at the multiplier output: dr = dt
If the PN sequence at the receiver is not synchronized properly to the received signal, the data cannot be recovered. pnr ≠ pnt the multiplier output becomes: dr = rx b . pnr = (dt . pnt ). pnr
In the receiver, detection of the desired signal is achieved by correlation against a local reference PN sequence. For secure communications in a multi-user environment, the transmitted data dt may not be recovered by a user that doesn’t know the PN sequence pnt used at the transmitter. Therefore: crosscorrelation Rc (t) = average (pnt . pnr) << 1 for all t
is required. This orthogonal property of the allocated spreading codes, means that the output of the correlator used in the receiver is approximately zero for all except the desired transmission.
A pseudo-noise sequence pnt generated at the modulator, is used in conjunction with an M-ary PSK modulation to shift the phase of the PSK signal pseudo randomly, at the chipping rate Rc (=1/Tc) a rate that is an integer multiple of the symbol rate Rs (=1/Ts). The transmitted bandwidth is determined by the chip rate and by the baseband filtering. The implementation limits the maximum chip rate Rc (clock rate) and thus the maximum spreading.
Direct Sequence Spread Spectrum (DSSS) For logic 0 = same PN code For logic 1 = inverted PN code
XOR Logic in spread spectrum method
Direct Sequence Spread Spectrum
Spreading by multiplication- also shows relation between bit and chip durations
Bit rate of PN code is called the chip rate, which is 10 times or more than the data bit rate. The clock rate is provided for generating each bit of a PN code (with one clock cycle one bit of PN code comes out). The longer the chip, the greater the probability that the original data can be recovered, and the more bandwidth required. Even if one or more bits in the chip are damaged during transmission, it can be recovered the original data by using statistical techniques without the necessity for retransmission. To an unintended receiver, DSSS signals are received as lowpower wideband noise. In a typical DSSS system a double balanced mixer is driven by the PN code to switch a carrier’s phase between 0 and 180 degree. This is known as BPSK.
Direct Sequence Spread Spectrum Using BPSK Example
Approximate Spectrum of DSSS Signal
Advantages of DSSS Modulation
The DSSS allows greater capacity by allowing multiple access communication. Several users can occupy the same frequency spectrum simultaneously, and frequency bands can be reused without separation distance of users. Resistant to multipath fading in ground based mobile radio communication.
To simplify the influence of interference, the spread spectrum system is considered for baseband BPSK communication (without filtering).
The received signal rx b consists of the transmitted signal tx b plus an additive interference i. rx b = tx b + i = dt . pnt + i To recover the original data dt the received signal rx b is multiplied with a locally generated PN sequence pnr (that is pnr = pnt and synchronized). The multiplier output is therefore given by: dr = rx b . pnt = dt . pnt . pnt + i . pnt Due to the property of the PN sequnence: pnt . pnt = +1 for all t The multiplier output becomes: dr = dt + i . pnt
Spectra of desired received signal with interference: (a) wideband filter output and (b) correlator c orrelator output after despreading. despreading.
The greater the processing gain of the system, the greater will be its ability to suppress in-band interference.
After despreading, the data component dt is narrow band (Rs) whereas the interference component is wideband (Rc). By applying the dr signal to a baseband (low-pass) filter with a bandwidth just large enough to accommodate the recovery recovery of the data signal, most of the interference component i is filtered out. The effect of the interference is reduced by the processing gain (Gp).
The essence behind the interference rejection capability capability of a spread spectrum system: the useful signal (data) gets multiplied twice by the PN sequence, but the interference signal signal gets multiplied only once.
Multiplication of the received signal with the PN sequence of the receiver gives a selective despread of the data signal (smaller bandwidth, higher power density). The interference signal is uncorrelated with the PN sequence and is spread.
Frequency Hopping Spread Spectrum
Data signal is modulated with a narrowband carrier signal that hops from frequency to frequency as a function of time over a wide band of frequencies Relies on frequency diversity to combat interference This is accomplished by multiple frequencies, code selection and Frequency Shift Keying methods
Frequency Hopping Spread Spectrum
FHSS divides the available bandwidth into N channels and hops between these channels according to the PN sequence. A pseudo-noise sequence pnt generated at the modulator. An M-ary FSK modulation to shift the carrier frequency of the FSK signal pseudo randomly, at the hopping rate Rh. The transmitted signal occupies a number of frequencies in time, each for a period of time Th (=1/Rh), referred to as dwell time. At each frequency hop time the PN generator feeds the frequency synthesizer a frequency word FW which dictates one of the frequency positions f hi. Transmitter and receiver follow the same frequency hop pattern.
Hopping pattern is known to both transmitter & receiver. In order to properly receive the signal, the receiver must be set to the same hopping code and listen to the incoming signal at the right time and correct frequency. The net effect is to maintain a single logical channel if synchronizing sender and receiver properly. Unintended receiver see FHSS to be short shor t time impulse noise.
When the transmitter and receiver are synchronized, the users are unaware that the transmitter and receivers are changing the frequencies rapidly hence it provides anti jamming characteristics. As a frequency hopping transmitter typically hops over tens to thousands of frequencies per second ( the hop rate), the time it stays on a particular channel ( the dwell time) is very short and as a result the signal would appear as a burst of interference.
The type of spread spectrum in which carrier hops randomly from one frequency to another is called Frequency Hop (FH) spread spectrum.
FHSS are of two types:1 Slow frequency hopping 2 Fast frequency hopping
Slow and Fast FHSS
Frequency shifted every Tc seconds
Duration of signal element is Ts seconds
Slow FHSS has Tc Ts Fast FHSS has Tc < Ts Generally fast FHSS gives improved performance in noise (or jamming)
Slow Frequency Hopping (SFH) :
One or more data symbols are transmitted within one Frequency Hop. Advantage:
Coherent data detection is possible.
Disadvantage:
If one frequency hop channel is jammed, one or more data bits are lost. So we are forced to use error correcting codes.
