- 1 detected energies will be identically zero. Non-Coherent Frequency-Hopped Systems where lij is the energy detector output for the i-th M-ary symbol on the j-th diversity transmission. The linear sum metric of (2.57) is maximum likelihood for a Rayleigh fading channel [9, pp. 533-540], and it is asymptotically opti- mum for the additive white Gaussian noise channel in the limit of large EIN However, in general it is suboptimum, and we will later see that there is a signal-to-noise ratio reduction as a result of the non-coherent energy combining implicit in the A;'s. 2.3.3.1 Partial-Band Noise Jamming We will now analyze the performance of the FH/MFSK system with diver- sity utilizing the soft decision metric with side information in a worst case partial-band noise environment (refer back to Section 2.2.1). Suppose sym- bol 1 is sent: conditioned on the j-th chip being jammed, e1j is a non-central chi-square random variable while the remaining M - 1 eii's are central chi- square random variables, each with two degrees of freedom. The probabil- ity density functions of the eii's have the form of (2.8) and (2.9), with EJN1 replaced by Ec/N' = pKE/mN, = (see (2.21)). Let H denote the event that the j-th chip is hit by partial-band noise, with = (H, H,...,Hm) , representing the event that all m diversity transmis- sions are jammed; since the H}'s are independent, Pr{H} = p m (2.58) With perfect side information, an M-ary symbol error requires that H occur: (2.59) Adopting the bounding approach of [2], the M-ary problem of (2.59) is first reduced to a binary problem by applying the union bounding technique and noting that the A;'s are identically distributed for i # 1: (2.60) This expression is then reduced to the statistics of a single diversity trans- mission by using (2.57) and the Chernoff bound [2, (9), with L = m and Eb replaced by Es = KEb]: (2.61) Coding Countermeasures which should be minimized over the Chernoff parameter 1. The factor of 1/2 in the Chernoff bound is justified under certain conditions [10] 