COHERENT DIRECT-SEQUENCE SYSTEMS
1.8 OTHER CODING METRICS FOR PULSE JAMMING
In the previous section, the ideal Rake receiver was approximated by a receiver that estimated the slowly varying multipath fading envelopes.
Earlier, we had examined a similar ideal case in Section 1.4 where we con- sidered various decoding metrics for DS/BPSK signals agianst pulse jam- ming. The ideal metric for this case is the soft decision metric iwth jammer state information given by
(1.217) whereZis the jammer state random variable with probabilities
(1.218) andc(Z) is a weighting function such that c(0) is chosen as large as possi- ble and c(1)⫽1. Under these conditions, it was previously shown in Chapter 4, Part 1, that the channel parameter for this case is given by
(1.219) which when translated to its equivalent computational cutoff rate R0 via (1.92) is illustrated as curve (1) in Figure 1.8.
When using the same soft decision metric, i.e., (1.217) without jammer state information, we observed in Chapter 4, Part 1, that the appropriate met- ric weighting then becomes c(0)⫽c(1)⫽ 1. For this case, we obtain (see Chapter 4, Volume I)
(1.220) which has the equivalent computational cutoff rate R0⫽0for all Es/N0as illustrated by curve (3) in Figure 1.8. The implication of this result is that the soft decision metric without jammer state information has no protection against a jammer who concentrates his available power in a very narrow
D⫽min
lⱖ0 e⫺2lEs3rel2EsNJ>r⫹1⫺r4
⫽re⫺rEs>NJ
D⫽min
lⱖ0 r exp5⫺2lEs⫹l2EsNJ>r6 Pr5Z⫽06⫽1⫺r
Pr5Z⫽16⫽r m1y,x;Z2 ⫽c1Z2yx
D⫽ 14e11⫺e2 D⫽ c L
L⫹ET>NJ
dL
pulse. Stated another way, a receiver that uses a soft decision metric with no jammer state information has, in the presence of an optimized pulse jam- mer, an unbounded probability of error. One way of getting around this intol- erable degradation in the absence of jammer state information is to use hard decision decoding. This was shown, however, to result in a relatively poor performance, as can be witnessed by examining curve (4) in Figure 1.8.
In this section, we first examine other metrics that allow soft decision decoding to be used with an optimized pulse jammer and no jammer state information. Following this, we reexamine the behavior of soft decision decoding with jammer state information and the necessary modifications to the metric when, indeed, the estimate of the jammer state Zis not perfect, e.g., in the presence of background noise.
Perhaps the simplest modification of the soft decision receiver to allow operation in the presence of a pulse jammer is to clamp the input to the decoder at either a fixed or variable level, the latter case requiring an intel- ligent controller in the decoder. Clamping the decoder input level provides a means for limiting the potentially large excursions in this level caused by narrow pulse jammers and as such prevents the decoder’s decision metric from being dominated by these extreme signal level swings when the jam- mer is present. Mathematically speaking, the operation of clamping is equiv- alent to first passing the decoder input ythrough a zero mean non-linearity with transfer function
(1.221) wherey⬘now represents the actual decoder input and kⱖ0 is a parameter which is either fixed or is allowed to vary with Es/NJand the pulse jammer’s strategy, i.e.,r.
In the absence of jammer state information, El-Wailly [34] has shown that using the normalized maximum-likelihood metric m(y⬘,x) ⫽ y⬘x/NJ, the channel parameter Dis given by
(1.222) whereQ(x) is again the Gaussian probability integral and
(1.223) B⫽11⫹k222rEs>NJ⫺l22Es>1rNJ2 ,
A⫽11⫺k222rEs>NJ⫺l22Es>1rNJ2 ,
⫹el2Es>1rNJ23Q1A2⫺Q1B2 4 64,
⫹Q322rEs>NJ1k⫹12 4e2l11⫹k2Es>NJ D⫽min
lⱖ0 e⫺l2Es>NJ3 11⫺r2⫹r5Q322rEs>NJ1k⫺12 4e2l11⫺k2Es>NJ
y¿⫽ •k2Es ; y 7 k2Es
y; 0y0 ⱕk2Es
⫺k2Es ; y 6 ⫺k2Es
Intuitively, one would expect that a clamped (fixed or variable) soft deci- sion receiver characterized by (1.221) would outperform a hard decision decoding metric and at the same time not be totally vulnerable to the opti- mum pulse jammer as is the unclamped soft decision metric (i.e.,k ⫽q).