Slow Frequency Hop Spread Spectrum Using MFSK (M=4, k=2)
Fast Frequency Hopping (FFH) :
One data bit is divided over more Frequency Hops. Fast frequency hopping where one complete, or a fraction of the data symbol, is transmitted within the duration between carrier hops. Consequently, for a binary system, the frequency hopping rate may exceed the data bit rate. Advantage:
Error correcting codes are not needed. Diversity can be applied. Every frequency hop a decision is made whether a 0 or a 1 is transmitted, at the end of each data bit a majority decision is made.
Disadvantage:
Coherent data detection is not possible because of phase discontinuities. The applied modulation technique should be FSK or MFSK.
Fast Frequency Hop Spread Spectrum Using MFSK (M=4, k=2)
As with the frequency hopper, the receiver must use a circuit to adjust its clock rate so that the receiver’s PN code is at the same point in the code as the transmitter and hence the hopping pattern is also same. A tracking circuit is necessary to maintain the synchronism once it has been attained.
Example
Binary data is transmitted through an additive white Gaussian noise (AWGN) channel with SNR=3.5 dB and bandwidth B. Channel coding is used to ensure reliable communications. Then: i. What is the maximum bit rate that can be transmitted? ii. If the bit rate is increased to 3B, how much must the channel SNR be increased to ensure reliable transmission?
Example
A binary data stream of 4 digits [1011] is spread using an 8-chip code sequence C(t)= [01 10 10 01]. The spread data phase modulates a carrier using binary phase shift keying. The transmitted spread-spectrum signal is exposed to interference from a tone at the carrier frequency but with 30 degrees phase shift. The receiver generates an in-phase copy of the code sequence and a coherent carrier from a local oscillator.
i. Determine the baseband transmitted signal.
ii. Express the signal received. Ignore the background noise.
iii. Assuming negligible noise, determine the detected signal at the output of the receiver.
Other spread spectrum modulation techniques are:
Time Hopping Spread Spectrum (THSS))
Hybrid methods
Chirped Spread Spectrum (CSS)
Direct sequence system is an averaging system, while frequency hopping, time hopping and chirping systems are avoidance systems.
Spread-Spectrum Theoretical Justification
SS is apparent in the Shannon and Hartley channel-capacity theorem:
S C W log 2 1 N
C is the channel capacity in bits per second (bps), which is the maximum data rate for a theoretical bit-error rate (BER).
W is the required channel bandwidth in Hz,
S/N is the signal-to-noise power ratio.
S log 2 e log e 1 W N C
By Changing bases,
S 1.44 log e W N C
By logarithmic expansion,
C S 1.44 W N W
NC S
(approximation for the wider bandwidths)
From the derived relationship it can be clearly seen that a desired Signal to Noise Ratio for a fixed data rate C can be achieved by increasing the transmission bandwidth.
If data rate is 32 kbps and SNR is 0.001(-30 dB) then with the help of Shannon's theorem, we can find out the necessary bandwidth for SSM signal.
W
NC
32 103 1000
22 MHz
1.44S 1.44 for data rate of 32 kbps, operation with lower value of SNR of 30dB is achievable by spreading the signal over a bandwidth of 22 MHz. By using a much wider bandwidth than that of the original data is possible to maintain the data capacity without increasing the transmitter output power.
The interaction and the interdependence between two time (or frequency) varying signals are defined by the correlation function derived from the comparison of the two signals. The comparison of a signal with itself is described as the autocorrelation function. On the other hand, the cross-correlation is a measure of similarity between two autonomous signals. The correlation processing forms the basis upon which optimum detection algorithms in digital communication systems are derived.
Autocorrelation
Partial Autocorrelation
Cross correlation
It is the procedure by which the matching of the signal is done with itself. The auto correlation should be maximum for the DS or PN codes with itself so that correct PN signal can be identified from the number of co-existing signals.
Note that on a normalized basis it has a maximum value of 1 that repeats itself every period, but in between these peaks, the level is at a constant value of (1/ N c). If N c is a very large number the autocorrelation function is a very small in this region.
For PN sequences the autocorrelation has a large peaked maximum (only) for perfect synchronization of two identical sequences. The synchronization of the receiver is based on this property.
The (normalized) autocorrelation of the spreading waveform (PN signal) c(t) is represented mathematically as
R c ( )
1 T code
0 c(t )c(t )dt
T code = N ctchip is the code period and
τ
= represents a time shift variable.
The Partial Autocorrelation is similar to the autocorrelation but integrated only over a portion, may be over a message bit duration. This is done to avoid long processing time for matching purpose. If the partial sequence is auto correlated then obviously the full sequence will be auto correlated because of uniqueness of the code and that is the concept behind the partial auto correlation.
Different signals have different spreading codes. The cross correlation between two codes i and j is
R c ( )
1
0 c i (t ) c j (t )dt
T code Which is equal to the autocorrelation if i = j. When the cross-correlation Rc(τ) is zero for all τ, the codes are called orthogonal. It is desirable to have poor cross correlation between two different codes so that unwanted code will be rejected easily by the receiver of the system.
Randomness of PN sequence is tested by following code properties observed over one full period. Balance property: In each period of the sequence, number of binary ones differ from binary zeroes by at most one digit.
