Error Probability for FHMDPSK in Multitone Jamming, Fast Rician Fading,
L.J. Mason · 1995
A method is presented for the calculation of M- ary DPSK error probability in partial-band multitone jam- ming and white gaussian noise. Both unfaded and faded signals and jamming tones may be analyzed by the method. The fading considered is frequency-flat, time selective, Ri- cian fading. The results have application in mobile, slow frequency-hopped, military, satellite systems. The method uses the Pawula-Rice-Roberts method for calulating the prob- ability distribution function of the phase angle between two vectors when they are perturbed by noise. The worst case error probability in partial-band jamming with fading is effi- ciently computed using Houston's approach. A fundamental parameter, independent of the actual jammer power being used on the system, is identified. The effect of the ratio of the direct signal energy to the faded (indirect) signal energy is shown by example, as is the effect of fading bandwidth and Doppler shift. For practical values of the parameters, jam- mer fading was not found to be significant. Results for M = 2, 4, and 8 are given. The system considered is a slow frequency-hopped spread spectrum system which has part of the transmission band jammed by a series of tones. It is assumed that the jam- mer knows the frequencies of the frequency-hopping pat- tern and the total number, Nt, of possible frequencies used, but not the pattern itself. Consider full-band jamming in which the energy of each tone is ZJ. Then, the total energy expended in one symbol time is ETOT = N~EJ. ETOT is assumed fixed for the jammer under consideration. Thus, for - a system with a fixed number of hop frequencies, Nt, EJ = ETOT/N~ is a constant. Now consider partial-band jamming in which the jammer may choose to expend its total energy, ETOT, by using NJ < Nt tones. The fraction of the band jammed is a = Nj/Nt. The total energy ex- pended per tone is now EJ = ETOT/NJ. Thus, the energy of each tone in partial-band jamming may be expressed as EJ = EJ/CY. Equivalently, if Es is the energy of the signal expended in a symbol period, then YJ = E~/EJ = ayJ where yJ = Es/EJ is the average signal-to-jamming ratio (SJR). A fundamental question is: given that the jammer has available a fixed average SJR, yJ, what is the proper combination of a and YJ to use such that the error proba- bility is maximized?