The degree to which this observation is true is demonstrated by the follow- ing illustrations and discussion.
Figure 1.19 is an illustration of Das computed from (1.222) versus Es/NJ for a fixed clamping level of k⫽1 and the worst case jammer who chooses hisrto maximize Dfor each value of Es/NJ. Also illustrated for purposes of comparison is the corresponding result for the hard decision decoding met- ric (with no jammer state information) which from Chapter 4, Volume I, is given by
(1.224) As a reminder, the curve for the unclamped soft decision decoder (with no jamming state information) for worst case pulse jamming would simply be a horizontal line corresponding to D⫽1 for all Es/NJ.
When the clamping level is allowed to vary, then assuming that the receiver contains an intelligent controller that can choose the value of kso as to minimize Dfor each value of r, Figure 1.20 illustrates the corresponding performance in terms of a three-dimensional plot of DversusEs/NJandr. Alternately, suppose it is assumed that the receiver has no intelligence, i.e.,
D⫽ 24e11⫺e2 ; e⫽rQ122rEs>NJ2.
Figure 1.19. Channel parameter versus signal-to-noise ratio for worst case jammer (no jammer state information). (Reprinted from [35].)
it fixes kat some value between .01 and 1.0, but the jammer, on the other hand, is intelligent and assumed to know k, in which case, he chooses rto maximizeD. For this scenario, Figure 1.21 illustrates DversusEs/NJandk.
Clearly, much larger values of Dresult for this case as compared with those in Figure 1.20.
We now return to a consideration of soft decision decoding metrics with jammer state information that derive their benefit from the fact that the con- tribution to the metric in those time intervals where the jammer is absent is very heavily weighted compared to that in the intervals where the jammer is known to be present. Up until now, we have assumed that knowledge of the presence or absence of the jammer during a given time interval was per- fect. With, for example, additive background noise, the estimation of Z, the jammer state parameter, will not be perfect. We now examine how to suit- ably modify the soft decision decoding metric and the impact on its perfor- mance as a result of having imperfect jammer state information.
Figure 1.20. Channel parameter versus rand signal-to-noise ratio for best clamp- ing level. (Reprinted from [35].)
Figure 1.21. Channel parameter versus signal-to-noise ratio and clamping level for worst case jammer. (Reprinted from [35].)
DS/BPSK transmission over a channel with background additive white Gaussian noise of power spectral density N0and a pulse jammer with para- metersNJ⫽J/Wandr, is equivalent to sending a BPSK signal over an addi- tive white Gaussian noise channel with noise spectral density
(1.225) whereZis again the jammer state parameter with probabilities as in (1.218).
If perfect jammer state information were possible, then the metric of (1.217) would be used with
(1.226) Since, as mentioned above, the background noise perfect jammer state information is not possible, then the metric of (1.217) is still appropriate except that now Zis replaced by an estimate of the jammer state. The manner in which the estimate is obtained and its statistics are the sub- ject of the following discussion.
As in the Rake receiver illustrated in Figure 1.18, an estimate of the jammer state might be based on the previous BPSK transmission.6Another approach would be to base the estimate on the amplitude of the received signal in the current interval as follows:
(1.227) Basically, this receiver assumes that when the channel output yis close to one of the two signal terms, and , then the channel contains only background noise. Substituting (1.226) in (1.217) then gives the metric examined by El-Wailly and Costello [35]:
(1.228)
It should be noted that the metric of (1.228) can also be regarded as one that assumes no jammer state information since indeed the jammer state information comes directly from measurements on the observable (i.e., the channel output) rather than from an external source.Thus, combining (1.227)
m1y,x;Zˆ2 ⫽ à yx N0
; if Zˆ ⫽0 yx
N0œ ; if Zˆ ⫽1.
