Performance analysis of TCM with generalized selection combining on Rayleigh fading channels

We derive the computational cutoff rate, R/sub 0/, for coherent trellis-coded modulation (TCM) schemes on independent identically distributed (i.i.d.) Rayleigh fading channels with (K, L) generalized selection combining (GSC) diversity, which combines the K paths with the largest instantaneous signal-to-noise ratios (SNR) among the L available diversity paths. The cutoff rate is shown to be a simple function of the moment generating function (MGF) of the SNR at the output of the (K, L) GSC receiver. We also derive the union bound on the bit error probability of TCM schemes with (K, L) GSC in the form of a simple, finite integral. The effectiveness of this bound is verified through simulations.     

Performance evaluation of generalized selection diversity systems over Nakagami‐m fading channels

The generalized selection combining (GSC) scheme that adaptively combines a subset of M strongest paths out of L available diversity paths finds applications in several wideband receivers and broadband wireless communications. In this paper, exact closed‐form expressions for the moment generating function (MGF), the probability density function (PDF) and the cumulative density function (CDF) of the GSC(M, L) output signal‐to‐noise ratio (SNR) in independent and identically distributed (i.i.d) Nakagami‐m fading channels are derived while the fading index is a positive integer. These expressions hold for any M and L and provide a comprehensive framework for performance analysis including the derivation of closed‐form formulas for the average symbol error probability (ASEP) of a broad class of binary and M‐ary modulations, mean combined SNR and the outage probability of GSC(M, L) receiver structures. When the Nakagami‐m fading index is not an integer, the MGF of GSC(M, L) output SNR is derived as an (M − 1)‐fold infinite series. With this MGF, analytical expressions for both the outage probability and error rates can be readily obtained. An easily programmable recursive solution of the MGF of GSC(M, L) output SNR is also outlined for both the positive integer and noninteger fading severity index cases. Copyright © 2002 John Wiley & Sons, Ltd.     

Unified error probability analysis for generalized selectioncombining in Nakagami fading channels

We study generalized selection combining (GSC) schemes in independent Nakagami fading channels, where N diversity branches with the largest instantaneous signal-to-noise ratios (SNRs) are selected from the total of L (N⩽L) branches and then coherently or noncoherently combined. We propose two different techniques to derive the moment generating function (MGF) expressions for the GSC output SNR in generalized Nakagami fading channels, where there are distinct and noninteger fading severity parameters, as well as different average SNRs in different diversity branches. For arbitrary fading severity parameter m_k, k=1, ···L, the MGF expression is given in a summation of N-dimensional definite integrals with the limits independent of SNR or channel parameters, and therefore can be evaluated very efficiently with numerical methods. Furthermore, for integer m_k closed-form MGF expressions are derived. Specializations of our results to Rayleigh channels and independent identically distributed (i.i.d.) Nakagami channels are presented, which are either new or equivalent to previously published results. Using the newly derived MGF expression, we provide a unified error probability analysis for many coherent and noncoherent modulation/detection schemes     

Optimum spreading bandwidth for selective RAKE reception overRayleigh fading channels

Building on the developments in the performance analysis of generalized selection combining (GSC), this paper examines the optimum spreading bandwidth for a fixed-complexity GSC diversity receiver operating over independent identically distributed Rayleigh paths. For this purpose, the study considers three performance criteria: (1) average combined signal-to-noise ratio (SNR) at the GSC output; (2) average bit error probability (BEP); and (3) outage probability of the instantaneous combined SNR at the GSC output. For the average BEP criterion, results are presented for both coherent and noncoherent combining. For the average combined SNR and some instances of the average BEP optimization problem, an accurate approximate estimate of this optimum bandwidth in the form of a solution of a transcendental equation is provided. In other cases, where the optimization is not easily tractable in an analytic fashion, a numeric-search procedure is used to find this optimum bandwidth for different performance criteria and system parameters of interest. Finally, simplified rule-of-thumb-type formulas are also presented as a good reference for picking the optimum spreading bandwidth given a set of system parameters and a particular performance criterion of interest     

Performance of generalized selection combining receivers in K fading channels

In this letter, we present a moment generating function (MGF) based performance analysis of generalized selection combining (GSC) receivers operating over independent and identically distributed (i.i.d.) K fading channels. Analytical expressions for the marginal MGF of the signal-to-noise ratio of a single diversity branch for integer plus one-half values of the fading parameter are obtained and used to efficiently evaluate the average error probability of GSC receivers.     

