 codes. Basically, this approach is to create a super channel as in Figure 4.21, where the super channel begins at the input to a convolutional encoder and ends at the Viterbi decoder output. The convolutional encoder with the Viterbi decoder that forms part of the super channel is the "inner code." The inner code reduces the error probabilities of the first coding channel (see Figure 4.1) which consists of the modulator, radio channel, and demodulator. This then forms a new coding channel for another "outer code" which can further reduce the error rate. In general, it is difficult to analytically obtain the bit error statistics of the super channel for a specific convolutional code. In the following example, some simulation results assuming an ideal additive white Gaussian noise radio channel [14] are shown. These simulations show that burst lengths out of Viterbi decoders can be modeled by a geometric probability distribution. Figures 4.23 and 4.24 show how decoded bit errors out of a Viterbi decoder tend to occur in bursts of various lengths with BPSK or QPSK modulation with three bit quantization. Generally, the decoded bits are error free for a while and then when a decoding bit error occurs the errors occur in a burst or string of length Lb with a probability distribution that is geometric [14], Pr{Lb=m}p1-p)m-1; m=1,2, (4.125) M-ARY REED-SOLOMON Q-ARY (OUTER) CONVOLUTIONAL Codes INTERLEAVER SYMBOLS ENCODER TO BITS (INNER) ENCODER CODING CHANNEL M-ARY REED-SOLOMON BITS TO VITERBI (OUTER) DEINTERLEAVER Q-ARY (INNER) DECODER SYMBOLS DECODER EQUIVALENT REED-SOLOMON CODE SUPER CHANNEL Figure 4.22. Reed-Solomon outer code. BURST. code; Figure 4.23. LENGTH OF ERROR BURST, Lb Figure 4.24. Histogram of burst lengths; Viterbi decoded constraint length 10, rate 1/3 convolutional code; E/ = 0.75 dB General Analysis of Anti-Jam Communication Systems where p=1/1 (4.126) and Lb is the average burst length in data bits. The waiting time, W, between bursts has the empirical distribution Pr{W=n}q(1-q)n--1 = K + K = 2, (4.12