⫺2Es
2Es
Zˆ ⫽ e0; if 0y0 ⱕb2Es
1; if 0y0 7 b2Es .
Zˆ
Zˆ Zˆ
c112⫽ 1 N0œ . c102⫽ 1
N0
N0e⫽ •N0; if Z⫽0 N0œ ⫽¢ N0⫹ NJ
r ; if Z⫽1
6This estimate can be done before deinterleaving at the receiver to make the channel memoryless.
and (1.228), we get the alternate form
(1.229)
For the metric of (1.229), El-Wailly and Costello [35] have shown that the channel parameter Dis given by
(1.230) where
(1.231) and
(1.232) Figure 1.22 illustrates the performance of a receiver using the metric of (1.229) in terms of Dversusrwithbas a parameter for Es/NJ⫽5 dB and a background-to-jammer-noise-spectral-density ratio N0/NJ ⫽ 1.0. Also shown for comparison are the corresponding results for the ideal soft deci- sion metric with unknown and perfectly known jamming state information.
We observe that the performance corresponding to (1.230) is much improved over that of the unknown jammer state case but not as good as when the jammer state is perfectly known. Also, a value of b⫽2 seems to give best performance.
Suppose that we again use the metric of (1.228) but now the jammer state estimate is provided by means external to the decoder. In particular, assume that the probability of error in Zis described by the false alarm and missed detection probabilities
(1.233) i.e., the probability of estimating the state of the jammer depends on the state of the jammer. Typically, we would want to choose PMDVPFAsince
PMD⫽Pr5Zˆ ⫽Z6 when Z⫽1 PFA⫽Pr5Zˆ ⫽Z6 when Z⫽0 F1t2 ⫽Q3t22>N⫺l212Es>Nˆ2 1N>Nˆ2 4.
I3L,H,Nˆ ,N4 ⫽e⫺1Es>Nˆ232l⫺1N>Nˆ2l243F1L2⫺F1H2 4,
⫹I3 11⫹b22Es , ⫹q,N0œ,N0œ4 6,
⫹I3 11⫺b22Es , 11⫹b22Es , N0,N0œ4
⫹r5I3⫺q,11⫺b22Es , N0œ,N0œ4
⫹I3 11⫹b22Es , ⫹q,N0œ,N04 6
⫹I3 11⫺b22Es , 11⫹b22Es , N0,N04 D⫽min
lⱖ0 ⱖ 11⫺r25I3⫺q,11⫺b22Es , N0œ,N04 m1y,x;Zˆ2 ⫽ à
yx N0
; if 0y0 ⱕb2Es
yx
N0œ if 0y0 7 b2Es
the effect on the metric of missing a jammed channel symbol is much more severe than that produced by assuming the jammer is present when indeed he is not. For this scenario, the channel parameter Dhas been shown to be given by [35]
(1.234) which is illustrated in Figure 1.23 versus randPFAforPMD⫽10⫺8,Es/NJ⫽ 5 dB, and N0/NJ⫽1.0.
Finally, it should be obvious that one could employ a metric that combines the averages of clamping the decoder input level with a jammer state esti- mate provided by external means. Such metrics have been considered in [35]
and their performance analyzed by the general analysis techniques of Chapter 4, Part 1.
In conclusion, we leave the reader with the thought that while many other metrics for the pulse jamming channel with additive background noise are theoretically possible, from a practical standpoint, one wants to select a metric that is easy to implement and robust in the sense that the
⫹rPMDe⫺32l⫺1N0œ>N02l24Es>N0.
⫹r11⫺PMD2e⫺32l⫺l24Es>N0œ
⫹ 11⫺r2PFAe⫺32l⫺1N0>N0œ2l2Es>N0œ
D⫽min
lⱖ011⫺r211⫺PFA2e⫺32l⫺l24Es>N0
Figure 1.22. Channel parameter vs.rfor system with estimate based on signal level and with Es/NJ⫽5 db and N0/NJ⫽1.0. (Reprinted from [35].)
worst case jammer does not do much more harm than the baseline con- stant power jammer.