Effects of correlated fading on level crossing rates and average fade durations with predetection diversity reception

Abstract

General expressions for the level crossing rate (LCR) and average fade duration (AFD) are obtained for several diversity combining schemes employing two-branch predetection reception of correlated Rayleigh fading signals. These expressions are obtained from joint and conditional probability density functions (PDFs) of the received signals, and lead to a unified treatment. This simplified method contrasts with the characteristic function approach used in previous investigations. Numerical results are presented for a space-diversity system using horizontally spaced antennas at a mobile station. It is shown that while the angle between the antenna axis and the direction of vehicle motion does not appear in the cumulative distribution function (CDF) of the combined output signal envelopes, it affects the LCR and AFD when the two fading signals are correlated. When the two antennas parallel with the direction of vehicle motion are used, the LCR can be reduced below the value obtainable from signals which fade independently. When the two antennas are perpendicular to the direction of vehicle motion, the AFD is loosely dependent on the antenna spacing and, provided the antenna spacing is not too small, is approximately half that for the no-diversity case. p(a | b) = conditional PDF of a with b given Re (.) = real part of complex value Rj^t) = time-varying signal strength envelope for Nth branch R(t) = time derivative of time-varying signal strength envelope z, = complex amplitude of ith multipath wave zf = complex conjugate of complex ith multipath wave | Zj | = modulus of complex ith multipath wave <z,> = ensemble average of complex ith multipath wave = mean matrix = inverse of matrix M -transpose of matrix M = angle between antenna axis and direction of vehicle motion = phase of ith multipath wave = phase difference between ith and ;th multipath waves = wavelength = complex crosscorrelation function between ith andjth branches = average received signal power = variance of R = time delay = average fade duration (AFD) at envelope level R M A/" 1 a Ot X Pill) List of principal symbols d = distance between space-diversity antennas e, = received amplitude of ith multipath wave erfc (.) = complementary error function f D = maximum Doppler frequency J o (.) = zero-order Bessel function nib = mean value of R N/{ = level crossing rate (LCR) at envelope level R n = conditional level crossing rate p(a, b) = joint probability density function (PDF) of a and bPaper 5770F (E8), first received 4th September 1986 and in final revised form 26th August 1987 Dr. Adachi is with the

A unified approach to average LCR and AFD performance of SC diversity in generalized fading environments

“…(12), i.e., the equivalent Qð0; bÞ ¼ 0:5e À0:5b 2 has been utilized. With L ¼ 2, dual-branch SC, then last equation coincides with the results in [37]. By the way of combining (T8) shown in Table 1 with (11), the AFD can be obtained as…”

Second-order statistics of maximal-ratio and equal-gain combining in Hoyt fading

“…Several works have addressed the second-order statistics of diversity-combining systems, including the following. In [5], LCR and AFD expressions of selection combining (SC), maximal-ratio combining (MRC), and equal-gain combining (EGC) for dual-branch diversity in correlated Rayleigh channels were presented. The case of M independent identically distributed (iid) Nakagamim channels was solved in [6] for SC, MRC, and EGC.…”

Level crossing rate and average fade duration for pure selection and threshold selection diversity‐combining systems

“…In PSC and for M"1 and for M"2, the formulas deteriorate into the expressions already available in the literature (e.g. References [1,2,4]). In TSC and for the threshold set to be very low (say 2 "0) or very high (say 2 PR), the formulas deteriorate into those for the one-branch case, as expected.…”

“…Lee [1] investigated the second order statistics of the equal gain combining (EGC) scheme deriving the level crossing rate and the average fade duration in an approximate manner. Adachi [2] approached this problem in a less restrictive way, also investigating the maximal ratio combining (MRC) and the Pure Selection schemes, but in all cases for two branches only. More recently, Yacoub et al [3] solved this problem for the general case of EGC and MRC schemes.…”

“…An arbitrary number of M independent branches for PSC and two independent branches for TSC are assumed. The analytical results are thoroughly validated by reducing the general case to some special cases, for which the solutions are known [1,2,4], and by means of simulation for the more general case. The plots presented here are also compared with those of the general solution recently published for the EGC and MRC [3] and an agreeable coherence is found.…”