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Multiple access interference in MIMOCDMA systems under Rayleigh fading: statistical characterization and applications
 Khalid Mahmood^{1},
 Syed Muhammad Asad^{1},
 Muhammad Moinuddin^{2},
 Azzedine Zerguine^{3}Email author and
 Lahouari Cheded^{4}
https://doi.org/10.1186/s136340160338y
© Mahmood et al. 2016
 Received: 9 June 2015
 Accepted: 17 March 2016
 Published: 4 April 2016
Abstract
A major limiting factor in the performance of multipleinputmultipleoutput (MIMO) code division multiple access (CDMA) systems is multiple access interference (MAI) which can reduce the system’s capacity and increase its bit error rate (BER). Thus, a statistical characterization of the MAI is vital in analyzing the performance of such systems. Since the statistical analysis of MAI in MIMOCDMA systems is quite involved, especially when these systems are fading, existing works in the literature, such as successive interference cancellation (SIC) or parallel interference cancellation (PIC), employ suboptimal approaches to detect the subscriber without involving the need for MAI statistics. The knowledge of both MAI and noise statistics plays a vital role in various applications such as the design of an optimum receiver based on maximum likelihood (ML) detection, evaluation of the probability of bit error, calculation of the system’s capacity, evaluation of the outage probability, estimation of the channel’s impulse response using methods including the minimum meansquareerror (MMSE), the maximum likelihood(ML), and the maximum a posteriori probability (MAP) criterion. To the best of our knowledge, there is no existing work that explicitly evaluates the statistics of the MAIplusnoise in MIMO fading channels. This constitutes the prime objective of our proposed study here. In this work, we derive the expressions for the probability density function (PDF) of MAI and MAIplusnoise in MIMOCDMA systems in the presence of both Rayleigh fading channels and additive white Gaussian noise. Moreover, we evaluate the probability of the bit error rate in the presence of optimum reception using a ML receiver. Our theoretical findings can provide a reliable basis for both system design and various performance analyses of such systems. Our simulation results show that the theoretical findings are very well substantiated.
Keywords
 Probability Density Function
 Outage Probability
 MIMO System
 Rayleigh Fading Channel
 Multiple Access Interference
1 Introduction
In spite of their numerous advantages, MIMOCDMA systems suffer from a major drawback in MAI which can reduce their capacity and increase their BER, thus resulting in a degraded system performance. Hence, a statistical analysis of MAI becomes a very important factor in the performance analysis of these systems.
In CDMA systems, each user is assigned a unique spreading code. These orthogonal codes should ideally provide perfect isolation from other subscribers so as to maintain errorfree communication among all users. However, in reality, the orthogonality between these codes is difficult to preserve due to asynchronism and channel delay spread at the receiving end. While both asynchronism and channel delay spread exist on the uplink, only channel delay spread can be seen on the downlink of the channel. A correlation receiver, which in a multipath channel turns into a Rake receiver, cannot perfectly separate the signals in the case of multiple users. This lack of separation leads to MAI causing a system performance degradation, which may lead to unacceptable error performances for moderate user loads.
Most of the reported research work is done on characterization of singleinput singleoutput (SISO) CDMA systems and is based only on approximations, such as the standard Gaussian approximation (SGA) [1], improved Gaussian approximation (IGA) [2] and simplified IGA (SIGA) [3]. The central limit theorem is applied in SGA to get an approximate sum of additive white Gaussian noise processes (AWGNP). These approximations are widely used because of their ease of application but can exhibit major drawbacks. For example, the SGA overestimates the system’s performance, thus making the use of approximations an even more severe problem when the number of users is small [2]. The Standard Hermite polynomial error correction method was employed in [4] to improve the accuracy of SGA. In [5], the improved Gaussian approximation (IGA) method was used based on both the derivation of the conditional characteristic function of MAI and on bounds on its error probability for binary directsequence spreadspectrum multiple access (DSSSMA) systems. In the case where the number of users is low, IGA has outperformed the SGA [2] but at the cost of an increased in computational complexity which makes it a major limitation for this method. The IGA was later simplified and renamed as the simplified IGA (SIGA) in [6].
Another approach adopted by researchers to find the BER of the DS/SSMA scheme is to ignore MAI completely. Most of these techniques are basically an extension of previously studied intersymbol interference (ISI) techniques. Some of these techniques are moment space method, characteristic function method, moments method, and the approximate Fourier series method. It has been reported in the literature that these techniques are superior to the central limit theorembased techniques in approximating the BER but involve higher computational costs. A study of the signaltonoise ratio (SNR) of Rician fading channels at the correlator receiver’s output was done by [7]. The BER performance of the DSCDMA system in frequency nonselective Rayleigh fading channels for deterministic sequences using the SGA approach was evaluated in [8]. The characteristic function (CF) technique was utilized to assess the performance of the spread spectrum multiple access (SSMA) scheme in an AWGN environment in [9]. The CF method used to evaluate the performance of the DSSSMA scheme in multipath fading channels with multipath intersymbol interference was applied in [10] without taking into account the MAI effect. An approximate Fourier series technique was utilized in [11, 12] to evaluate the BER performance in selective and nonselective Rayleigh fading environments. System degradation caused by an imperfect chip and phase synchronization were also assessed in this technique.
