115) and a general coded error bound of the form (1.116) For the simple repeat m code described in Chapters 3 and 4, Volume I, the bit error bound is (1.117) where D is given by (1.114). For the constraint length K = 7, rate R = 1/2 convolutional code the bit error bound is given by 1404D14 + 11633D16 + (1.118) for the same D where Ts is the coded symbol time. Coherent Direct-Sequence Systems For the case of Rayleigh fading D is given by D = min - (1.119) Es = o2Ts is the average coded symbol energy. Figure 1.12 1/2 shows con- where bit error bounds for several repeat codes and the K = 7,R where = R is volutional the code rate in bits per coded symbol. Note that coding with coded code when there is Rayleigh fading. Here Es = REb interleaving Repeat m Code Baseline K =7, , = Convolutional 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Eb/Nj (dB) Figure 1.12. Coded bit error probability-soft decision. Slow Frequency Non-Selective Fading Channels combats fading much like it did for the worst case pulse jammer. Indeed, pulse jamming is similar to a fading channel where the signal-to-noise ratio can change in time. The results illustrated above assume the soft decision metric m(r,x) = rx (1.120) as implied in (1.111). A hard decision channel with ideal interleaving results in a binary symmetric channel where the crossover probability would depend on the specific amplitude term A. Here the crossover probability is (1.121) which is the same as (1.101) with T replaced by Ts, the coded symbol time. Now consider the fading channel with ideal interleaving and the usual hard decision metric when there are only two binary sequences of length N as shown in (1.107). Assuming x is transmitted and the channel output binary sequence is y = (y1,y2, yn), the probability that the decoder incorrectly chooses X is the probability that (1.122) The hard decision pairwise probability of error is thus Chernoff bounded X)=Pr(w(y,x)zw(y,x)x} (1.123) where l > 0 and the Hamming distance between two elements is a special case of the Hamming distance 