Consider a typical PN code
0001 0011 0101 111---- seven zeros and eight ones meets balance condition
Run length property:
Run lengths of zeros and ones should be (Run property):
Half of all run lengths should be unity One - quarter of all run lengths should be of length two One - eighth of all run lengths should be of length three A fraction 1/2n of all run lengths should be of length n for all finite n Consider the same code again
number of runs = 8 000 1 00 11 0 1 0 1111 3 1 2 2 1 1 1 4
Autocorrelation property:
Auto correlation function of a maximal length sequence is periodic and binary valued. The autocorrelation function, equal to 1 if sequence are same, and -1/N otherwise
Rc
1 N c
No. of (a) - No. of (d) in comparision of one full period
We can state autocorrelation function as
0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 _____________________________ d a a d d a d a d d d d a a a
Where, (a) = agreements (matching) (d) = disagreements (mismatching)
Rc
1 15
Aperiodic Periodic
Aperiodic PN sequence
does note repeat itself in a periodic fashion. usually assumed that the sequence has a value of zero outside its stated interval. inter val. an autocorrelation to be N c for no shift and 0 or 1 with a shift of one or more bits. bi ts. Such sequences also called Barker sequences. Few value of N c are possible, Specifically, they are N c = 1, 2, 3, 4, 5, 7, 11 and 13. No larger sequences have been found and it has been hypothesized. too short for the spread spectrum system. can be used for the synchronization purposes under some conditions
A seven digit aperiodic Barkers sequence has the form for m +1,+1,+1, -1,1,+1,-1. Determine the autocorrelation for this code. Does Does a cyclic shift of this sequence (e.g.+1,+1, -1,-1,+1,-1, +1) have the same correlation properties?
For Barker sequence
Theoretically Autocorrelation function
N c k
R c (k ) a n a n k n 1
= N c for k =0 =0
= 0 or 1 for k 0 where, k is is the number of shifts in bit positions.
For this particular sequence for k=0 +1 +1 +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 +1 -1 _______________________ ______________________ _ +1 +1 +1 +1 +1 +1 +1 = N c (summing up all the values) For one shift i.e. k =1 +1 +1 +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 +1 -1 __________________________ ______________________ ____ 0 +1 +1 -1 +1 -1 -1 0 = 0 For two shifts i.e. k = 2 +1 +1 +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 +1 -1 __________________________ ______________________ ____ 0 0 +1 -1 -1 -1 +1 0 0 = -1
Checking for the cyclic shift
For one cyclic shift i.e. k =1 +1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 -1 -1 +1 __________________________ -1 +1 +1 -1 +1 -1 -1 = -1 (summing up all the values)
Periodic PN sequences
Maximum Length sequences
Walsh Hadamard sequences
Gold codes/sequences
Kasami codes etc.
A Simple Shift Register Generator (SSRG) has all the feedback signals returned to a single input of a shift register (delay line). The SSRG is linear if the feedback function can be expressed as a modulo-2 sum (xor).
The feedback function f(x1,x2, … ,xn) is a modulo-2 sum of the contents xi of the shift register cells with ci being the feedback connection coefficients (ci=0=open, ci=1=connect). An SSRG with L flip- flops produces sequences that depend upon register length L, feedback tap connections and initial conditions. When the period (length) of the sequence is exactly Nc = 2L -1, the PN sequence is called a maximum-length sequence or simply an m-sequence. An L-stage shift register and a few EX-OR gates can be used to generate an m-sequence of length 2L -1. The number of 1-s in the complete sequence and the number of 0-s will differ by one. Note (L = P = no of shift registers)
It is interesting to note that, multiple m-sequences exist for a particular value of L > 2. The number of available msequences is denoted by The numerator is known as the Euler number, i.e. the number of positive integers, including 1, that are relatively prime to L and less than (2L -1). For example, if L = 5, it is easy to find that the number of possible sequences = 30/5 = 6.
Lines that run from the output of one register within the LFSR into XOR gates that determine input to another register within the LFSR. These are chosen based on the primitive polynomial. A polynomial of degree n that has the form: 1 + … + x n, where (…) are zero or more terms with a coefficient of 1. xn and 1 are always present.
Reciprocal Generating Functions
The reciprocal of a primitive polynomial is primitive The reciprocal of an irreducible polynomial is irreducible Hence the reciprocal of a MLSGF is another MLSGF The reciprocal of a polynomial of degree N is
Table of MLSGF’S
For an n-bit LFSR, you need to discover a primitive polynomial associated with it to implement it. Tap tables are available for such list taps as :
N = 3, Taps at 0, 1, 3 ( this corresponds to 1 + x + x3)
The short length sequences are N c = 7, 15, 31, 63, 127, 255….. The reason for using shift register codes is that the period of the PN sequence can easily be made very large by increasing the number of register stages, so that apparently it looks random to the users. LFSR can be implemented in two ways:
1. Fibonacci implémentation 2. Galois implémentation
Fibonacci implémentation
Linear feedback shift registers can be implemented in two ways. The Fibonacci implementation consists of a simple shift register in which a binary-weighted modulo-2 sum of the taps is fed back to the input. For any given tap, weight g i is either 0, meaning "no connection," or 1, meaning it is fed back. Two exceptions are g0 and gm, which are always 1 and thus always connected.
Figure 2. Galois implementation of LFSR.
Galois implémentation
The Galois implementation consists of a shift register, the contents of which are modified at every step by a binary-weighted value of the output stage. Careful inspection reveals that the order of the Galois weights is opposite that of the Fibonacci weights. Given identical feedback weights, the two LFSR implementations will produce the same sequence.
The initial state of the Fibonacci form is called the initial fill which comprises the first p bits output from the generator, while the initial state of the Galois generator must be adjusted appropriately to attain the equivalent initial fill. the initial states of the two implementations must necessarily be different for the two sequences to have identical phase. It should be noted that, in some industries, the Fibonacci form LFSR is referred to as a simple shift register generator (SSRG), and the Galois form is referred to as a multiple-return shift register generator (MRSRG) or modular shift register generator (MSRG). Find:::::Which form is faster …Galois or Fibonacci?
A given set of feedback connections can be expressed in a convenient and easy-to-use shor form, with the connection numbers being listed within a pair of brackets. In doing so, connection g0 is implied, and not listed, since it is always connected. Although gm is also always connected, it is listed in order to convey the shift register size (number of registers). A set of feedback taps for a Galois generator is denoted as [f 1, f 2, f 3, ..., f J] g
where subscript J is the total number of feedback taps (not including g 0), f 1 = m is the highest-order feedback tap (and the size of the LFSR), and f j are the remaining feedback taps. The g subscript signifies the Galois LFSR form.