For a given SNR, the BER dependency on the number of users is analyzed in [13], where a closedform expression for the CF of MAI for asynchronous operation in a Rayleigh fading environment was obtained. An expression for a single integral for overall BER is derived therein.
The conditional CF of MAI together with bounds on the probability error rate for the DSSSMA scheme are obtained here, whereas [14] derived only the average probability of error at the correlation receiver’s output for binary as well as quaternary synchronous and asynchronous DSSSMA schemes which use a random signature sequence.
The probability density function (PDF) of MAI for a synchronous downlink CDMA in an AWGN case was derived in [15]. This result is then used to derive the conditional probability density function of the MAI, ISI, and noise in multicarrier code division multiple access scheme (MCCDMA) provided that the fading environment is known. The PDF of both the MAI and the inter carrier interference is also derived while assuming that the channelfading effect in CDMA system is known.
A new unified approach to MAI analysis in fading environments was presented in [16], assuming that the channel phase is either known or has been perfectly estimated. Random behavior of the channel fading is also included in [16] to get realistic results for the PDF of MAI and noise. Also, the analysis does not make any simplifying assumptions on the MAI and provides a complete statistical characterization of MAI.
Accurate statistical analysis of MAI in MIMOCDMA systems has not received much attention from researchers, as indicated by the few published studies available in the literature, mainly because the computational complexities it involves. So, to alleviate these complexities, researchers in the past either used some strong assumptions, for example Gaussian assumption for interference in MIMO system [17], or suboptimal approaches to detect the subscriber without involving the need for MAI statistics such as in SIC [18] and PIC [19]. In [20, 21], other types of spreading codes are used. These codes, known as chaotic codes, are nonlinear with good correlation properties. In [20], the MAI is analyzed using the Gaussian distribution approximation and in [21] the performance was analyzed under a fading channel scenario in the MIMO case, where in contrast to existing works, an exact characterization of the MAI in a Rayleigh fading environment is developed for MIMOCDMA systems. Consequently, explicit closedform expressions for both the PDFs of MAI and MAI plus noise were derived for a Rayleigh fading channel. Recently in [22], the approach of [16] was used to analyze the MAI in MIMOCDMA systems which was later used in [23] to design minimum meansquareerror (MMSE) estimate of fading channels in the presence of MIMO systems. Although, the work in [22] has provided the derivation for the MAI statistics and the work in [23] has provided MMSE estimator design based on this statistics, but there can be many other interesting applications of the derived MAI statistics in practical scenarios which these work failed to provide. In this present work, we aimed to provide more comprehensive treatment of the MAI statistics in MIMOCDMA systems. More importantly, we have provided many interesting applications in Section 5 such as the design of optimum receiver, derivation of probability of bit error rate and the derivation for the probability of outage in MIMOCDMA systems in the presence of Rayleigh flat fading in addition to the design of MMSE estimator. Moreover, we have also highlighted some future research directions/challenges that can be dealt via utilizing the proposed framework of MAI statistics analysis.
1.1 Main contributions
 1.
Expressions for the PDF of MAI and MAI and noise in MIMOCDMA systems in the presence of both Rayleigh fading channel and additive Gaussian noise are derived.
 2.Three important applications that utilize the newlyderived statistics of MAI are provided and analyzed.
 (a)
In the first application, an optimum receiver based on ML detection criterion is developed. For this, a new expression for the probability of BER, which is of prime importance in the study of any communication system, is also derived.
 (b)
In the second application, a new expression for the minimum meansquareerror (MMSE) channel estimation of Rayleigh fading using our derived MAI statistics, is derived.
 (c)
In the final application, we derive the expression for the probability of outage for MIMOCDMA systems
 (a)
 3.
Although our analysis deals with only a Rayleigh fading channel, our methodology is nevertheless generalized and can be applied to other fading channels.
 4.
In the last section, we also highlighted some future applications for MIMOCDMA systems that can employ the proposed framework of designing MAI statistics.
1.2 Paper organization
The paper is organized as follows. Section 2 describes the system model used for the analysis. Section 3, where the analysis of the PDF of MAI is carried out is followed by Section 4 which describes the analysis of the PDF of MAIplusnoise. Two important and pertinent applications that utilize the analyses, namely the optimum receiver design using the ML criterion and the MMSE channel estimation, are presented in Section 5. Section 6 presents simulation and numerical results to corroborate the theoretical findings of this work. Finally, Section 7 concludes the paper.