Error analysis of generalized selection combining over relay channel

Cooperative communication is a recently popular concept which allows single-antenna devices to benefit from spatial diversity. The performance analysis of cooperative communication using generalized selection combining (GSC) over independent not necessarily identically distributed Nakagami-m fading channels is presented and compared with that of the conventional maximal ratio combining (MRC) and selection combining (SC) schemes. With the aid of Padé approximants theory, new closed-form expression is derived for the moment-generating function (MGF) of the GSC output signal-to-noise ratio (SNR). MGF is an important tool for researching the system performance. In this paper, the average bit-error probability is accurately approximated using the well-known MGF approach. Numerical results show that the proposed mathematical analysis is accurate and that for the more severe fading cases, the GSC receivers are closer to the optimum MRC receivers.     

Performance of generalized selection combining over Weibull fading channels

The literature is relatively sparse in performance analysis of diversity combining schemes over Weibull fading channels, despite the fact that the Weibull distribution is often found to be suitably fit for empirical fading channel measurements. In this paper, we capitalize on some interesting results due to Lieblein on the order statistics of Weibull random variables to derive exact closed‐form expressions for the combined average signal‐to‐noise ratio (SNR) as well as amount of fading (AF) at a generalized selection combining (GSC) output over Weibull fading channels. We also use some simple AF‐based mappings between the fading parameters of the Weibull distribution and those of the Nakagami, Rice, and Hoyt distributions to obtain the approximate but accurate average SNR and AF of GSC over these types of channels. The mathematical equations are validated and illustrated by some numerical examples for scenarios of practical interest. Copyright © 2006 John Wiley & Sons, Ltd.     

Generalized selection combining with double threshold and performance analysis

In this paper, we propose a new diversity combining scheme to save power, which is called as the generalized selection combining with double threshold (DT‐GSC). It selects the branch whose SNR is above an input threshold to combine, and this process will keep running until the combined output SNR is larger than an output threshold or until all paths are examined. The values of both thresholds are required to be predetermined on the basis of the practical communication conditions. For comparing the complexity of various combining schemes, we will show the mathematical formulas of the average number of path estimation and the average number of combined branches. Moreover, we will also compare the average bit‐error‐ratio performance of the proposed DT‐GSC with absolute threshold GSC (AT‐GSC) and output threshold MRC (OT‐MRC). Numerical examples and simulation results show that the proposed DT‐GSC leads to a lower complexity than the conventional AT‐GSC and OT‐MRC while it has a satisfactory performance.Copyright © 2012 John Wiley & Sons, Ltd.     

General switch‐and‐examine relaying for demodulate‐and‐forward cooperative diversity

Two new demodulate‐and‐forward schemes of multi‐relay cooperative diversity with switch‐and‐examine relaying (SER) are analyzed. To reduce relay usage and enhance bandwidth efficiency, the two new cooperative diversity schemes employ a switch‐based relay selection. The proposed schemes consume less communication resource than regular relaying schemes, such as the selection combining (SC) or maximal ratio combining (MRC) schemes that always use all relays, and also achieve better performance than distributed switch‐and‐stay schemes. In the first scheme, the decision statistic for relay usage and selection is based on the signal‐to‐noise ratio (SNR). In the second scheme, the log‐likelihood ratio (LLR) of received signals is used for the decision of relay usage and selection. With the two SER schemes, the bit error probability (BEP) of binary phase shift keying (BPSK) and the average number of used paths are derived and expressed in closed‐form for the independent and identically distributed (i.i.d.) Rayleigh fading channels. Numerical and simulation results are presented for performance illustrations. According to the numerical results, the LLR‐based SER not only achieves a lower BEP but also consumes less relay resource than the SNR‐based SER. Furthermore, the LLR‐based SER scheme even outperforms the corresponding SNR‐based SC scheme for a range of average SNR. Copyright © 2014 John Wiley & Sons, Ltd.     

Performance of M-PSK with GSC and EGC with Gaussian weighting errors

Using a moment-generating function (MGF)-based approach, we study the performance of M-ary phase-shift keying (M-PSK) with generalized selection combining (GSC) and equal gain combining (EGC) in fading channels (including Rayleigh, Rician, Nakagami-m, and Nakagami-q fading) with independent and identically distributed (i.i.d) branches. Analytical expressions for the error and outage probabilities, the signal-to-noise-ratio (SNR) statistics, and the channel capacity of M-PSK diversity receivers are derived, taking into account the effects of Gaussian weighting errors and all relevant system and channel parameters. Unlike the case of perfect channel-state information (CSI), the outage probability for the case of imperfect channel estimation (ICE) is not only a function of the normalized SNR with respect to the SNR threshold, but also a function of the operating SNR itself. The SNR loss of the M-PSK GSC and EGC receivers due to ICE and the relation between the receiver input and output SNRs for ICE are derived. Our results show that, even with ICE, GSC and EGC are effective in improving the output SNR and significantly reduce the error floor and the channel-capacity loss caused by ICE.     

