Angeles, 1975. [13] G. D. Forney, Jr., Concatenated Codes, Cambridge, MA: M.I.T. Press, 1966. [14] L. J. Deutsch and R. L. Miller, "Burst statistics of Viterbi decoding," TDA Progress Report 42-64, Jet Propulsion Laboratory, Pasadena, California, pp. 187-193, May-June 1981. [15] R. Blahut, Theory and Practice of Error Control Codes, Reading, MA: Addison- Wesley, 1983. [16] E.R. Berlekamp, "The technology of error-correcting codes," Proc. IEEE, vol. 68, pp. 564-593, May 1980. APPENDIX 4A. CHERNOFF BOUND Figure 4A.1 shows two functions (4A.1) ex(t) = e " all t. (4A.2) For any 1=0 clearly u(t) ex(t). (4A.3) Next consider a random variable X with probability density function px(x) for all X. Then Pr(x = o)= E{elx}. (4A.4) Hence, for any l > 0. (4A.5) This is the Chernoff bound [1]. In the application of the Chernoff bound in this chapter, the random vari- able X has the form (4A.6) Appendix 4B. Factor of One-Half in Error Bounds Figure 4A.1. Comparing u(x) and e where the terms in this sum are independent of each other. Thus, (4A.7) This is used as a bound for the pairwise error probability P(x X) in Section 4.2. APPENDIX 4B. FACTOR OF ONE-HALF IN ERROR BOUNDS Based on the results of Jacobs [8] and Hellman and Raviv [9], general error bounds on the average error probability for cases where there are two hypotheses are derived. Special cases of these bounds result in a factor of one-half times the usual Bhattacharyya bound and the Chernoff bound. Let Z be a continuous random variable that can have one of two proba- bility densities: H2:f2(z),-oo<x (4B.1) General Analysis of Anti-Jam Communication Systems where a priori probabilities for these two hypotheses are denoted 1Pr{H1and2Pr{H = An arbitrary deterministic decision rule is characterized by the following decision function: given an observed value Z of the random variable Z, let (4B.2) In terms of this decision function define conditional error probabilities Pg = Pr{decide H3H H (4B.3) PE2 = Pr{decide H |H2} (4B.4) The average error probability is = - + (4B.5) Oft