974 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008 Error-Rate Analysis of FHSS Networks Using Exact Envelope Characteristic Functions of Sums of Stochastic Signals Y. Maghsoodi and A. Al-Dweik, Senior Member, IEEE Abstract—In this paper, we present a novel approach to ana- lytically 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 sig- nals, 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 condi- tions, 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 am- plitude 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 al- ternative 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. Index Terms—Characteristic function (CF), envelope density, exact formula, frequency hopping (FH), generalized Ricean, jamming, orthogonal frequency-division multiplexing (OFDM), stochastic sinusoids. I. I NTRODUCTION T HE SYMBOL-ERROR probability P e in frequency- hopping spread-spectrum (FHSS) systems using M -ary frequency shift keying (MFSK) has extensively been consid- ered in the literature [1]–[4]. Recently, a new bandwidth- efficient modulation scheme was proposed as an alternative for the MFSK in the FHSS networks, which is a hybridization of orthogonal frequency-division multiplexing (OFDM) and noncoherent (NC) M -ary amplitude shift keying (MASK) [5], Manuscript received October 11, 2006; revised June 10, 2007, August 12, 2007, and August 31, 2007. The review of this paper was coordinated by Dr. D. Marabissi. Y. Maghsoodi is the Director of Scinance Analytics (e-mail: ym@ ScinanceAnalytics.com). A. Al-Dweik is with the Communications Engineering Department, Etisalat University College, Sharjah, U.A.E. (e-mail: dweik@fulbrightweb.org). Digital Object Identifier 10.1109/TVT.2007.909293 [6]. The P e of the FHSS–MASK system was evaluated by using approximations [5]. The main difficulty that prevents the accurate evaluation of P e for such systems is the lack of exact or accurate knowl- edge of the probability density function (pdf) of the sufficient statistics when the transmitted symbol is jammed. To address this problem, in this paper, we derive the exact formulas for the envelope pdf of the general sum of n stochastic sinusoids of the form A i cos(ωt + φ i ), where i =1,...,n, each with random amplitude and phase, and apply the formulas to deduce the exact analytical expressions for P e in the presence of intentional interference, such as multitone jamming (MTJ), or noninten- tional interference, such as multiple-access interference (MAI). The results are then applied to the evaluation of the overall P e for FHSS–MASK systems. To obtain the exact envelope pdf, in Section II, we apply the envelope separation theorem (EST) formula to derive formulas for the exact envelope characteristic function (EECF). In the important cases of interest, the calculation of the pdf via the inversion of the CF turns out to be equivalent to the Fourier transform (FT) of the Bessel functions. This leads to simpler new exact expressions. To deal with noisy channels, these methods are further applied to derive a new formula for the exact density of the envelope of noisy stochastic sinusoids (EDENSS) in Theorem 2. The conditional form of this pdf is also useful in many applications including error-rate studies. Previously, it was given by the Ricean distribution [7], [14], which was only verified for the cases where n =1, and the phase was assumed to be uniform and independent of the amplitude. Here, in Corollary 1, we present the formula for the generalized Ricean envelope density (GRED), which is shown to be valid under the most general assumption that just a general joint density exists, and we also allow n ≥ 1. The frequency-hopped spread-spectrum signals are primarily used in communication systems that require antijamming (AJ) projection and in code-division multiple-access (CDMA) sys- tems where many users share a common bandwidth. In both ap- plications, an interfering signal is injected in the desired-signal spectrum, causing distortion to the transmitted signal. In the case of the AJ systems, the interference is intentional, where the jammer’s goal is to hit the transmitted signal; the interference in this case is referred to as jamming. For the CDMA systems, the interference is nonintentional, and it is a consequence of collisions between signals coming from different users that are simultaneously accessing the channel using the same hopping frequency (HF). In this case, the interference is called MAI. 0018-9545/$25.00 © 2008 IEEE