The set of feedback taps for the equivalent Fibonacci generator is denoted as [f 1, m-f 2, m-f 3, ..., m-f J] f where the f subscript signifies the Fibonacci LFSR form. Note that subtracting the feedback tap numbers from m is equivalent to reversing the order of the feedback taps As an example,
consider an LFSR of size m = 8 with feedback connections at g 8, g6, g5, g4, and implied g0. The feedback taps are specified as [8, 6, 5, 4]g for the Galois form, and [8, 8-6, 8-5, 8-4]f = [8, 2, 3, 4]f = [8, 4, 3, 2]f for the Fibonacci form. Note that the taps are customarily arranged in a descending order.
For example, we may be asked to find all sets of maximal-length feedback taps for an LFSR with m=3 registers. We do this as follows:
The length of the m-sequences will be N=23-1=7. We know that the solution lies in all the primitive factors of polynomial X 7+1. We use modulo-2 linear algebra to find the prime factors to be
The primitive polynomials are those factors whose order is the same as the register size, m = 3. Of the three prime factors, the last two meet this criterion. Thus we see that there are exactly two sets of m-sequence feedback taps, [3, 1]g and [3, 2]g.
The feedback connections, or the corresponding realized primitive polynomial, are represented by the notation [ p, h( p-1) ( p-1), ..., h1]
Where, zero-entries are not written explicitly.
It can be summarized that, the code generator outputs a binaryvalued sequence. This sequence repeats itself every N c elements. One period of the sequence, thus N c chips, is called the pseudo noise or spreading code. In DS-CDMA practice, the sequence is transmitted mostly as a bipolar waveform, called the spreading waveform.
Suppose the clock frequency is 10 MHz, hence tchip=10-7 seconds. Suppose a shift register with 32 stages is used for generating the PN sequence then compute the time to complete one cycle of the sequence. Total number of chips in a period N c = 232-1 = 4.29 x 109 =>
N c x tchip = 4.29 x 109 x 10-7
= 4.29 x 102 seconds = 429 seconds
Draw a [3,1] PN code generator which realizes the polynomial and generates [3,1] codes. Also draw the waveforms for the m-sequence for 3 periods.
Bracket [3,1] for ML code generation can be interpreted as follows.
The [3, 1] code has register length n = 3, the code length N c = 23 -1 = 7
The primitive polynomial h(x) for this case is h(x) = x3 + x + 1 (polynomial suggests the stages from which the output is fed back. As can be seen h1 = 1, h2 = 0 and h3 = 1.
[3,1] ML code generator Fibonacci implementation of LFSR. Galois implementation of LFSR.
Waveform of the [3,1] code sequence
ML LFSR sequences have desirable autocorrelation properties for spread spectrum applications.
a spread-spectrum receiver can "easily" recognize code synchronization between the received code and the locally generated (identical) code by examining their correlation value, and a receiver locked to the dominant wave, only sees little interference from delayed waves, which may be present in a multipath channel
It satisfies:
Balance properties Run length property autocorrelation
Walsh code sequences are obtained from the Hadamard matrix which is a square matrix where each row in the matrix is orthogonal to all other rows, and each column in the matrix is orthogonal to all other columns. The Hadamard matrix Hn is generated by starting with zero matrix and applying the Hadamard transform successively.
If perfectly synchronized with respect to each other, W-H codes are perfectly orthogonal. That is, W-H are optimal codes to avoid interference among users in the link from base station to terminals. The Hadamard-Walsh codes are generated in a set of N = 2n codes with length N = 2n. The generating algorithm is simple:
The rows (or columns) of the matrix HN are the Hadamard-Walsh codes.
In each case
the first row (row 0) of the matrix consist entirely of 1s and each of the other rows contains N/2 0s and N/2 1s. Row N/2 starts with N/2 1s and ends with N/2 0s.
The distance (number of different elements) between any pair of rows is exactly N/2. For H8 the distance between any two rows is 4, so the Hamming distance of the Hadamard code is 4. The Hadamard-Walsh code can be used as a block code in a channel encoder: each sequence of n bits identifies one row of the matrix (there are N =2n possible rows).
All rows are mutually orthogonal for all rows i and j.
The code of user 1 is the first column, i.e., (1, 1), the code of user 2 is the second column, i.e., (1, -1). Clearly (1, 1) is orthogonal to (1, -1). This matrix can be extended using a recursive technique.
Gold codes/ Gold Sequences
The autocorrelation properties of the m-sequences cannot be bettered. But a multi-user environment (CDMA) needs a set of codes with the same length and with good cross-correlation properties. These are constructed by EXOR-ing two m-sequences of the same length with each other. Thus, for a Gold sequence of length N c = 2 p-1, one uses two LFSRs, each of length 2 p-1. If the LFSRs are chosen appropriately, Gold sequences have better cross-correlation properties than maximum length LSFR sequences. The codes generated are of the same length as the two base codes which are added together, but are nonmaximal.