Exact Error Rate Analysis of Output-Threshold Generalized Selection Combining (OT-GSC)

Generalized selection combining (GSC) is one of the most widely investigated low-complexity diversity techniques. Recently, output-threshold GSC (OT-GSC) was proposed as a power-saving variant of traditional GSC scheme. In this paper, we present an exact performance analysis of OT-GSC over Rayleigh fading channels. In particular, we derive the cumulative distribution function (CDF), probability density function (PDF) and moment generation function (MGF) of the combined SNR with OT-GSC. Then, we apply these results to study the outage probability and average error rate performance of OT-GSC. This analysis allows for a thorough and accurate comparison of OT-GSC with other variants of GSC, such as minimum selection GSC.     

Efficient performance evaluation for generalized selection combining on generalized fading channels

The authors propose an efficient moment generating function (MGF)-based method to evaluate the performance of generalized selection combining (GSC) over different fading channels. Employing a recently proposed method which is, however, only applicable to GSC diversity with independent and identically distributed branches, they derive a general MGF expression for the GSC output signal-to-noise ratio (SNR) for generalized fading channels, where the channel statistics in different diversity branches may be nonidentical or even distributed according to different distribution families. The resulting MGF expression is applicable to the analysis of the error probability, the outage probability, and the SNR statistics for GSC in a number of wireless communications scenarios with generalized fading. Numerical examples are presented to illustrate the application of the new analysis.     

Theoretical diversity improvement in GSC(N,L) receiver with nonidentical fading statistics

The study on generalized selection combining (GSC(N,L)) diversity systems that adaptively combines a subset of N paths with the highest instantaneous signal-to-noise ratios (SNRs) out of L available diversity paths has both theoretical and practical importance in the design of low-complexity receiver structures for cellular wideband CDMA, indoor millimeter-wave and ultra-wideband communications. This paper presents a novel mathematical framework to tackle the problem at hand by deriving a single integral expression for the moment generating function (mgf) of the GSC(N,L) output SNR when the L resolvable multipaths are independent with nonidentical fading statistics. The mgf is then used to unify the performance evaluation of a broad range of digital modulation/detection schemes in practical wireless channels.     

Performance evaluation of triple-branch GSC diversity receivers over generalized-K fading channels

In this letter, a detailed performance analysis of generalized selection combining GSC(2,3) receivers operating over independent but not necessarily identically distributed (n.i.d.) generalized-K (KG) fading channels is presented. For this class of receivers, a novel closed-form expression for the moments of the output signal-to-noise ratio (SNR) is derived. This result can be afterwards used to evaluate the outage probability and the average symbol error probability of different signal constellations. Various performance evaluation results are also presented and compared to equivalent simulation ones.     

Finger assignment schemes for RAKE receivers with multiple-way soft handover

We propose and analyze new finger assignment techniques that are applicable for RAKE receivers in the soft handover (SHO) region. Specifically, extending the results for the case of two-base station (BS), we consider the multi-BS situation, attack the statistics of several correlated generalized selection combining (GSC) stages, and provide closed-form expressions for the statistics of the output signal-to-noise ratio (SNR). By investigating the tradeoff among the error performance, the average number of required path estimations/comparisons, and the SHO overhead, we show through numerical examples that the new schemes offer commensurate performance in comparison with more complicated GSC-based diversity systems while requiring a smaller estimation load and SHO overhead.     

The effect of Gaussian error in selection diversity combiners

Selection diversity combining (SDC) is one of the simplest and most commonly implemented diversity mechanism for mitigating the detrimental effects of deep fades experienced on wireless channels. While SDC improves the mean combined signal‐to‐noise ratio (SNR) over that of a single branch with increasing diversity order, its main advantage is the reduction of the probability of deep fades. The effect of Gaussian errors in the branch gain estimates on the SDC receiver performance is investigated by deriving new closed‐form expressions for the probability density function, cumulative distribution function and the moment generating function of the combiner output SNR statistic. Mathematical expressions for quantifying the degradation in the mean combined SNR, outage probability and the average symbol error rate of a broad class of binary and multilevel modulation schemes owing to imperfect branch SNR estimates in Rayleigh fading are also derived. It is shown that combiner errors affect the mean combined SNR negligibly in comparison to their effect on the deep fades. Copyright © 2001 John Wiley & Sons, Ltd.     

