For FH/MFSK signals there are three well-known types of codes one can use that have M-ary code symbols. Reed-Solomon codes [21] are natural block codes to use where the decoders generally require a hard decision M-ary channel.M-ary alphabet convolutional codes include those found by Trumpis [18] and dual-Kcodes9[20] where M2K. These codes can be decoded using the Viterbi algorithm with all metrics discussed in this chapter.
Today, however, most codes used in practice are binary codes, especially binary convolutional codes. These have been used primarily with coherent BPSK and QPSK modulations with one-bit (hard decision), two-bit, and three-bit quantized channel outputs. For binary convolutional codes LSI Viterbi decoders chips and sequential coder chips have been developed. It seems reasonable to exploit these developments and use binary convolu-
1
M1 c ° a
L k1
2PkB1ML2 a
M jL1
Pj¢
2
1d. 1
M1 c a a
M k1
2Pk1L2b21d Dq1J2 1
M1 c a
M k1a
M j1
2Pk1L2Pj1L21d Pl1L2 à
Pl; l1, 2,p, L a
M jL1
Pj
ML ; lL1,p, M.
652 Non-Coherent Frequency-Hopped Systems
9These are convolutional codes using the Galois Field GF(2K).
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tional codes with the FH/MFSK signals. Here we examine how the M-ary input channel with outputs of Menergy detector samples can be converted into a binary input channel with one-bit, two-bit, and three-bit outputs that will be suitable for binary codes with corresponding decoders.
For M2Kthere are Kbits associated with each M-ary channel input sym- bol. For example, with K3 (M8), we have a channel with eight input symbols and the output samples as seen in Table 2.23.
Between the M-ary input and the Menergy detector outputs there exists MFSK modulation, frequency hopping, the real channel, dehopping and M energy detectors. To create a binary input channel with one-bit output, we merely make a hard decision as to which M-ary symbol was sent. For exam- ple, for M8 above if e4is the largest energy detector output then the chan- nel output hard decision bits would be 011. Here the probability of a bit error is denoted dqif the signal hopped into the q-th sub-band. If the bits are inter- leaved before converting them into M-ary symbols and then deinterleaved, the hard decision bits obtained are the outputs of the usual binary symmetric
Other Coding Metrics 653
Figure 2.131. List metric cutoff rates for broadband noise jammer.
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channel (BSC) with crossover probability dq, which is the uncoded bit error probability for the FH/MFSK system.
For the above BSC a binary code can be used with a decoder that uses jammer state information (JSI) with metric given by (2.317) and the result- ing coding parameter given by (2.318). Without JSI the decoder metric becomes (2.325) with the resulting coding parameter given by (2.327) and (2.328).
Suppose next we want a binary input channel with two-bit outputs. Here the hard decision with 1-bit quality measure suggested by Viterbi [27] and discussed in Section 2.7.2 can be used. For M8 there are 16 outputs here of the form (m, 1) and (m, 0) for M1, 2, . . . , 8. We can now convert these into two-bit binary channel outputs by taking, for example, the output (4, 1) and converting it into three two-bit binary outputs of the form
(2.393) where recall m4 corresponds to the 011 bits. Similarly, if (4, 0) is the out-
14, 12S10, 12 11, 12 11, 12
654 Non-Coherent Frequency-Hopped Systems
Figure 2.132. List metric cutoff rates for worst case tone jammer.
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put of the channel in Section 2.7.2, then we have the conversion
(2.394) With interleaving and deinterleaving the result is a binary input channel with two-bit outputs as shown in Figure 2.136 where the transition probabilities can be obtained using the type of analysis given in Section 2.7.2 (see Viterbi [27]).
This approach can be extended to three-bit outputs for each input bit by using three threshold levels g1g2g31 where (2.351) is generalized as
(2.395) y à
1m, 112, em g1 max
km ek 1m, 102, g1 max
km ek 7 emg2 max
km ek 1m, 012, g2 max
km ek 7 emg3 max
km ek 1m, 002, g3 max
km ek 7 emmax
km ek. 14, 02S10, 02, 11, 02, 11, 02.
Other Coding Metrics 655
Figure 2.133. List metric cutoff rates for worst case tone jammer.
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Again for the M8 example we have the conversion
(2.396) where the hard decision bits are followed by two “quality” bits. With inter- leaving and deinterleaving there are three-bit outputs for each binary input into the channel which can be used in the usual three-bit soft decision decoder for binary codes.
