between two vectors, (1.124) w(y,x) = w(y1,X1) + w(y2, X2) + (1.125) Note that -{ probability E(An) with E(An) (1.126) with probability Coherent Direct-Sequence Systems for those d components where X # Xn. Thus, = X n (1.127) (1.128) where (1.129) is the averaged crossover probability. Defining (1.130) we have all 1=0. (1.131) The choice of l that minimizes D(A) satisfies (1.132) giving the parameter D = min D(A) = V46(1- E) = = (1.133) The Rayleigh fading amplitude statistics has (1.134) For the repeat m code with the bit error bound given by (1.117) and the con- straint length K=7,rate R = 1/2 convolutional code with the bit error bound (1.118) where D is given above by (1.133), the coded bit error probabilities are shown in Figure 1.13. Again note that coding can overcome the degra- dation due to fading. In practice quantized amplitude histograms of a fading channel may be available through measurements. Suppose, for example, the measured quan- tized amplitude values area1,a2,...,aL with measured probabilities = l=1,2,...,L. (1.135) Slow Frequency Non-Selective Fading Channels Repeat m code K = 7, r =1/2 Convolutional m = 20 m = 10 Eb/N, (dB) Figure 1.13. Coded bit error probability-hard decision. For this channel the uncoded bit error probability is = L piQ ( INTO) NJ (1.136) For soft decision decoding (1.114) becomes D = min 120 e12N, EN (1.137) Coherent Direct-Sequence Systems while for hard decision decoding D is given by (1.133) where (1.138) With this empirically measured fading, the impact of coding with ideal interleaving can be computed. 1.6.3 Continuous Jammer with Coding-Fading Estimate With no coding, the optimum demodulator with a fading channel is the same as that for the non-fading channel. Thus, for the uncoded case in Section 1.6.1 knowledge of the fade term A does not change the performance. This is not true, however, for coded DS/BPSK systems with fading (see Proakis [33]). Consider the same coding case described in Section 1.6.2 except that we now assume ideal estimates4 for the fade values