Spectral efficiency for selection combining RAKE receivers over Weibull fading channels

Abstract

Novel closed-form expressions for the probability density function and the average output signal-to-noise ratio at the output of a selection combiner in Weibull fading are derived. Using these expressions, the spectral efficiency of a direct sequence code division multiple access system is analytically obtained and performance evaluation results are presented.

Introduction

Spectral efficiency is of primary concern in the design of modern mobile radio systems and the utilization of the associated communication channels. Spectral efficiency (SE) is defined as the data rate per unit channel bandwidth for a specified average transmitted power and a fixed bit-error-rate (BER) value. With respect to this, the maximum spectral efficiency of a communication channel of bandwidth W would be equal to C/W, where C denotes the Shannon capacity [1] of this channel.

For time-invariant channels, a variety of modulation and coding schemes have been investigated in order to further increase the already achieved spectral efficiency, while, for single-user channels, Shannon capacity can readily be used to estimate the upper bound of the achievable spectral efficiency, in bits/s/Hz. Commonly, in any communication system, physical channels can be shared among all users according to one of three basic accessing schemes: time division multiple access, frequency division multiple access, and code division multiple access (CDMA). In an ideal additive white Gaussian noise (AWGN) environment, the theoretically achieved spectral efficiency of a radio system utilizing any of those multiple access techniques will be the same, since it does not matter whether the available signal space (time and bandwidth) is divided into time-slots, frequencies, or codes. However, in mobile radio, physical channels exhibit randomly time-varying characteristics that result in signal fading and substantial capacity degradation. Hence, the theoretical equivalence of the three multiple-access schemes does not exist and the spectral efficiency achieved by each of them depends on the physical model of the fading channel, i.e. on what is known about the particular channel. For example, if nothing is known for the fading statistics of a channel then its countable capacity (in bits/s) will be dictated by the minimum (due to fading) signal-to-noise ratio (SNR), γ_min, and will thus tend to zero as ${\gamma }_{\mathit{min}}\to 0$. On the other hand, if fading statistics is known, then an “average” capacity formula can be derived after the distribution of the fading SNR γ for a fixed transmission rate [2].

One of the simplest and yet most efficient techniques to overcome the destructive effects of fading in wireless communication systems is diversity. For all diversity techniques the receiver has to process the obtained diversity signals in a fashion that maximizes the system's power efficiency. There are several diversity reception methods employed in communication receivers including equal gain combining (EGC), maximal ratio combining (MRC), selection combining (SC), and a combination of MRC and SC, called generalized selection combining (GSC). MRC is the optimal combining scheme but it comes at the expense of increased complexity since knowledge of all channel parameters, which usually affect the received signal, is required. EGC provides an intermediate solution for improved overall performance and low implementation complexity. For GSC(L, L_c), the L_c strongest signal(s) out of L available diversity branches are optimally combined. However, when L is large, GSC will give poorer error performance than MRC or EGC. SC is the least complicated since the processing is performed only on one of the diversity branches. Traditionally, in SC, the combiner will choose the branch with the highest SNR, which corresponds to the strongest signal, if equal noise power is assumed among the branches [3].

The performance of diversity receivers has been studied extensively in the past for several well-known fading channel models, such as Rayleigh, Rice, Nakagami-m and Nakagami-q, assuming independent or correlative fading [3]. However, another well-known fading channel model, namely the Weibull model, has not yet received as much attention, despite the fact that it provides an excellent fit to experimental fading channel measurements for both indoor [4] and outdoor environments [5]. Only very recently, the topic of communications in Weibull fading channels has begun to receive renewed interest. For example, Alouini and Simon have presented an analysis for the evaluation of GSC diversity receiver's performance [6].

In this paper, the probability density function (PDF) and the average SNR at the output of a SC diversity receiver in Weibull fading environment, is derived analytically in closed-form expressions, with arbitrary parameters for the channel's severity of fading as well as the number of diversity branches. Using these expressions and following the methodology presented in [7], the achievable SE of a non-cooperative direct sequence code division multiple access (DS-CDMA) system, assuming SC RAKE reception, is obtained.

The remainder of this paper is organized as follows: in Section 2, the statistical analysis of the SC output SNR in Weibull fading is performed. Following this, in Section 3, the average SE in DS-CDMA systems is obtained. In Section 4, numerical results are presented and finally, concluding remarks are presented in Section 5.