Spectral efficiency of dual diversity selection combining schemes under correlated Nakagami-0.5 fading with unequal average received SNR

The spectral efficiency results for different adaptive transmission schemes over correlated diversity branches with unequal average signal to noise ratio (SNR) obtained so far in literature are not applicable for Nakagami-0.5 fading channels. In this paper, we investigate the effect of fade correlation and level of imbalance in the branch average received SNR on the spectral efficiency of Nakagami-0.5 fading channels in conjunction with dual-branch selection combining (SC). This paper derived the expressions for the spectral efficiency over correlated Nakagami-0.5 fading channels with unequal average received SNR. This spectral efficiency is evaluated under different adaptive transmission schemes using dual-branch SC diversity scheme. The corresponding expressions for Nakagami-0.5 fading are considered to be the expressions under worst fading conditions. Finally, numerical results are provided to illustrate the spectral efficiency degradation due to channel correlation and unequal average received SNR between the different combined branches under different adaptive transmission schemes. It has been observed that optimal simultaneous power and rate adaptation (OPRA) scheme provides improved spectral efficiency as compared to truncated channel inversion with fixed rate (TIFR) and optimal rate adaptation with constant transmit power (ORA) schemes under worst case fading scenario. It is very interesting to observe that TIFR scheme is always a better choice over ORA scheme under correlated Nakagami-0.5 fading channels with unequal average received SNR.     

Soft handover overhead reduction by RAKE reception with finger reassignment

We propose and analyze in this paper a new finger assignment technique that is applicable for RAKE receivers when they operate in the soft handover (SHO) region. This scheme employs a new version of generalized selection combining (GSC). More specifically, in the SHO region, the receiver uses by default only the strongest paths from the serving base station (BS) and only when the combined signal-to-noise ratio (SNR) falls below a certain pre-determined threshold, the receiver uses more resolvable paths from the target BS to improve the performance. Hence, relying on some recent results on order statistics we attack the statistics of two correlated GSC stages and provide the approximate but accurate closed-form expressions for the statistics of the output SNR. By investigating the tradeoff among the error performance, the path estimation load, and the SHO overhead, we show through numerical examples that the new scheme offers commensurate performance in comparison with more complicated GSC-based diversity systems while requiring a smaller estimation load and SHO overhead.     

SNR of generalized diversity selection combining with nonidenticalRayleigh fading statistics

A closed-form expression for the average signal-to-noise-ratio (SNR) of generalized diversity selection combining, using the m largest (in instantaneous SNR) diversity signals, for arbitrary m, assuming that the Rayleigh fading statistics on each diversity branch are identically, independently distributed (i.i.d.), already exists in the literature. In this paper, a similar closed-form expression, but for nonidentically distributed statistics, is derived, This expression specializes to known results, such as the average SNRs of maximal-ratio combining with either i.i.d. or non-i.i.d. diversity statistics and that of conventional selection combining (CSC) with i.i.d. diversity statistics. In addition, it provides, for the first time, a simple closed-form solution to the combined SNR of CSC with non-i.i.d. diversity statistics. Further, the closed-form solution for the SNR of selecting the m, largest diversity signals has negligible computational complexity     

Multi-antenna Decode and Forward Relay Selection over Nakagami-m Fading Channels

This paper analyzes the performance of opportunistic relay under aggregate power constraint in Decode-and-Forward (DF) relay networks over independent, non-identical, Nakagami-m fading channels, assuming multiple antennas are available at the relay node. According to whether instantaneous Signal-to-Noise Ratio (SNR) or average SNR can be exploited for relay selection, two opportunistic relay schemes, opportunistic multi-antenna relay selection (OMRS) and average best relay selection (ABRS) are proposed. The closed form expressions of outage probability and error performance for binary phase shift keying (BPSK) modulation of OMRS and ABRS are determined using the moment generating function (MGF) of the total signal-to-noise ratio (SNR) at the destination. Simulations are provided to verify the correctness of theoretical analysis. It is observed that OMRS is outage-optimal among multi-antenna relay selection schemes and approaches the Beamforming (BF) scheme known as theoretical outage-optimal very closely. Compared with previous single-antenna Opportunistic Relaying (OR) scheme, OMRS brings remarkable performance improvement obtained from maximum ratio combining (MRC) and beamforming, which proves that multiple antennas at the relays could provide more array gain and diversity order. It also shows that the performance of ABRS in asymmetric channels is close to OMRS in the low and median SNR range.