LFSR 1 Gold Code Out
LFSR 2
A balanced Gold code sequence is one in which the number of ones exceeds the number of zeros by 1 Number of balanced and unbalanced Gold codes for odd N
Any relative shift of the Gold code generator sequences such that the initial 1 of one sequence corresponds to a 0 of the second sequence will result in a balanced Gold code
Kasami code
Kasami sequences are one of the most important types of binary sequence sets because of their very low cross-correlation and their large number of available sets There are two different sets of Kasami sequences, Kasami sequences of the ‘small set’ and sequences of the ‘large set’ A procedure similar to that used for generating Gold sequences will generate the ‘small set’ of Kasami sequences with M = 2n/2 binary sequences of period N = 2n/2 + 1 In this procedure, we begin with an m-sequence ‘a’ and we form the sequence a’ by decimating ‘a’ by 2n/2 + 1 It can be verified that the resulting sequence a’ is an m-sequence with period 2n/2 - 1
Example: Let n=10, therefore, N=2n - 1 = 1023 (length of ‘a’) The decimation value is 2n/2 + 1 = 33 which is used to create a’ 1023/33 = 31 which will be the length of a’ If we observe 1023 bits of sequence ‘a’, we will see 33 repetitions of the 31-bit sequence which we will call sequence ‘b’ Now taking 1023 bits of sequence ‘a’ and ‘b’ we form a new set of sequences by adding (modulo-2 addition) the bits from ‘a’ and the bits from ‘b’ and all 2n/2 – 2 cyclic shifts of the bits from ‘b’ By including ‘a’ in the set, we obtain a set of 2n/2 = 32 binary sequences of length 1023 All the elements of a ‘small set’ of Kasami sequences can be generated in this manner
X4 + X + 1
a
a =
1 1 1 1 0 1 0 1 1 0 0 1 0 0 0
q = 2n/2 + 1 = 5 m = 2n/2 - 1 = 3 Where,
a’ = b =
1 1 0
q = decimation value
1 1 0 1 1 0 1 1 0 1 1 0 1 1 0
m = period of
Repeats
1
xor b = a
•
•
2
3
4
a’
5
0 0 1 0 1 1 1 0 1 1 1 1 1 1 0
Kasami codes are generated by cyclically shifting ‘ a’ 2n/2 -2 = 2 times
The autocorrelation and cross-correlation functions provide excellent properties, as good or better, than Gold Codes The ‘large set’ of Kasami sequences is generated in a similar manner with the addition of another register The two registers are a preferred pair as in Gold Code and therefore when combined with the decimated sequence, produce all the associated Gold Codes and the Kasami sequences for an even larger set of sequences
These sequences have lengths that are prime numbers of the form N c = 4q-1= prime, where, q is an integer. Since prime numbers are fairly distributed and since half of the prime numbers can form quadratic residue sequences, there are many more sequences of this type than there are m – sequences. It implementation is little complex as not generated directly by using shift registers. There are some values of m for which both q-r and m sequences may exist, e.g. N c= 3, 7, 31, 127, 8191, 131071 ….However except N c =3 and 7, q-r and m sequences are distinct. q-r sequences have the same correlation properties as msequences have.
Hall sequences also have a length that is a prime number described by N c = 4q-1= 4r 2+27 = prime, where, both q and r are integers. Since both the conditions are to be satisfied there are very few Hall sequences. It is having same properties as q-r sequences, but there is no particular advantage of using such sequences.
The twin-prime sequences are ones in which the length of the sequence is defined by N c = p p( p p+2), where both p p and p p+2 are prime numbers. The only advantage of twin prime sequences is that it is possible to achieve some lengths that are not readily achieved by any of other form otherwise the implementation is not simple.
In synchronous DS-CDMA, the link performance is affected by Multi-User interference the asynchronous multipath interference, arising from the delayed signals from the other users as well as user himself
The user capacity of a synchronous CDMA system is limited by the number of different codes. With a spread factor N c and K users, perfectly orthogonal codes are chosen if K <= N c . A bound for the maximum normalized cross correlation Rmax (at zero time offset) between user codes can be given as shown formula. For Walsh-Hadamard codes N c = K , so Rmax =0, and for Gold codes N c = K - 1, so Rmax=1/N c ( K / N c) 1 2 R max K 1
Simplified diagram for biphase modulation (transmitter)
For biphase modulation a MOD 2 adder is used For quadriphase modulation two MOD 2 adders are used with two alternate chips available from PN code generator. Two balanced modulators are fed with 90 degree phase shifted carriers (just like QPSK generation). Adding both the signals the SSM output is available.
Receiver diagram
Receiver for a spreaded signal must perform three distinct functions;
Detection of presence of signal, despreading or carrier removal and demodulation using PN sequence.
Detection of signal and despreading operations can be either active or passive.
Active method
Passive method
involves searching for the signal’s presence in both time and frequency domain
only require that the signal be searched for in frequency i.e. only carrier
tracking the sequence after it has been acquired
The despreading is accomplished in a matched filter rather than a correlator.
demodulating the signal i.e.PN sequence again.
demodulating the signal i.e.PN sequence again.
despreading the signal with the correlator
passive methods may be preferred when the sequence is short or when used as an aid to acquisition.
Active methods are preferred when the sequence is too long and the processing gain is very large.
Spectral Density Bandwidth Processing Gain
PN signal means the waveform waveform generated due to PN sequence The frequency domain representation of a time domain pulse is a sinc ( sin x/x) form of envelope.
PN code and data both are pulsed / square wave nature
it is straight forward to show that the spectral density of it i t is, 2 2 sin ( f f 0)t chip sin sin ( f f 0)t chip sin t chip S ( f ) 2 ( f f )t chip 0 ( f f 0)t chip
f o represents the center frequency of the sync shaped spectrum
The bandwidth B bandwidth B s of a PN signal can be defined as the frequency increment between the two zeroes of the spectral density that are closest closes t to the center frequency. i.e. 2 / t chip. The message is binary/square wave it will have similar spectral density but density but centered on zero. Thus the message signal spectral density is
sin f t m S m ( f ) t m f t m
2
As we know that processing gain is the term which gives the amount of spreading of the message signal.
Considering,
Bs = Bandwidth of the PN signal = 2/ t chip
Bm = Bandwidth of the message signal = 1/t m
Processing gain is defined as
Processing Gain = Bs/ Bm = 2 tm/ tchip
If the chip rate of a DSSS transmitter is 20 MHz, the message bit rate is 10 kbps. Find out the Processing Gain achieved finally, if the biphase modulation is used.
Chip rate is 20 MHz i.e. tchip = 1/ 20 *106
Bandwidth of the PN signal B s =2/ tchip = 40 *106
the message bit rate Bm = 1/tm = 10 kbps=104
Processing Gain = Bs/ Bm = 2 tm/ tchip = 4000
In dB Processing gain can be represented as 10 log10 (4000) = 36 dB
The receiver for DSSS system is a specially designed RAKE receiver. What is RAKE Receiver? Or How RAKE receiver works?
A RAKE receiver consists of a bank of correlators, each of which correlate to a particular multipath component of the desired signal. The correlator outputs may be weighted according to their relative strengths and summed to obtain the final signal estimate.
