Error-rate analysis of FHSS networks using exact envelope characteristic functions of sums of stochastic signals

In this paper, we present a novel approach to analytically study and evaluate the error probability of frequency-hopping spread-spectrum (FHSS) systems with intentional or nonintentional interference over the additive white Gaussian noise and Rayleigh-fading channels. The new approach is based on the derivation of new formulas for the exact envelope characteristic function (EECF) of the general sum of n stochastic sinusoidal signals, with each of the n signals having different random amplitude and phase angle. The envelope probability density function (pdf) is obtained from the characteristic function (CF), which, in the important cases of interest, is shown to also give simpler formulas in terms of the Fourier transform (FT) of the Bessel functions. Previously, the Ricean envelope density had only been verified for the very special case where n = 1, and the phase is uniform and independent of amplitude. Here, a new formula for the exact density of the envelope of noisy stochastic sinusoids (EDENSS) is presented, which leads to the generalization of the Ricean envelope-density (GRED) formula under the most general conditions, namely, n ≥ 1 signals, dependent amplitudes, and phases having an arbitrary joint pdf. The EDENSS and GRED formulas are applied to compute the pdfs needed in noncoherent detection under noise. The derived formulas also lead to the exact formulas for the error probability of the FHSS networks using M-ary amplitude shift keying (MASK) without setting limits on the number of interferers or the symbol alphabet. The power of our EECF and FT methods is further demonstrated by their ability to give an alternative derivation of the exact general envelope-density (EGED) formula, which has previously been reported by Maghsoodi. The comparative numerical results also support the analytical findings.