where the quantizer forces a decision on each transmitted coded symbol. This results in the Binary Symmetric Channel (BSC) where the coded symbol error probability is = = Pr{n = VE}} -((2) (4.36) where (4.37) Conventional Coherent BPSK QUANTIZER (a) General BPSK Coding Channel (b) Hard Decision BPSK Coding Channel Figure 4.3. BPSK coding channels. General Analysis of Anti-Jam Communication Systems For the hard decision channel the maximum-likelihood metric is (4.38) Hence, = ee + (1 - 8)e - (4.39) = V4s(1 - 8). (4.40) In general for binary symbols (M = 2), R = 1 - log2(1+D) bits/symbol. (4.41) Figure 4.4 shows R versus E/No for the soft and hard decision detectors. Note that there is roughly a 2 dB difference for most values of E/No. -16 -14 -12 -10 -8 -6 -4 -2 O 2 4 6 8 10 12 14 16 18 20 Es/No (dB) Figure 4.4. R for hard and soft decision detectors. Conventional Coherent BPSK Figure 4.5a. K = 7, R = 1/2 binary convolutional code-BPSK. The most commonly used code for coherent BPSK and QPSK modula- tions is the constraint length K = 7 rate R = 1/2 bits per coded bit convo- lutional code found by Odenwalder [10]. This code has the bit error bound2 36D10 + 211D12 + 1404D14 + 11633D16 + (4.42) Another common code is the K = 7,R = 1/3 convolutional code also found by Odenwalder where the bit error bound is given by + 22D17 + 44D18 + (4.43) Figure 4.5 shows these two bit error bounds for the hard and sot decision 2The factor of one-half applies to all maximum-likelihood metrics considered here. General Analysis of Anti-Jam Communication Systems Figure 4.5b. K = 7, R = 1/3 binary convolutional code-BPSK. detectors where the maximum-likelihood Viterbi decoders [2] are used for these convolutional codes. Note that, regardless of the code used, the same difference in Eb/No between hard and soft decision detectors occurs as seen in Figure 4.4 for R versus E/No. The difference in ENo for fixed value of R0 directly translates to the difference in Eb/No for the corresponding bound on bit error proba- bility where E/ and Eb/No 