riable X. (2A.1) is distinguished from the conventional Chernoff bound by the factor of 1/2 and the absence of the restriction that the Chernoff parameter l be non-neg- ative. (In fact, the condition of (2A.2) guarantees that the minimizing 1 > 0.) Since (2A.2) must be satisfied for every non-negative X, it cannot apply to discrete-valued random variables which are asymmetric about X = 0. We will now prove that (2A.2) is valid for the particular cases of interest. Consider the reception of FH/MFSK signals with m diversity chips per M- ary symbol over the partial-band noise channel. Assume non-coherent energy detection of the diversity chips, linear summation of the detected chip energies for each symbol, and perfect jamming state side information. Let the random variable x above represent the detection metric: (2.61) shows that X is the difference of a non-central and a central chi-square random vari- able, each with 2m degrees of freedom. In particular, since the received sig- nal energy over all m chips is KEb, where M = 2K, and the side information Appendix 2A assumption requires that all m chips be jammed with noise of power spec- tral density Nj/r for an error to occur (designated by the event H in (2.61)), (2A.3) where the y{'s and Z¿'s are independent Gaussian random variables, statisti- cally defined by (2A.4) From [13], we can express the probability density function of x as W-k/2,m+(k-1)/2(4bxx) x 2 0 Wk/2,m+(k-1)2(-4bx) (2A.5) x < 0 (k+m-1)! where and W(.) is the Whittaker function defined by B - a + 1 / 0 0 (2A.6) Since m > 1 and k > 0 in (2A.5), the constraint on the indices of the Whittaker function in (2A.6) is satisfied. Now we need to demonstrate that (2A.5) is compatible with (2A.2); that is, we must prove that (2A.7) Non-Coherent Frequency-Hopped Systems A sufficient condition is simply m! Wk/2,m+(k-1)/2(4bx) (k+m-1)! W_k/2,m+(k-1/2(4bx); k,x0. (2A.8) Substituting (2A.6) into (2A.8), we need to SOW that (2A.9) However, since m 1, the integrand is non-negative whenever y > 0. Consequently, (2A.9