 at the i-th frequency. That is, ej is the normalized energy out of the i-th frequency detector. We denote the M modulation symbols by {1,2,...,M} and a general input symbol into the modulator by X. Assuming X = i, which corresponds to the i-th frequency tone being transmitted, e1, e2, , em are independent random variables where ri=Vei has probability density functions [2], (4.60) for ri j0 where Io() is the modified Bessel function of the first kind and zero-th order (this is a Rician density function) and (4.61) for and j i i these are Rayleigh density functions). 4.7.1 Uncoded With no coding, the maximum-likelihood decision rule is: given channel out- put y given by (4.59), choose the transmitted symbol corresponding to the largest value of li; = 1,2, M. This is equivalent to using a metric (4.62) Conventional Non-Coherent MFSK ( ) 2 ( ) 2 10/4" sinomt 2/4 2 COSWM + ( ) 2 Figure 4.9. Non-coherent MFSK detector. General Analysis of Anti-Jam Communication Systems Note that, for the uncoded case, the decisions are mode on a symbol-by-sym bol basis without regard to sequences. Thus, we do not require the additiv- ity property of metrics for uncoded case. For this uncoded case, the probability of making a symbol decision error is [11] Ps=Pr{x # x I(E,/No)/(1+1) (4.63) When M = 2k, we can associate K data bits to each modulation symbol. Denoting Eb as the energy per data bit we have is=KEb (4.64) and the bit error probability [10] For the special case of binary symbols (BFSK), (4.65) becomes (4.66) Applying the general coded error bound to the special case of uncoded non-coherent MFSK with metric m(y,x) = ex yields D(A) = (4.67) where (4.68) and when x # X, (4.69) provided 21, , 1. Since 1 is a free variable, substituting 1 in place of 21 results in the form, (4.70) The pairwise error probability is, thus, x) < D (4.71) Conventional Non-Coherent MFSK yielding the symbol error probability Ps = 1 < ej for some j # ix=i} 2 1/2 (M - 1)D. (4.72) This bound is based on the union bound with the factor of one-half apply- in