In CDMA spread spectrum systems, the chip rate is typically much greater than the flat fading bandwidth of the channel – Conventional modulation and receiver techniques require an equalizer to undo the ISI between adjacent symbols – CDMA spreading codes are designed to provide very low correlation between successive chips
Propagation delay spread in the radio channel provides multipath signal at the receiver – If multipath components are delayed in time by more than one chip duration (1/Rc), they appear like uncorrelated noise at a CDMA receiver, and equalization is not required
-> RAKE type correlator receiver can be used!!!
RAKE receiver, used specially in CDMA cellular systems, can combine multipath components – To improve the signal to noise ratio (SNR) at the receiver – Provides a separate correlation receiver for each of the multipath signals – Multipath components are practically uncorrelated when their relative propagation delay exceeds one chip period
The basic idea of A RAKE receiver was first proposed by Price and Green and patented in 1956
CDMA receivers may combine the time delayed versions of the original signal transmission in order to improve the signal to noise ratio at the receiver. RAKE receiver does just this — it attempts to collect the time-shifted versions of the original signal by providing a separate correlation receiver for each of the multipath signals.
Multipath Channel
Due to reflections from obstacles a radio channel can consist of many copies of originally transmitted signals having – different amplitudes, phases, and delays
Multipath can occur in radio channel in various ways – Reflection, diffraction, scattering
The RAKE receiver uses a multipath diversity principle – It rakes the energy from the multipath propagated signal components M-ray multipath model can be used – Each of the M paths has an independent delay, t, and an independent complex time variant gain, G – t(t) is transmitted signal – r(t) is received signal
M-finger RAKE Receiver
RAKE receiver utilizes multiple correlators to separately detect M strongest multipath components Each correlator detects a time-shifted version of the original transmission, and each finger correlates to a portion of the signal, which is delayed by at least one chip in time from the other fingers The outputs of each correlator are weighted to provide better estimate of the transmitted signal than is provided by a single component Demodulation and bit decisions are then based on the weighted outputs of the M correlators
RAKE Receiver Blocks
Matched filter – Impulse Response Measurement – Largest peaks to RAKE fingers – Timing to delay equalizer – Tracks and monitors peaks with a measurement rate depending on speeds of mobile station and on propagation environment
Code Generators – PN codes for the user or channel
Correlator – Despreading and integration of user data symbols
Channel Estimator – Channel state estimate – Channel effect corrections
Phase Rotator
Delay Equalizer
– Phase correction – Compensates delay for the difference in the arrival times of the symbols in each finger
Combiner
– Adding of the channel compensated symbol – Multipath diversity against fading
RAKE Receiver Requirements
RAKE receiver has to know Multipath delays -> time delay synchronization Phases of the multipath components -> carrier phase synchronization Amplitudes of the multipath components -> amplitude tracking Number of multipath components -> RAKE allocation Time delay synchronization is based on correlation measurements Delay acquisition Delay tracking by feedback loops
Number of available fingers depends on the channel profile and the chip rate The higher the chip rate, the more resolvable paths are there A very large number of fingers lead to combining losses and practical implementation problems The main challenges for RAKE receivers operating in fading channels are in receiver synchronization
The order of diversity achieved by the rake receiver depends on the number of resolvable paths L with L = round( max / tchip) + 1
Where, max is the maximum delay spread of the multipath channel Partial Correlation of PN Sequences at the rake receiver can also be applied.
Transmitted coded signal for data signal {1,0,1} is,
Received multiple copies of transmitted signal on different paths.
The use of a correlation window in the receiver is proposed which is equal to the full period of the spreading code. However, the spreading sequence used by the transmitter is longer than the period used by the correlator for detection. The extension consists of a cyclic prefix. In an OFDM symbol the cyclic prefix is a repeat of the end of the symbol at the beginning. The purpose is to allow multipath to settle before the main data arrives at the receiver. The length of the cyclic prefix is often equal to the guard interval. In the prefix example of an m-sequence {1, 1, 1, – 1, 1, – 1, – 1}, the transmitter creates the spreading code { – 1, – 1, 1, 1, 1, – 1, 1, – 1, – 1}. The receiver correlates during the same window, for every branch of the rake, but uses different, cyclically shifted sequences.
Due to cyclic prefix, all ISI terms disappear influencing the BER performance but at the same time:
Cyclic prefixes waste some transmit bandwidth and energy Correlated fading of the signal amplitudes in the various diversity branches of the rake degrades the performance. The best performance is achieved with uncorrelated fading.
Mainly three partitions reconfigurable hardware
dedicated hardware
Dataflow oriented tasks that operate on a word-level coarse data stream are executed using the reconfigurable hardware Bit-level data processing tasks that execute continuously are mapped onto dedicated hardware resources
DSP
is used to execute the control-flow and synchronization tasks
the complex multiplication of the aligned incoming data with scrambling codes.
multiplication of the corresponding spreading code with the real and imaginary part of the descrambled data
channel coefficients are calculated on the basis of a specific sequence of pilot signals.
Calculates the channel coefficients that are used for the channel correction then transferred to the reconfigurable hardware
Interference Rejection
Antijam Characteristics
Energy and Bandwidth Efficiency
Near Far Problem and Power Control
Interference Rejection
Narrowband and wideband interference rejection
The SNR is an important parameter for a spread spectrum system because it decides the detectability of the signal as well as the output signal quality.
Maximum SNR will occur when there is no delay spread.
output SNR will be, -> P r is the received power, SNR ou t =
P N / 2 t r
o
-> N o is the noise power spectral density -> tm is the message bit duration
m
SNR depends upon the ratio of received power to noise spectral density and bit duration.
Now the mean square value of the output interference signal is given by J is the total interference power and J 2 j PG is the processing gain PG This is the case when interfering signal bandwidth is less than transmission bandwidth. The interference power can be reduced by a factor equal to processing gain. consider the situation in which interference bandwidth B j is greater than spread spectrum bandwidth Bs. In this case J B j s 2 B j PG 2
-> J is the interference power in its total bandwidth -> the factor
B s J 2 B j
represents simply that portion of the
interference power that exists in the receiver bandwidth.
