cessing complexity and speed required in channel decoding. To achieve small error rates in practice, Forney [13] proposed using a concatenation of codes where two layers of coding are used. This is illustrated in Figure 4.21 where the coding channel is the same as shown enclosed in dotted lines in Figure 4.1. The encoder and decoder of Figure 4.1 now become the "inner" encoder and decoder in Figure 4.21. To create a memoryless super channel interleaving and deinterleaving of the symbols that are encoded and decoded by the inner code is often required. 4.10.1 Binary Super Channel Typically the inputs to the inner encoder and the inner decoder outputs are binary symbols. With interleaving and deinterleaving, the resulting super channel becomes a memoryless binary symmetric channel (BSC) with crossover probability E. Here E is the coded bit error probability for the inner code with the coding channel. Previously examples were presented where this bit error probability has a bound. (4.111) where R0 is determined by the coding channel (see Figure 4.1) and given as a function of EJN, where now E is the energy of each coded symbol of the inner code. If we denote the energy per super channel binary symbol by Es Ec=REs (4.112) where R is the inner code rate in bits per symbol. SUPER CHANNEL OUTER ENCODER INTERLEAVER INNER ENCODER CODING CHANNEL Angeles OUTER DECODER DEINTERLEAVER INNER DECODER Figure 4.21. Concatenation of codes. Systems Concatenation of Codes The BSC super channel such as sketched in Figure 4.21 has a parameter like D which is denoted as D5. Thus, Ds = V4s(1 - 8) (4.113) and the cutoff rate is Rs=1-log21 Ds) (4.114) which is measured in bits per super channel bit. Next, consider an outer code characterized by Bs() with the resulting final bit error bound (4.115) Note that if Rs is the outer code rate then the overall rate measure in data bits per coding channel symbol is RRR bits/coding channel symbol and the energy per bit is given by Eb = RE E (4.116) As an example, suppose that with 8-ary FSK there is