are related by (4.2) with N1 replaced by No. One can view the uncoded case as a special case of coding where the code rate is R = 1. For this special case, for any transmitted sequence the maxi- mum-likelihood receiver bases its decision on a symbol-by-symbol basis. That is, the optimum decision rule is to decide which particular symbol in a sequence was transmitted based only on the corresponding output symbol. This is the form of all memoryless channels with no coding since each chan- nel output symbol depends only on the corresponding channel input sym- Conventional Coherent BPSK bol and all channel input symbols are independent of each other. With cod- ing, of course, there are a restricted number of channel input sequences, thus introducing dependence among coded channel input sequences. For the uncoded case, the symbol-by-symbol optimum decision rule is identical to the hard decision rule shown in Figure 4.3(b) where the exact bit error probability is Pb = o(v) No (4.44) Note that these decisions assume the soft decision metric of (4.32). For the corresponding result using the general bound Pb < 1D (4.45) Figure 4.6 shows both the exact bit error probability and this special case of Exact Bound Eb / No (dB) Figure 4.6. BPSK bit error probability. General Analysis of Anti-Jam Communication Systems the general coded bit error bound. For most values of Eb/No, there is approx- imately a 1 dB difference between the exact uncoded bit error probability and the bound. 4.5 DS/BPSK AND PULSE JAMMING The previous results for the white Gaussian noise channel also apply to the case where there is continuous jamming of a direct sequence (DS) spread BPSK signal. The only difference is that N1 replaces No in E/No given by (4.3). Jamming, however, can take many forms. To illustrate the application of the general bounds presented in this chapter, consider the following example. Suppose there is a pulse jammer with average power J/p for a fraction p of the time and zero power for a fraction 1 - p of the time. Assume that when