opt at this optimum code rate as a function of n, Figure 2.105 shows that Mopt decreases monotonically with n, finally reach- ing unity at n = n* II 92, for which Ropt = .76. = Remember that (2.183) rep- resents the average performance over all possible code sets {si}, with the obvious implication that some particular codes must perform even better. Generalizing these results for arbitrary system parameters, we know that for sufficiently large block lengths and optimized code rates, codes exist that are powerful enough to defeat any kind of non-adaptive jamming without the need for additional diversity redundancy. It remains for information theorists to determine how to generate these codes. We can derive closed form solutions for n* and the corresponding Ropt* Independent of the kind of jamming, (2.183) tells us that EIN, is bounded Coding Countermeasures WC NOISE JAMMING Pb==10-5 BLOCK CODE RATE R' BITS/CHANNEL USE Figure 2.104. Performance of FH/4-ary FSK system employing (n, nR'/2) random block code with optimum diversity in worst case partial-band noise. For a given block length n over the 4-ary field, the required Eb/N1 to achieve Pb = 10-5 is minimized at R' = Ropt. WC NOISE JAMMING Pb=10-5 R'=ROPT n = 92 BLOCK LENGTH n Figure 2.105. Variation in optimum diversity with block length n for FH/4-ary FSK system with (n, k) random block code and optimized code rate, when P = 10-5 and signals are received in worst case partial-band noise. For n >n III 92,mopt = 1,imply- ing that diversity is not required. Non-Coherent Frequency-Hopped Systems by an expression that is proportional to where, for convenience, A = M - 1 B = M(2Pb)1/n (2.184) Differentiating f(R') with respect to R' and setting this result equal to zero shows that Ropt must satisfy the transcendental constraint (2.185) for a given n (remember that B depends on n). If we further require that the bound on Mopt in (2.183) be unity at n = n*, we have the additional con- straint (2.186) For this particular value of n, (2.185) simplifies to (2.187) in bits