When a pseudonoise (PN) balanced modulation is superimposed on an FH/QPSK signal, each jammer tone of power J0is then spread over a band- width equal to the PN chip rate Rc. Let Rs⫽1/Tsbe the information symbol
Figure 1.14. Worst case rversusEb/NJ—FH/QASK-16 (noise jamming).
Figure 1.15. Worst case PbversusEb/NJ—FH/QASK-16 (noise jamming).
rate and Rc/Rsrepresent the processing gain associated with the PN spread- ing,herein taken to be a larger integer I. It follows from the assumption of a large processing gain that the spread tone jammer possesses a fairly flat power spectral density within the data modulation bandwidth and is now caused to behave like a white noise jammer of spectral density
(1.87) Using (1.4) and (1.5), we can rewrite (1.87) as
(1.88) Since the hop frequency slots (assumed to be contiguous) must now be Rc wide to accommodate the PN modulation, in terms of the total hop fre- quency band Wssand the number of hop slots Nin that band, we then have (1.89) Combining (1.88) and (1.89) gives
(1.90) which is identical to (1.63) (the case of FH/QPSK in the presence of partial- band noise jamming) independent of the chip rate Rc.
The above discussion concerns the spectral characteristics of the spread tone jammer which, as concluded, resembles a white noise jammer with iden- tical spectral density. We shall now discuss the assumptions and theoretical adjustments under which the spread tone jammer can also be treated as a Gaussiannoise interference.
Consider an FH/PN/QPSK demodulator, similar to the one in Figure 1.2, where the despreading process now also includes a PN code correlator fol- lowing the frequency dehopper. Let prepresent the PN code length (in num- ber of chips). Accounting for the effects of the spread jammer only (i.e., neglecting thermal noise), it follows that the decision variable zIfor the in- phase channel (see (1.11)) becomes
(1.91) where
(1.92) andc(t) is the ⫾1-valued PN waveform. In deriving (1.91), ideal PN code synchronization at the receiver has been assumed. Recalling that the pro- cessing gain is the ratio of the PN chip rate to the data symbol rate, or equiv- alently Ts/Tc ⫽ I, then it follows that the integral in (1.92) amounts to a partial correlation of the PN code, starting from some random phase, pro-
CPN⫽ 冮1i⫺12TiTs
s
c1t2dt zI⫽aiB
S
2Ts⫺ 2J0 sin uJCPN
NJœ ⫽ J>Wss
r ⫽ NJ
r , N⫽Wss>Rc. NJœ ⫽J>rNRc. NJœ ⫽J0>Rc⫽J0Tc.
vided that the number Iof integrated chips does not equal the code length p. Phrased differently, the conclusions to follow hold when 1 VIVp(or in a weaker sense when 1 VImodulopVp). This is because, as is well known, the full-period integration of a PN code equals the constant 1/p, in which case, no randomness about CPNexists. On the other hand, when pis very large, successive code chips can be considered almost independent, iden- tically distributed ⫾1-valued random variables, in which case the condition IVpwould provide an approximate binomial distribution for the random variableCPN. The additional constraint IW1 then causes this binomial dis- tribution to behave like a Gaussian distribution.
Arguments similar to the above were made in Chapter 1 in connection with the evaluation of the performance of pulse-jammed direct-sequence spread-spectrum systems. A more rigorous treatment of the validity of the Gaussian assumption for CPNhas been examined in [6] for a variety of PN and Gold codes, with sufficient evidence that it holds, at least approximately, for a wide class of such codes.
Let us now return to (1.91). Conditioned on the (0, 2p)-uniformly dis- tributed phase uJ,zIis, according to the above, a Gaussian random variable whose conditional variance is
(1.93) Hence, when the jammer is present, the average bit error probability for the Ichannel (identically for the Qchannel) is given by
(1.94) where (1.18), (1.87), (1.90), and (1.93) have been used in arriving at this result. Finally, multiplying (1.94) by rgives the average bit error probabil- ity of FH/PN/QPSK in the presence of partial-band multitone jamming, namely,
(1.95) Although (1.95) can be used in assessing the effect of a spread tone jam- mer on system performance, we shall now indicate an even simpler way of evaluating the tone interference effect by means of converting the spread tone jammer to an equivalentAWGN interference, where the equivalence is understood in terms of its effect on bit error probability.Let repre- sent the one-sided power spectral density of AWGN interference whose sta- tistical characteristics remain the same after the PN despreader. Clearly, the bit error probability in this case is given by
(1.96) PbJ
n ⫽Qa B
2Eb
N0J
b.
N0J
Pbt ⫽ 2r
p冮0p>2QaB rEb
NJ
1 sinuJ
bduJ. PbtJ⫽ 1
2p冮02pQaB STs
2TcJ0
1
0sinuJ0bduJ⫽ 2
p冮0p>2QaB rEb
NJ 1 sinuJ
bduJ
var5zI>uJ6⫽TsTcJ0 sin2uJ.
Equations (1.94) and (1.96) have been plotted in Figure 1.16. The abscissa is the signal-to-jammer power ratio SJR in decibels where, for the spread tone case (SJR)t⫽ Eb/NJwhile, for the noise case, . A careful examination of Figure 1.16 reveals that, for values of (SJR)tup to
approximately 12 dB, the difference between the
signal-to-jammer ratios which achieve the same performance is, to a high degree of accuracy, linearly increasing with , the slope of the line being 0.2. It is therefore concluded that, given a spread tone jammer of
1SJR21tdB2
1SJR21tdB2⫺ 1SJR21ndB2
1SJR2n⫽Eb>N0J
Figure 1.16. QPSK bit error probability versus SJR(db)for AWGN and tone jam- ming and spread tone “equivalent” AWGN jamming.
, an “equivalent”AWGN jammer can be devised for which (1.97) If the SJR’s are measured in ordinary numbers rather than decibels, then (1.97) implies that
(1.98) From (1.96) and (1.98), it then follows that the spread tone jammer can be conveniently thought of as an “equivalent” AWGN jammer with corre- sponding bit error probability
(1.99) Equation (1.99) has been plotted in Figure 1.16 (dashed lines), from which the high degree of agreement with the exact expression for can be witnessed. The range of applicability of (1.99) is, for the current purposes, more than adequate since, in the frequency bands where the jammer is pre- sent, the system is forced to operate at high bit error probabilities (this is especially true when the jammer strategy, i.e., choice of r, has been opti- mized). From (1.99), it follows that the overall average bit error probability (accounting for partial-band jamming) is given by
(1.100) The worst case jammer can be found from (1.100), with the result
(1.101)
with corresponding
(1.102)
A comparison of (1.102) with (1.73) indicates that the worst tone jammer for FH/PN/QPSK is slightly less effective than the worst noise jammer for FH/QPSK.
Pbmax
t ⫽ à
0.0789 Eb>NJ
; Eb>NJⱖ0.9220
Qa
B2aEb
NJ
b0.8b; Eb>NJ 6 0.9220.
rwc⫽ • 0.9220 Eb>NJ
; Eb>NJⱖ0.9220
1; Eb>NJ 6 0.9220
Pbt ⫽rQa
B2arEb
NJ
b0.8b.
PbJ
t
PbJ
t ⫽Qa
B2arEb
NJ
b0.8b. 1SJR2equivn ⫽ 11SJR2t20.8. 1SJR2equivn 1dB2⫽ 10.82⫻ 1SJR21tdB2. 1SJR21tdB2