308 IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 11, NOVEMBER 1999 Virtual Path Analysis of Selective Rake Receiver in Dense Multipath Channels Moe Z. Win, Senior Member, IEEE, and Zoran A. Kostic, Senior Member, IEEE Abstract—We develop an analytical framework to quantify the effects of spreading bandwidth on spread-spectrum systems oper- ating in dense multipath environments. Closed-form expressions for the mean and variance of the total Rake receiver output signal-to-noise ratio are derived. The proposed problem is made analytically tractable by transforming the physical Rake paths into the virtual path domain. Index Terms— Dispersive channels, diversity methods, fading channels, maximal-ratio combining, Rake receiver, spreading bandwidth, spread-spectrum techniques, selection diversity, vir- tual path technique. I. INTRODUCTION S PREAD-SPECTRUM (SS) systems have the ability to re- solve closely spaced multipath components. The detection of SS signals in a multipath environment leads to a Rake receiver [1]. One version of the Rake receiver consists of multiple correlators (fingers). The equivalent matched filter version of the receiver involves a matched front-end processor (MFEP) (matched only to the transmitted signature waveform) followed by a tap delay line and a combiner. We use the term all Rake (ARake) receiver to describe the receiver with unlimited resources (correlators or taps) so that it can, in principle, combine all of the resolved multipath components for the best possible performance. For a dense multipath channel, the number of resolvable multipath com- ponents increases with the spreading bandwidth. However, the number of multipath components that can be utilized in a typical Rake combiner is limited by power consumption and design complexity [2]–[4]. Complexity and performance issues have motivated studies of selective Rake (SRake) receivers. The SRake is a reduced- complexity multipath combining receiver, which selects the best paths (from available diversity paths) and com- bines the selected components using maximal-ratio combining (MRC) [1]. For a given transmission bandwidth and for a typical power delay profile (or alternatively the worst one), two fundamental questions related to the SRake receiver design are as follows: 1) How does the average SNR at the output of the SRake receiver increase with an increasing number of fingers? Manuscript received October 27, 1998. The associate editor coordinating the review of this letter and approving it for publication was Prof. C. D. Georghiades. The authors are with the Wireless Systems Research Department, Newman Springs Laboratory, AT&T Labs—Research, Red Bank, NJ 07701-7033 USA (e-mail: win@research.att.com). Publisher Item Identifier S 1089-7798(99)09920-2. 2) What is the behavior of the SNR fluctuation at the output of the SRake receiver as a function of the number of fingers? Typical studies on the performance of code-division multiple-access (CDMA) systems as a function of the number of Rake fingers assume and study the performance gain as increases [5], [6]. These studies were made for an ARake receiver, and do not address the performance improvement vs. receiver complexity tradeoff. However, as pointed out in [6], a more realistic model is the Rake receiver tracking the strongest out of paths. In fact, quasi- analytical/experimental analysis of the ultrawide bandwidth SRake receiver in [3] shows that a high diversity order can be achieved even with a single path (SP) receiver. Recently, both theoretical and simulation-based methods have been used to investigate the effects of multipath fading [7]–[11]. A closely related problem of hybrid selection/maximal-ratio combining (H-S/MRC) was considered in a more general diversity setting [12], [13]. The principal contributions of this paper are in discovering a methodology and deriving closed form expressions for the mean and variance of the SRake combiner for arbitrary and (i.e., arbitrary spreading bandwidth). Third-generation CDMA cellular systems will operate around 4–5 Mchips/s. The designers will face the question of how many fingers should be active in a Rake receiver. The results of this paper provide a part of the necessary inputs for making that decision. II. SRAKE RECEIVER PERFORMANCE ANALYSIS In this section, the theory developed in [13] for an H-S/MRC receiver is applied to the study of a SRake receiver. A. The SRake Receiver We model the instantaneous SNR of the MFEP output samples ’s as continuous random variables with mean The instantaneous output SNR of the SRake receiver is where . The quantity is the ordered , i.e., Note that the possibility of at least two equal is excluded, since almost surely for continuous random variables . 1 We consider a wide-sense stationary uncorrelated scattering (WSSUS) channel with frequency-selective fading. The chan- nel is modeled as a 2-D complex circular Gaussian process with zero mean and constant power delay profile. Several researchers have previously considered this type of delay 1 In our context, the notion of “almost sure” or “almost everywhere” can be stated mathematically as [14] if then 1089–7798/99$10.00  1999 IEEE