1.51 Low Probability of Intercept (LPI)
1.5.2 Independent Interference Rejection and Multiple-Access Operation
The ability of a SS system to reject independent interference is the basis for the multiple-access capability of SS systems, so called because several SS systems can operate in the same frequency band, each rejecting the interfer- ence produced by the others by a factor approximately equal to its energy gain. This asynchronous form of spectrum sharing is often called spread- spectrum multiple-access(SSMA) or code-division multiple-access(CDMA).
As an illustration of SSMA operation, consider a transmitted signal xT(t), (1.63) and an interfering signal xt(t),
(1.64) impinging on a receiver which is frequency synchronous with the transmit- ted signal and which computes the inner product of the received signal with a reference modulation mR(t). We assume without loss of generality that mT(t),mI(t), and mR(t) are unit power waveforms, i.e., the time-averaged and ensemble-averaged value of 0mj(t)02is one for j⫽T,I,R. The output of the receiver correlator is the sum of two terms, corresponding to the desired inner product and the interference inner product . These terms for the k-th correlation interval ((k ⫺ 1)T,kT), normalized to unit input power, are
(1.65)
(1.66) wherefeis the phase tracking error in the receiver (see Figure 1.3), and ⌬
vI1k2⫽ 1mI,mR2⫽ 冮1k⫺kT12TmI1t⫺tI2mR*1t2ej2p¢tdt ej1fI⫺fT⫹fe2,
vT1k2⫽ 1mT,mR2⫽ 冮1k⫺kT12TmT1t2mR*1t2dt#ejfe,
2PIvI
2PTvT
xI1t2⫽Re52PImI1t⫺tI2ej12p1fc⫹¢2t⫹fI26, xT1t2⫽Re52PTmT1t2ej12pfct⫹fT26, MMF>Th, 0f0 ⫽MF, 0d0 ⫽M
d僆d f僆f
is the carrier frequency offset between the interference and the transmitted signal. The average signal-to-interference energy at the input to the corre- lator is simply
(1.67)
and the frequency-synchronous correlator’s output signal-to-interference measurement ratio is
(1.68) Here8⭈9denotes discrete time averaging over the parameter k.
The above calculation is based on the supposition that outputs from both the in-phase and quadrature channels of the correlator are necessary, i.e., both the real and imaginary parts of yT(k) are necessary in the reception process. If instead reception is phase-synchronous, then yT(k) is real, and on the average only half of the interference power contributes to the distur- bance of Re{yT(k)}. Hence, for phase-synchronous detection,
(1.69) The key to further analysis is the evaluation of the interference level at the correlator output.
(1.70) The variable tIdenotes an arbitrary time shift of the interference modula- tion relative to the reference modulation, this shift being inserted to model the fact that the interference is assumed asynchronous with respect to all clocks generating the transmitted signal. Stored reference modulations (e.g., mI(t) if the interference is SSMA in nature) are generally cyclo-stationary, i.e., they may be made stationary by inserting a time-shift random variable (e.g.,tI) which is uniformly distributed on a finite interval. Hence, we assume thattIis uniformly distributed on a finite interval which makes mI(t⫺tI) a wide-sense stationary random process. This assumption and the further assumption that any random variables in mI(t) are independent of any in MR(t), effectively allow us to treat asynchronous multiple-access interference and jamming identically in average power calculations.
⫻mR*1t2mR1t¿2dt dt¿f.
E5 0vI1k2 026⫽Ee冮冮1k⫺kT12TmI1t⫺tI2mI*1t¿⫺tI2ej2p¢1t⫺t¿2
SIRout1f sync2⫽2#SIRout1freq sync2.
SIRout1freq sync2⫽ 冓E5 0vT1k2 026冔 冓E5 0vI1k2 026冔 #SIRin. SIRin⫽¢ 冓EePT冮1k⫺kT12T0mT1t2 02dtf冔
冓EePI冮1k⫺kT12T0mI1t2 02dtf冔 ⫽
PT
PI
,
Averaging (1.70) over tIgives
(1.71) whereRI(ⴢ) and SI(ⴢ) are the autocorrelation function and PSD respectively of the unit power process mI(t⫺tI). That is,
(1.72) andMRk(f) is the Fourier transform of the reference modulation mR(t) in the k-th correlation interval.
(1.73) Certainly when the correlation time Tis not a multiple of the reference mod- ulation generator’s period (and this must be the case to avoid problems with simple repeater jammers), then 0MRk(f)02will vary with k.
Using the fact that the modulation forms are normalized to unit energy, and combining (1.65)—(1.68) and (1.71), demonstrates that the improvement in signal-to-interference ratio achieved in the correlation calculation is
(1.74)
Again we emphasize that (1.74) applies both to asynchronous multiple- access interference and independent jamming.
