IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 6, DECEMBER 2014 557 Analysis of M -PSK With MRC Receiver Over κ − μ Fading Channels With Outdated CSI Pawan Kumar, Student Member, IEEE, and P. R. Sahu, Member, IEEE Abstract—Average-symbol-error rate (ASER) and diversity gain of an L-branch maximal ratio combining (MRC) receiver for M-ary phase-shift keying (M-PSK) modulation are analyzed over κ − μ fading channels with outdated channel state informa- tion using moment generating function (MGF) based approach. Analytical expressions for the MGFs of the exact and asymp- totic signal-to-noise ratio at the output of the MRC receiver are obtained using which analytical expressions for the ASER and diversity gain are derived. The effects of κ, μ, and L on the ASER and diversity gain are analyzed. Index Terms—ASER, diversity gain, κ − μ fading, M-PSK, MRC, outdated channel state information (CSI). I. I NTRODUCTION R APID phase fluctuation in fading channels causes syn- chronization loss in Costas loops used for phase recovery of M -ary phase shift keying (M -PSK) signals resulting in poor error-rate performance of the M -PSK communication system [1]. Alternate methods such as pilot-tone assisted modulation, pilot-symbol assisted modulation (PSAM) and pre-survivor- processing technique which provide embedded references to assist signal recovery in the receiver have been proposed in [1]. PSAM technique requires periodic insertion of pilot symbols in a data frame and for a certain number of received frames the channel estimates of pilot symbol positions are fed to an inter- polator to provide estimates of fading at symbol rate [2]–[4]. Sinc interpolator, minimum mean square error (MMSE) inter- polator and Gaussian interpolator can be used for interpolation purpose. For MMSE interpolator, the effect of imperfect chan- nel estimates (ICE) with Rayleigh, Rician and Nakagami-m fading channels for different modulation schemes including M -PSK have been studied in [4]–[6]. Other interesting works related to channel estimation problem with M -PSK modula- tions can be found in [7]–[9]. However, these analyses are lim- ited to basic fading channel models only. The κ − μ distribution is a generalized fading model that characterizes mobile commu- nication signals having a line-of-sight (LoS) component in non- homogeneous environments and closely fits to experimental data compared to generalized form of Nakagami-n (Rice) and Nakagami-m distributions [10]. Diversity combining technique is used in fading channel receivers to improve its perfor- mance. Among known linear diversity combining schemes Manuscript received April 10, 2014; revised August 31, 2014; accepted September 1, 2014. Date of publication September 8, 2014; date of current version December 17, 2014. The associate editor coordinating the review of this paper and approving it for publication was Y. Chen. P. Kumar is with the Department of Electronics and Electrical Engineer- ing, Indian Institute of Technology Guwahati, Assam 781039, India (e-mail: kpawan@iitg.ernet.in). P. R. Sahu is with the School of Electrical Sciences, Indian Institute of Technology Bhubaneswar, Odisha 751013, India (e-mail: prs@iitbbs.ac.in). Digital Object Identifier 10.1109/LWC.2014.2355849 maximal ratio combining technique (MRC) possesses best performance [11]. In this letter, assuming MMSE channel estimation the effects of ICE on the average-symbol-error rate (ASER) and diversity gain of an L-branch MRC receiver are analyzed for κ − μ fad- ing channels. In fact, it is assumed here that the statistics of true channel gain and its estimates are identical. Hence, it becomes appropriate to use the term outdated channel state information (CSI) instead of ICE [12]. The analysis uses moment generating function (MGF) based approach to derive mathematical expres- sions for ASER and diversity gain. The main contributions of the work presented here can be enumerated as: Derivation of expressions for (a) exact and asymptotic MGFs of the output signal-to-noise ratio (SNR) of MRC; (b) exact and asymptotic ASER for M -PSK modulation system, which are not available in literature to the best of our knowledge; (c) diversity gain at finite SNR; and (d) asymptotic diversity gain. Based on the above obtained analytical expressions the effects of κ, μ, L and estimation error parameters on the system performance are analyzed and discussed. II. SYSTEM MODEL Let d(i) be the transmitted symbol in the ith interval with energy E s = E[|d(i)| 2 ], with {d(i) ∈ e (j2πk/M) ,k = 0, 1,...,M − 1} for M -PSK modulation where E[|·| 2 ] de- notes the statistical averaging. For slow frequency-nonselective fading channel, the received complex baseband signal at an L-branch diversity receiver in the ith symbol interval can be given as [6] r(i)= c(i)d(i)+ n(i), (1) where the elements of the noise vector n(i)=[n 1 (i),n 2 (i), ...,n L (i)] T are independent and identically distributed (iid) zero-mean complex Gaussian processes with power spec- tral density E[|n l (i)| 2 ]= N 0 for l =1, 2,...,L. And c(i)= [c 1 (i),c 2 (i),...,c L (i)] T is the channel-coefficient vector for L branches and its each element is κ − μ distributed. The κ − μ distribution models the non-homogeneous fading channels hav- ing LoS components. The envelope, |c l | of κ − μ fading chan- nels can be modeled in terms of in-phase and quadrature-phase components as [10] |c l | 2 = m  j=1 (X j + p j ) 2 + m  j=1 (Y j + q j ) 2 , (2) where X j and Y j are zero mean Gaussian processes with E[|X j | 2 ]= E[|Y j | 2 ]= σ 2 c , and p j and q j are mean of in-phase and quadrature-phase components. Each channel-coefficient element can be given as c l = c f,l + c d,l , where c f,l and c d,l are diffused and dominant components, respectively. The mean value of |c l | 2 is Ω c,l = E[|c l | 2 ]= δ 2 c,l + d 2 c,l , where δ 2 c,l = 2mσ 2 c,l and d 2 c,l = ∑ m j=1 (p 2 j + q 2 j ). The envelope and 2162-2337 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.