136 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 2, FEBRUARY 2005 New Results on the BER of Switched Diversity Combining over Nakagami Fading Channels Lei Xiao, Student Member, IEEE, and Xiaodai Dong, Member, IEEE Abstract— New closed-form bit error rate (BER) expressions are derived for multibranch switched combining (SWC) systems with independent Nakagami faded diversity branches having integer Nakagami m-parameters. Constellations considered in- clude BPSK, M-PSK, M-PAM and M-QAM. The analysis is also applicable to the generalized hierarchical PAM and QAM modulation formats. Index Terms— Switched combining, Nakagami fading, error performance. I. I NTRODUCTION S WITCHED combining (SWC) is simpler to implement than maximal ratio combining (MRC), equal gain com- bining (EGC) or selection combining (SC) because an SWC combiner only needs to monitor and process one of the L di- versity branches. This makes SWC attractive to cost-stringent applications. The pioneering papers, [1], [2], and [3], laid the analytical groundwork for the performance of coherent signal- ing with multibranch switched combining reception. Most of the previous studies on this topic considered either differential detection or noncoherent detection (see the references in [1]). In [1], new generic and exact analytical results were presented for the combiner output signal-to-noise ratio distribution and the performance of coherent constellations with dual-branch switch-and-stay combining (SSC) in Rayleigh, Nakagami and Ricean fading channels. In [3], the authors derived the output statistics and the error rates of multibranch switched diversity systems, including SSC and switch-and-examine combining (SEC), in general fading channels. In [4], a thorough study was presented for SSC in independent and correlated generalized fading channels with power imbalance. And recently in [5], the author derived generic formulae for the performance of nonideal reference-based dual predetection SSC systems in correlated Nakagami fading channels. Adopting the moment generating function (MGF) approach in essence, the results in [1], [3], and [4] for the bit error rate (BER) of coherent M -PSK, M -PAM and M -QAM in Nakagami fading channels were in the form of a single finite integral with incomplete Gamma functions as integrand. In this letter, we present a new analysis on coherent M -PSK, M -PAM and M -QAM with multibranch switched diversity combining in independent Nakagami fading channels having Manuscript received June 10, 2004. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. P. Cotae. Lei Xiao is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN (e-mail: lxiao@nd.edu). Xiaodai Dong is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada (e-mail: xdong@ece.ualberta.ca). Digital Object Identifier 10.1109/LCOMM.2005.02024. integer Nakagami m-parameters. In general, the analysis is applicable to any modulation formats whose error rates in additive white Gaussian noise (AWGN) channels can be expressed as a weighted sum of error functions. The resultant closed-form BER formulae are in the form of a sum of incomplete Gamma functions, thus avoiding the need for numerical integration. II. SWITCHED DIVERSITY COMBINING We denote the switching threshold as S T . For switch-and- stay combining, the probability density function (PDF) of the instantaneous signal-to-noise ratio (SNR) γ s at the output of the combiner for independent and identically distributed (i.i.d.) diversity branches, regardless of the number of diversity branches L, is [3] f SSC γs =  F s (S T )f s (γ s ), 0 <γ s <S T [1 + F s (S T )] f s (γ s ), γ s ≥ S T (1) where f s (·) and F s (·) are the PDF and cumulative distribution function (CDF) of the instantaneous SNR distribution on each diversity branch. In Nakagami fading, they have the expressions [3] f s (γ s )=  m Λ  m γ m−1 s Γ(m) e − mγs Λ (2) F s (γ s )=1 − Γ ( m, m Λ γ s ) Γ(m) (3) where m is the fading parameter, Λ is the average SNR, Γ(·) is the Gamma function, and Γ(·, ·) is the incomplete Gamma function [6, Section 8.35]. For integer m which covers most cases of interest in practice, the CDF has a simple form [6, eq. (8.352.2)] F (γ s )=1 − e − m Λ γs m−1  k=0 γ k s k!  m Λ  k . (4) For SEC, the PDF of the instantaneous SNR at the output of the combiner can be written as [3, eq. (35)] f SEC γs =  [F s (S T )] L−1 f s (γ s ), 0 <γ s <S T [Fs(ST )] L −1 Fs(ST )−1 f s (γ s ), γ s ≥ S T . (5) It is observed that the PDF of SSC does not depend on the number of diversity branches, L, and SEC have the same PDF as SSC when L =2 [3]. As the error performance in SSC can be obtained by setting L =2 in SEC, we only need to analyze SEC in the remainder of this letter. 1089-7798/05$20.00 c  2005 IEEE