Example 1.4. Suppose that a DS/BPSK communication system employs binary antipodal modulation and phase coherent reception over a symbol timeTs. Hence, in the k-th correlation interval,mT(t) in (1.45) is equal to c(t) or⫺c(t), and reception is performed by correlating the received signal with c(t) as given in (1.22). Then,
(1.75) where again Nc denotes the number of chips per data bit,F denotes the Fourier transform ooperation, and P(f) is the Fourier transform of the chip
⫽P1f2 a
kNc⫺1 n⫽1k⫺12Nc
cne⫺j2pfnTc, MRk1f2⫽Fe a
kNc⫺1 n⫽1k⫺12Nc
cnp1t⫺nTc2 f SIRout1freq sync2
SIRin ⫽
冓Ee ` 冮1k⫺kT12TmT1t2mR*1t2dt`2f冔
冮q
⫺q
SI1f⫺ ¢2冓E5 0MRk1f2 026冔df .
MRk1f2⫽ 冮1k⫺12TkT mR1t2e⫺j2pftdt.
RI1t2⫽E5mI1t⫹t⫺tI2mI*1t⫺tI26⫽ 冮⫺qqSI1f2ej2pftdf,
⫽ 冮⫺qqSI1f⫺ ¢2E5 0MRk1f2 026df,
E5 0vI1k2 026 ⫽ 冮冮1k⫺kT12TRI1t⫺t¿2ej2p¢1t⫺t¿2E5mR*1t2mR1t¿26dt dt¿
pulsep(t). Denoting the period of {cn} by Nand time-averaging the squared magnitude of MRk(f) over kgives
(1.76) wherePcis defined in (1.51) and Sm(f) is given by (1.52). This result can be verified analytically, despite the fact that Sm(f) is the PSD of the data-mod- ulated waveform mT(t) while (1.76) represents an average of short-term energy spectra of mR(t). However, this result is reasonable because averag- ing over the zero-mean independent data symbols, which leads to the result (1.52), breaks up the PSD calculation into a time average of short-term energy spectra.
The SIR improvement ratio for a DS/BPSK system is determined by sub- stituting (1.76) into (1.74) and using the fact that mT(t)⫽dmR(t) in a cor- relation interval.
(1.77) Since both SI(f) and Sm(f) have fixed areas and are non-negative, the worst- case interference PSD SI(f⫺ ⌬) in (1.77) is a Dirac delta function located at the frequency which maximizes Sm(f). Therefore,
(1.78) A well-designed spreading sequence {cn} should result in a PSD Sm(f) close to that of a purely random sequence, namely 0P(f)02/Tc. In keeping with our power normalizations, let’s assume that
(1.79) and hence, maxf0P(f)0⫽Tc. Therefore, a well-designed spreading sequence has maxfSm(f)⫽Tc, and from (1.78)
(1.80) This lower bound on signal-to-interference ratio improvement is the energy gain indicated in Example 1.1.
When the interference is spectrally similar to the transmitted signal, e.g., in the SSMA case, and both signals possess the spectra of a purely random
SIRout1f sync2
SIRin ⱖ 2Ts
Tc
⫽2Nc. p1t2⫽ e1, 0ⱕt 6 Tc
0, otherwise 0P1f2 0 ⫽ `sin1pfTc2 pf `, SIRout1freq sync2
SIRin ⱖ2c 1 Ts
max
f Sm1f2 d⫺1. SIRout1f sync2
SIRin ⫽2c 1
Ts 冮⫺qqSI1f⫺¢2Sm1f2dfd⫺1.
⫽TsSm1f2, 冓0MRk1f2 02冔⫽ 1
Pc a
Pc⫺1 k⫽0
0MRk1f2 02
binary sequence modulated on rectangular chip pulses (1.79), then
(1.81)
The above example illustrates the following points concerning SIR improvement calculations based on long-term energy averages:
1. The jammer can minimize the signal-to-interference ratio by transmit- ting a single tone at the frequency of the transmitter spectrum’s peak.
However, this is not the best jamming strategy if the receiver has an addi- tional notch filtering capability, or if the designer is interested in the more significant bit-error-rate (BER) design criterion.
2. The above SIR improvement calculation is valid even if mI(t)⫽mT(t), randomization by time shift tIbeing enough to make mT(t ⫺ tI) and mT(t) quite distinct on the average. This suggests that several systems could use identical SS waveforms, provided the probability that they arrive in near synchronism at a receiver (for any reason, natural or jam- motivated) is virtually zero.
3. Cross-correlation functions and cross-spectra between asynchronous interference and desired signal are not a specific factor in the signal-to- interference ratio improvement based on long-term averages.
Clearly the use of long-term averages in a SIR-based figure-of-merit has led to these simple results.
Similar results can be achieved for other forms of SS modulation. the DS example was particularly simple because the receiver used only one corre- lator. Corresponding analyses of SS systems using higher dimensional sig- nal sets, e.g., FH/FSK, must consider the total signal energy and total interference energy collected in a set of correlators.