132 IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 5, MAY 1999 Post-Detection Diversity in Nakagami Fading Channels with Correlated Branches Pierfrancesco Lombardo, Member, IEEE, and Gennaro Fedele, Member, IEEE Abstract—We derive a closed-form expression for the perfor- mance of the post-detection product detector combiner (PDC) operating on correlated branches in Nakagami fading. We con- sider the 2-DPSK signaling scheme and nonselective slow fading. The average bit error rate (BER) obtained with this scheme is compared to the ideal predetection MRC, showing limited loss. Moreover, the post-detection PDC is shown to outperform the selection diversity combiner (SDC) under the considered case of Nakagami fading. Index Terms— Bit error probability, differential phase-shift keying, diversity reception, nonselective fading channels. I. INTRODUCTION I N MOBILE communications the received signal amplitude can be affected by fade phenomena due to multipath propagation; a general model for the fading was proposed by Nakagami [1], under the name of -distribution. Performance improvement can be achieved by using an branch diversity receiver, though, in some practical mobile systems, branch correlation reduces diversity gain and must be accounted for in the system design. For independent branches, the predetection MRC is the optimum scheme and is frequently considered since its performance gives an upper bound for suboptimal combiners. In [2] we presented the BER of binary signaling on Nakagami channels with predetection MRC and correlated branches. In practice, the coherent detection is difficult to implement on faded channels, and signaling schemes are preferred, which operate with differential modulation and detection. Moreover, predetection MRC has a high implemen- tation complexity, therefore, it is of interest to analyze the performance of simpler diversity schemes. We consider the product detector [3]–[5]. A differential product is evaluated at each branch and the decision variable is the unweighted sum of the outputs of the differential products. This combiner will be referred to as post-detection product detector combiner (PDC), while in [4] it is named post-detection equal gain (EG) combiner and in [5] post-detection MRC. In this letter, we analyze the performance of 2-DPSK signals on -distributed fading channels using an branch post-detection PDC with generic branch correlation. II. SYSTEM MODEL The received signal is corrupted by an AWGN which is assumed statistically independent on each branch. In the Manuscript received January 16, 1998. The associate editor coordinating the review of this letter and approving it for publication was Prof. K.-C. Chen. The authors are with the InfoCom Department, Universit´ a di Roma “La Sapienza,” 00184 Rome, Italy (e-mail: pier@infocom.ing.uniroma1.it). Publisher Item Identifier S 1089-7798(99)04123-X. Fig. 1. Diversity receiver model. complex baseband model, the received signal at the th bit interval at the input of the th branch is (1) where represents the th transmitted signal. For binary PSK, and , being an unit energy pulse. is AWGN with one sided power spectral density in watts per hertz (W/Hz) units, is the fading phase shift, and is the fading amplitude which follows the Nakagami- pdf [1] (2) where is the Gamma function, the mean square value of and the fading severity. We consider nonselective and slow fading so that and remain constant over at least two consecutive bit intervals. As shown in Fig. 1, after matched filtering the signal on each branch, the post-detection PDC takes the unweighted sum of the differential detectors outputs, and its decision variable, which is tested for being positive or negative, is (3) where and are the AWGN components at the output of the matched filters in two consecutive bit intervals. The instantaneous SNR per bit , at the output of the combiner, is equal to the sum of the signal-to-noise ratios (SNR’s) [6], where is the SNR at the input of the detector in the th branch (4) 1089–7798/99$10.00  1999 IEEE