2 System model
where the random variable \(I_{mn}^{l}=\sum _{k=2}^{K}A^{k}b_{n}^{l,k}\rho _{n}^{k,1}\), which is nothing but MAI in AWGN environment, is shown to follow Gaussian behavior in [16], that is, \(I_{mn}^{l}\sim \mathcal {N}\left (0,{\sigma _{I}^{2}}\right)\) where \({\sigma _{I}^{2}}=\frac {A^{2}\left (K1\right)}{N_{c}}\). For the sake of tractability of the analysis, we have assumed the same behavior for random variable \(I_{mn}^{l}\) in the MIMO system.
3 PDF of MAI in MIMOCDMA systems
where C _{ n } is defined in (59) (see Appendix A) and K _{ μ }(z) is the Bessel function for imaginary arguments [24] and it is defined in (67). The PDF of MAI in (12) for the distinct scenario shows that the MAI experienced at any receiving antenna is a sum of Laplacian distributed random variables. It can be easily seen that by setting N=1 in the above, the PDF of MAI will reduce to a single Laplacian random variable, which is consistent with the result obtained in [16].
3.1 Variance of MAI
4 PDF of MAIplusnoise
where, for compactness purposes, the symbols \(\mathcal {A}_{n}=\frac {C_{n}}{\sigma _{U_{n}}}\exp \left (\mathcal {B}_{n}\right)\), \(\mathcal {B}_{n}=\frac {\sigma _{\eta }^{2}}{2\sigma _{U_{n}}^{2}}\) and \(\mathcal {C}_{n}=4\sigma _{U_{n}}^{2}\) have been introduced.
5 Important applications involving the utilization of the exact MAI statistics

design of an optimum MIMOCDMA receiver using the ML criterion.

design of MMSE channel estimator for Rayleigh fading in MIMOCDMA systems.

derivation of probability of outage for Rayleigh fading in MIMOCDMA systems.
5.1 Design of an optimum MIMOCDMA receiver using the ML criterion
Both the design and characteristics of optimum receivers in the presence of AWGN for different modulation methods have been extensively covered in the literature [26]. It is reported in the literature that the optimum detector for a AWGN channel is comprised of a correlation demodulator or a matched filter followed by an optimum decision rule, and is based on the maximum a posteriori probability (MAP) criterion when the a priori probabilities of the transmitted signal are unequal, and the ML criterion when these a priori probabilities are equal. Decisions based on any of the criteria used depend on the conditional PDF of the received vector at the output of either the matched filter or the correlator, whichever is used. In our analysis, the ML criterion is utilized, since we are using the case where the a priori probabilities of the transmitted signal are equal.
BER is defined as the number of bits received in error with respect to the total number of bits received at a particular receiver. BER performance is considered to be a very important criterion for CDMA systems as it determines the quality of transmission as well as the amount of data that can be transmitted per unit of bandwidth. Since all users contribute to the interference levels at the receiving side, the BER of each user increases when several users try to access the same channel simultaneously. Subsequently, the maximum number allowable of users is determined by the amount of interference which can be tolerated [27]. This section deals with the design of an optimum receiver in the presence of MAIplusnoise for a Rayleigh fading environment. The probability of error is derived here for the maximum likelihood receiver and the simulation results obtained here support our analytical finding.
where \(\psi = \frac {\sigma ^{2}}{N}\left [\left (2N1\right)!!\right ]^{1/N}\), \(\left (2N1\right)!!=\left (2N1\right)\left (2N3\right)\cdots 3.1\), \(\alpha =x/\sqrt {N}\), x is the normalized Rayleigh random variable and N is the number of transmitters.
There is no closedform expression for the above integral and so it was evaluated numerically.
5.2 Design of MMSE channel estimator for Rayleigh fading in MIMOCDMA systems
Although the integrals in (44) and (46) cannot be evaluated analytically, there are various numerical integration techniques to evaluate these numerically [31]. Hence, the estimate in (45) can be used in (37) to get the MMSE estimate of the Rayleigh fading channel for MIMOCDMA system.
6 Outage probability
It is well known that outage probability is used as a performance measure for a communication link when signal to interferenceplusnoise ratio (SINR) is a random quantity. This happens when we deal with random channel as in the case fading.
7 Simulation results
In this section, we present some simulation results to validate our theoretical findings. The simulation setup consists of two different scenarios of 2×2 and 4×4 MIMO systems. The CDMA system used here relies on the use of random signature sequences of length 31. The waveform chosen for the PN signatures are rectangular chip waveforms. The channel noise is taken to be an additive white Gaussian noise with an SNR of 20 dB. The Rayleigh channel is chosen to be flat and slowfading. The simulation results are discussed according to the various tasks in the ensuing subsections.