Another approach to obtaining a binary input channel with a quantized output is to consider the conditional probability of each transmitted bit given the Menergy detector outputs. In general let b1,b2, . . . ,bKbe the Kcoded bits that result in an M-ary symbol m with corresponding energy detector output em. Using Bayes rule (see Lee [33])
(2.397) Pr5bl0e6 Pr5e0bl6Pr5bl6
Pr5e6 . 14, 012S10012, 11012, 11012
656 Non-Coherent Frequency-Hopped Systems
Figure 2.134. List metric cutoff rates for worst case tone jammer.
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Figure 2.135. List metric cutoff rates for worst case tone jammer.
Table 2.23
Bits to symbol conversion,K3,M8.
Bits 8-ary Input Energy Detector Output
b1b2b3 m e
0 0 0 S1 e1
0 0 1 S2 e2
0 1 0 S3 e3
0 1 1 S4 e4
1 0 0 S5 e5
1 0 1 S6 e6
1 1 0 S7 e7
1 1 1 S8 e8
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Next observing that there are several possible M-ary symbols having the l- th bit,bl,
(2.398) Defining
(2.399) by symmetry
(2.400) Recall
(2.401) where p(ek0m,q) is given by (2.330) for the general slowly fading non-uni- form channel with jammer power distribution J. Defining
(2.402) the final form is
(2.403) Pr5bl0e6 G1e2 a
mm1bl2r1em2 r1ek2 N0bqJq
aqEN0bqJq
expe aqEek
1N0bqJq21aqEN0bqJq2f q
M
k1p1ek0m, q2 Pr5e0m, bl6 Pr5e0m6 Pr5m0bl6 •
2
M , mm1bl2 0, mm1bl2. m1bl2 5m: l-th bit is bl6, Pr5e6bl6 a
M m1
Pr5e0m, bl6Pr5m0bl6.
658 Non-Coherent Frequency-Hopped Systems
Figure 2.136. Binary input 2-bit output channel.
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where
(2.404) But Pr{bl0e} is the sum of exponential functions of em,mHm(bl) and for sig- nal-to-noise ratios of interest such a sum is dominated by the largest term.
Thus, using the approximation
(2.405) gives an approximate ML metric (taking logarithm) given by
(2.406) For the K3 (M8) example of Table 2.23
(2.407) and the metrics for each of the three coded binary inputs,
(2.408) The above metrics are approximate ML metrics with no quantizations.
They result in a binary input channel with real valued outputs. Various quan- tized outputs can be obtained for each coded input bit in the form
(2.409) where f() is typically a uniform quantizer function. Here the difference
(2.410)
mmaxm1bl02 em max
mm1bl12 em
f1 max
mm1bl02 em max
mm1bl12 em2 ylf1m1e, bl02m1e, bl1222
m1e, b312 max5e2, e4, e6, e86 m1e, b302 max5e1, e3, e5, e76 m1e, b212 max5e3, e4, e7, e86 m1e, b202 max5e1, e2, e5, e66 m1e, b112 max5e5, e6, e7, e86 m1e, b102 max5e1, e2, e3, e46
m1b312 52, 4, 6, 86 m1b302 51, 3, 5, 76 m1b212 53, 4, 7, 86 m1b202 51, 2, 5, 66 m1b112 55, 6, 7, 86 m1b102 51, 2, 3, 46
m1e, bl2 max
mm1bl2 em. Pr5bl0e6 G1e2 max
mm1bl2r1em2 G1e2
1 M q
M
k1a 1
N0bqJq e1>1N0bqJq2b
Pr5e6 .
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is treated like the correlator output of the coherent BPSK demodulator.With interleaving and deinterleaving we have a discrete memoryless binary input quantized output coding channel suitable for the usual binary codes with cor- responding decoders.
Although the above conversion of an FH/MFSK channel into a discrete memoryless binary input channel was based on a Rayleigh fading channel with receiver and jammer Gaussian noise statistics, it can certainly be used in other cases as well. It is a straightforward conversion and should be robust in the sense of being good for all cases of interest. Generally, for Gaussian noise and optical signals with M-ary pulse position modulation good binary convolutional codes with these converted binary input channels perform as well or better than M-ary input channels for roughly the same complexity (see Lee [33]). For very high speeds and large M, Reed-Solomon codes look attractive for the hard decision M-ary channels. Reed-Solomon codes at data rates over 100 Mbpsa exist for M28256 [21].