This power is again reduced by a factor equal to the processing gain. Considering the interference signal as a noise the SNR is again to be calculated as follows. SNR ou t =
P N / 2 t J / PG r
o
but
m
PG
t
m
Hence the output SNR becomes
B s Bm Bs
B
m
SNR out =
PG P ( N B / 2) J r
o
Now,
SNR in =
s
P ( N B / 2) J r
o
s
The new relationship can be written SNRout = (PG). SNRin
Here the losses in the system are not considered.
Another most important characteristics of spread spectrum is Antijam characteristic. SS System is having great rejection capabilities to reduce the effects of intentional jamming. This ability can be expressed in terms of jamming margin or sometimes called antijam (AJ) margin expressed in dB. It can be expressed as, SNRin (dB) = SNRout(dB) – PG (dB) Antijam margin can be defined as, Margin(AJ) = - SNRin (dB) – L (dB) o Where L = the losses in the system (dB)
The Antijam margin can be written as, Margin(AJ) = PG (dB) – L (dB) – SNRout (dB)
The output SNR of the spread spectrum receiver in presence of noise and interference is given as SNR ou t
P N / 2 t J / PG r
o
m
the energy associated with a message bit is
Eb = P r tm = Energy per bit the output SNR can be represented again with the following expression /
SNR ou t
E b N o E b .t P / 2 J / PG N o / 2 J / B s 1 / 2 J / N o B s N t r
o
m
m
Rewriting the above equation 1 J SNRou t 2 N o N o B s E b
Two important comments can be made from this result.
When the interference is narrowband, increasing the spreading improves the energy efficiency When the interference is wideband, such as white noise, increasing the spreading does not improve the energy efficiency because the interference power J increases directly with the signal bandwidth Bs.
To obtain the bandwidth utilization efficiency B/R B R
B s
2 R m
, where, Rm = the message bit rate Bs/2 = the equivalent energy bandwidth of the signal.
If the interference is narrowband compared to the bandwidth of the spread spectrum signal then combining the results 1 J 1 J / N o Rm SNR ou t 1 SNRou t 2 ( B / R ) N o 2 N o R m 2 B / R E b
If the interference is wide band compared to the bandwidth of the spread spectrum signal then
1 J o 1 SNRout N o 2 N o E b
The quantity J0 is the spectral density of the interfering signal.
Each user is a source of interference for the other users, and if one is received with more power, than that user generates more interference for the other users. It is important that the receiver gets the same power from each transmitter. The use of power control ensures that all users arrive at about the same power at the receiver, and therefore no user is unfairly disadvantaged relative to the others. Near far problem in spread spectrum system relates to the problem of very strong signals at the receiver swamping out the effects of weaker signals.
In short, the near-far problem is one of detecting and receiving a weaker signal amongst stronger signals
The receiver level fluctuates quickly due to fading at receiver. In order to maintain the strength of received signal level at BS, power control technique must be employed in CDMA systems.
Power control is implemented to overcome the near-far problem and to maximize the capacity of the system
– It tries to control the powers of the mobile stations in the system so that the received powers at the base station stay equal – It tries also to compensate the effects of slow fading and fast fading – There is no near-far problem in downlink due to a one-to-many situation
Power Control in CDMA
CDMA goal is to maximize the number of simultaneous users Capacity is maximized by maintaining the signal to interference ratio at the minimum acceptable Power transmitted by mobile station must be therefore controlled Transmit power enough to achieve target BER: no less no more
It uses dynamic output power adjustment of the transmitters. That is the closer transmitters use less power so that the SNR for all transmitters at the receiver is roughly the same. The two issues must be satisfied simultaneously:
1. The transmit power from each mobile must be controlled to limit
interference. 2. The power level should be adequate for a satisfactory voice quality.
The objective of the power control is to limit transmitted power on the forward and reverse links while maintaining link quality under all conditions.
Power control is capable of compensating the fading fluctuation. Received power from all MS are controlled to be equal. Near-Far problem is mitigated by the power control.
In frequency hopping systems, the transmitter changes the carrier frequency according to a certain "hopping" pattern. The meaning is the frequency is constant in each time chip, but changes from chip to chip. The advantage is that the signal sees a different channel and a different set of interfering signals during each hop.
Signal broadcast over seemingly random series of frequencies
Receiver hops between frequencies in sync with transmitter
Jamming on one frequency affects only a few bits This avoids the problem of failing communication at a particular frequency, because of a fade or a particular interferer.
Fast Frequency Hopping (FFH) One data bit is divided over multiple hops. i.e. frequency hopping rate is greater than message bit rate. In fast hopping, coherent signal detection is difficult, and seldom used. Mostly, FSK or MFSK modulation is used.
Slow Frequency Hopping (SFH) In this case one or more data bits are transmitted within one hop. i.e hopping rate is less than the message bit rate. An advantage is that coherent data detection is possible. Often, systems using slow hopping also employ (burst) error control coding to restore loss of (multiple) bits in one hop.
Consider, a fast hop system in which there are k frequency hops in every message bit duration tm. Thus the chip duration is tchip = tm /k k=1,2,3……. The number of frequencies over which the signal may hop is usually a power of 2. The frequency hopping is accomplished by means of a digital frequency synthesizer, which in turn is driven by PN code generator. The ML-sequence i.e. p chips will produce M=2 p frequencies for each distinct combination of these digits.
FHSS transmitter
1 bit from message and p-1 bits come from PN code generator. If a bit from message produces the smallest frequency change, then by itself it will produce a binary FSK signal. The p-1 bits from PN code generator then hop this FSK signal over the range of possible frequencies. There is a frequency multiplier K at the output of the system. It is to increase the bandwidth and thereby increase the processing gain.