7.1 Investigation on MAI statistics
In this section, we aim to investigate the effect of different parameters on the PDF of the MAI and MAIplusnoise, namely the effect of the number of transmitting and receiving antennas, length of the pseudorandomnoise (PN) sequence used, and the number of users in the system. Moreover, we validate the theoretically derived results for the PDF of MAI and MAIplusnoise by comparing them with various simulation experiments.
Kurtosis and variance of MAI in a 4×4 MIMO system with K=10
N _{ c }=31  N _{ c }=63  N _{ c }=127  N _{ c }=255  

Experimental kurtosis of MAI  4.02  4.11  4.16  4.21 
Experimental variance  1.1689  0.5736  0.2823  0.1411 
Analytical variance  1.1613  0.5714  0.2835  0.1412 
Experimental kurtosis of MAIplusnoise under different system’s capacity with N _{ c }=31
MIMO system  2×2  3×3  4×4 

K=4  3.77  4.28  4.63 
K=10  3.30  3.90  4.02 
K=15  3.18  3.80  3.94 
K=20  3.14  3.71  3.83 
7.2 Comparing proposed MAI statistics with the SGA
7.3 Probability of bit error rate
To validate the theoretical findings, simulations are carried out and results are discussed below. The simulation setup used random signature sequences of length 31 and rectangular chip waveforms. The measurement noise is taken to be additive white Gaussian with an SNR of 20 dB.
7.4 Channel estimation
7.5 Probability of outage
8 Conclusions
In this work, a thorough statistical analysis of MAI and MAIplusnoise in MIMOCDMA systems has been performed in the presence of Rayleigh fading channel. the major contribution of this work is the statistical characterization of MAI without relying on any Gaussian assumption as is usually reported in the literature. Consequently, the analysis results in new closedform expressions for the PDF of MAI and MAIplusnoise. It is found that the derived PDF of MAI is in the form of summation of Laplacian distributed random variables while the PDF of MAIplusnoise is found to be summation of generalized incomplete gamma functions. These PDFs are found to be a function of key factors such as the number of users, number of antennas, spreading code length, channel variance and noise variance. Moreover, the effect of these parameters on the PDF of MAI has been investigated through simulations whose results have shown a close agreement with the theoretical findings. As applications, we have demonstrated how the derived statistics can be utilized for designing the optimum coherent receiver, derivation for the expression of probability of bit error rate, derivation of MMSE channel estimator for Rayleigh fading, and derivation for the probability of outage in fading environment. For future work, our analysis can provide a platform for many interesting applications in MIMOCDMA systems. For example, one can utilize the expression for probability of outage to design a receiver that can optimization the system performance by minimizing the probability of outage. Another interesting application can be the design of optimum algorithm for MIMO antenna selection by joint minimization of receive probability of outage. In addition, our work can extended to frequency selective channels and asynchronous systems. Thus, we believe that our work will open new research directions to solve various challenging problems in MIMO communication systems.
9 Endnote
^{1} Note that \(\sigma _{U_{n}}^{2}\) is just an intermediate variable and it does not represent the true variance of the random variable \({U_{n}^{l}}\). The actual variance of the \({U_{n}^{l}}\) is equal to \(2{\sigma _{I}^{2}}\sigma _{h_{mn}}^{2}\) as \(E\left [(h_{mn}^{l})^{2}\right ]=2\sigma _{h_{mn}}^{2}\).
10 Appendix A
10.1 Evaluation of \(\frac {1}{2\pi }\int _{\infty }^{\infty }\frac {\exp \left (i\omega z\right)}{\prod _{n=1}^{N}\left (\omega ^{2}\sigma _{U_{n}}^{2}+1\right)}d\omega \)
To evaluate the integral in (11), we consider two different scenarios: (1) \(\sigma _{U_{n}}^{2}\) have distinct values for each n and (2) \(\sigma _{U_{n}}^{2}\) are equal for all n.
Scenario 1: For distinct \(\sigma _{U_{n}}^{2}\)
Finally, after substituting the result for the above integral in (60), the PDF of the MAI with distinct \(\sigma _{U_{n}}^{2}\) is found to be given in (12).
Scenario 2: For identical \(\sigma _{U_{n}}^{2}\)
Finally, the PDF of the MAI with identical \(\sigma _{U_{n}}^{2}\) is obtained after substituting the above solution and it is given in (12).
10.2 Derivation of Eq. (32)
Declarations
Acknowledgements
The authors acknowledge the support provided by the Deanship of Scientific Research at KFUPM under Research Grant SB111012.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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