Then the energy per bit is Eb=ATo. (1.100) Assuming the maximum-likelihood demodulator for the AWGN channel, then the uncoded bit error probability is bounded by (see 1.37) and (1.72)) (1.101) where as before N1 is given by J/Ws for the continuous jammer of average power J. Slow Frequency Non-Selective Fading Channels For a fading channel A is a random variable with some probability den- sity function PA() and the average uncoded bit error probability is bounded (1.102) For Rayleigh statistics where PA(a) = az0 (1.103) Bound Exact No Fading Fading Eb/Nj or Eb/Nj (dB) Figure 1.11. Uncoded bit error probability. Coherent Direct-Sequence Systems and the received signal energy has average value (1.104) the exact uncoded bit error probability is (1.105) The bound in (1.102) is (1.106) Figure 1.11 shows the uncoded bit error probabilities and their bounds with no fading versus Eb/N1 and with Rayleigh fading versus Eb/N - . Note that the effects of Rayleigh fading are like the worst case pulse jammer shown in Figure 1.5. 1.6.2 Continuous Jammer with Coding-No Fading Estimate With coding, assume ideal interleaving where each coded symbol is assumed to have an independent fading term. Consider two binary sequences of length N denoted (1.107) where x,xe{-1,1}. (1.108) Let the Hamming weight be H(x,x) = d and assume the fading amplitude sequence is A = (A1, A2, , AN) (1.109) where {An} are independent identically distributed random variables with common probability density function PA(). For the equivalent white Gaussian noise channel due to the constant power jammer, the unquantized channel outputs are rkxkAkVT2kk= 1,2,,N (1.110) where Ts is the coded symbol and Nk is zero mean with variance Nj/2. Assuming the soft decision metric m(r, x) = rx, the pairwise error proba- Slow Frequency Non-Selective Fading Channels bility that the receiver chooses X when x is sent is Chernoff bounded by (1.111) for all l > 0. Note that here (1.112) XK # XK Xk=xk (1.113) Defining (1.114) the pairwise error bound has the form Pr{x->}} D d (1.