Considering again fast hopping case, if M frequencies are separated by f 1=1/ tchip = k/ tm , then the signal bandwidth is given by
Bs = KMf 1 = KM/ tchip
So the processing gain PG = Bs/ Bm KM / t chip kKM / t m = kKM 1/ t m
1 / t m
Noncoherent frequency hopping The error correction is receiver made to improve the BER response.
The output of the mixer is also applied to early and late gates that produce an error signal to control the clock frequency. This keeps locally generated frequency hop pattern in synchronism with the incoming signal.
The time axis is divided into intervals known as frames and each frame is subdivided into M time slots. The total number of slots is decided on the basis of PN code or length of m-sequence. During each frame one and only one time slot is modulated with a message by any reasonable modulation method. The particular time slot that is chosen for a given frame is selected by means of a PN code generator .
All of the message bits accumulated in the previous frame are transmitted in a burst during the selected time slot. Frame duration T f , number of message bits k and message bit duration tm are related to each other by the equation T f = ktm
The width of each time slot in a frame is T f /M and
width of each bit in the time slot is T f /kM or simply tm/M .
Processing gain = Bs/ Bm =2 tm /(tm/M)= 2M for biphase modulation = tm /(tm/M) = M for quadriphase modulation
Interference among simultaneous users in time hopping system can be minimized by coordinating the times at which each user can transmit a signal. This also avoids the near far problem. In a noncoordinated system, overlapping transmission bursts will result in message errors and this will normally require the use of error correction coding to restore the message bits.
Waveforms showing time hopping spread spectrum signal formation in one frame duration
Self study……….
Comparison of Spread Spectrum Modulation Methods
Hybrid Spread Spectrum Systems
The standard IEEE 802.15.4a uses chirp spread spectrum standard. A chirp is a sinusoidal signal whose frequency increases or decreases over a certain amount of time. A chirp spread spectrum system utilizes linear frequency modulation of the carrier to spread the bandwidth. It is a very common technique in radar system but also used in communication systems. Consider, T = the duration of a given signal waveform and B = the bandwidth over which the frequency is varied. In this case the processing gain = B T .
It is also possible to use nonlinear frequency modulation and in some cases this may be desirable. Self inter symbol interference reduction is achieved in chirp spread spectrum communication systems.
Advantages:
Chirp Spread Spectrum uses its entire allocated bandwidth to broadcast a signal, making it robust to channel noise. Chirp Spread Spectrum is also resistant to multi-path fading even when operating at very low power. it does not add any pseudo-random elements to the signal to help distinguish it from noise on the channel, instead relying on the linear nature of the chirp pulse. Chirp Spread Spectrum is resistant to the Doppler effect, which is typical in mobile radio applications
Chirp spread spectrum transceiver
Examples………
Example 1
In a frequency hopping spread spectrum system for the generation of hopping frequencies suppose 3 bit PN code generator is used. If the carrier frequency is 8 KHz and frequency spacing is 0.5 kHz. Corresponding to code pattern 000 the frequency is 9.75 kHz. Find out all the hopping frequencies. Show all this frequencies on the frequency time plane.
Control Carrier Derivation of Output Codes Freque output freque ncy frequency (kHz) ncy (kHz) (kHz) 000 8 8+1.75 9.75 001 8 8+1.25 9.25 010 8 8+0.75 8.75 011 8 8+0.25 8.25 100 8 8-0.25 7.75 101 8 8-0.75 7.25 110 8 8-1.25 6.75 111 8 8-1.75 6.25
Example 2 A frequency hopping spread spectrum system is to have the following parameter. Message bit rate=2400 bps Hops per message bit =16 Frequency multiplication = 8 Processing gain 45dB (a) Find the smallest number of frequencies required if this number is to be a power of 2. (b) Find the bandwidth of the FH spread spectrum signal.
(a) Bm = 2400 bps k =16, K =8, M =?
Now, Processing gain 45 dB = 10 log10 PG PG = Bs/ Bm = 31623
Bs = 31623 x Bm =31623 x 2400 = 75.9 Mbps
For frequency hopping system
Bs = KMf 1 = KM/ tchip = kKM/ tm Bs = 8 x 16 x M x 2400/240 = 75.9
Mbps => M = 75.9 x 106/8 x 16 x 2400 = 247 But to make M power of 2 next selected value =256 = number of hopping frequencies (b) Bandwidth of the FH spread spectrum signal Bs = 8 x 16 x 256 x 2400 =78.64 Mbps
Now check for processing gain: PG = Bs/ Bm = 78.64 x 106/2400 = 32768 = 45.13 dB which is higher than the required, So condition for processing gain is satisfied.
Example 3
A certain time hopping spread spectrum system is allocated a maximum signal bandwidth of 8 MHz. Assuming the message bit rate after error correction is 3200 bps (a) find the number of bits necessary from the code generator to control the ON/OFF switching time. (b) If biphase modulation is used what is the processing gain.
PG= Bs/B m =2M for biphase ss modulation system
8 x 106/3200 = 2500 =2M = 33.97 dB M =1250 = total number of slots
It is not power of 2 but bandwidth must not exceed 8 MHz, we shall choose M =1024 (previous lowest power of 2) Now M=2 p => p=10 bits are required from the code generator to control the ON/OFF switching time. PG will be now 2M = 2048 =33dB (Note: This value is not much differed than the previous value of PG)
Example 4
Consider the spread spectrum system in which the chip rate is 107 chips per second and the message bit rate is 100 bps. If it is desired to obtain an output SNR of 25, which is about 15 dB and if losses are determined to be 2 dB, then find out antijam margin.
Processing gain PG = 2 / =2 x 107/ 100 = 2 x 105 = 53 dB
Margin (AJ) = 53- 2 -15 = 36 dB
i.e the desired output SNR can be obtained if the jamming signal is less than 36 dB.
Example 5
A direct sequence system has a PN code rate of 192 x 106 chips per second and a binary message bit rate at 7500 bps
If quadriphase modulation is used find PG. Assuming the received signal power is 4 x 10-14 watts and the one sided noise spectral density level N o is 1.6 x 10-20 W/Hz. Find the signal to noise power ratio in the input bandwidth of the receiver.
