Effective interference and effective bandwidth of linear multiuserreceivers in asynchronous CDMA systems

The performance of linear multiuser receivers in terms of the signal-to-interference ratio (SIR) achieved by the users has been analyzed in a synchronous CDMA system under random spreading sequences. In this paper, we extend these results to a symbol-asynchronous but chip-synchronous system and characterize the SIR for linear receivers-the matched-filter receiver the minimum mean-square error (MMSE) receiver and the decorrelator. For each of the receivers, we characterize the limiting SIR achieved when the processing gain is large and also derive lower bounds on the SIR using the notion of effective interference. Applying the results to a power controlled system, we derive effective bandwidths of the users for these linear receivers and characterize the user capacity region: a set of users is supportable by a system if the sum of the effective bandwidths is less than the processing gain of the system. We show that while the effective bandwidth of the decorrelator and the MMSE receiver is higher in an asynchronous system than that in a synchronous system, it progressively decreases with the increase in the length of the observation window and is asymptotic to that of the synchronous system, when the observation window extends infinitely on both sides of the symbol of interest. Moreover, the performance gap between the MMSE receiver and the decorrelator is significantly wider in the asynchronous setting as compared to the synchronous case     

Large-system performance analysis of blind and group-blindmultiuser receivers

We present a large-system performance analysis of blind and group-blind multiuser detection methods. In these methods, the receivers are estimated based on the received signal samples. In particular, we assume binary random spreading, and let the spreading gain N, the number of users K, and the number of received signal samples M all go to infinity, while keeping the ratios K/N and M/N fixed. We characterize the asymptotic performance of the direct-matrix inversion (DMI) blind linear minimum mean-square error (MMSE) receiver, the subspace blind linear MMSE receiver, and the group-blind linear hybrid receiver. We first derive the asymptotic average output signal-to-interference-plus-noise ratio (SINR) for each of these receivers. Our results reveal an interesting "saturation" phenomenon: The output SINR of each of these receivers converges to a finite limit as the signal-to-noise ratio (SNR) of the desired user increases, which is in stark contrast to the fact that the output SINR achieved by the exact linear MMSE receiver can get arbitrarily large. This indicates that the capacity of a wireless system with blind or group-blind multiuser receivers is not only interference-limited, but also estimation-error limited. We then show that for both the blind and group-blind receivers, the output residual interference has an asymptotic Gaussian distribution, independent of the realizations of the spreading sequences. The Gaussianity indicates that in a large system, the bit-error rate (BER) is related to the SINR simply through the Q function     

Asymptotic normality of linear multiuser receiver outputs

This paper proves large-system asymptotic normality of the output of a family of linear multiuser receivers that can be arbitrarily well approximated by polynomial receivers. This family of receivers encompasses the single-user matched filter, the decorrelator, the minimum mean square error (MMSE) receiver, the parallel interference cancelers, and many other linear receivers of interest. Both with and without the assumption of perfect power control, we show that the output decision statistic for each user converges to a Gaussian random variable in distribution as the number of users and the spreading factor both tend to infinity with their ratio fixed. Analysis reveals that the distribution conditioned on almost all spreading sequences converges to the same distribution, which is also the unconditional distribution. This normality principle allows the system performance, e.g., the multiuser efficiency, to be completely determined by the output signal-to-interference ratio (SIR) for large linear systems.     

Linear multiuser detectors for incompletely known symmetric signals in CDMA systems

In this paper, we consider a synchronous code-division multiple-access (CDMA) system with a multiuser receiver. All users are assumed to have symmetric signature sequences, but the presence of a subset of the users is unknown to the receiver. We first calculate the signal-to-interference ratio (SIR) in this environment for the matched-filter receiver, the decorrelating receiver, and the linear minimum mean-square error (MMSE) detector. We then identify the user capacity for a single-class system, and the effective bandwidth for a multiple-class system. The result is compared to the case of random sequences and of optimum sequences. For symmetric sequences, the effective bandwidth cannot be expressed by a scalar as in , because two constraints have to be satisfied simultaneously to satisfy the SIR requirement. We introduce a two-dimensional (2-D) vector notion of effective bandwidth with and without unknown users. For both the decorrelator and the MMSE detector, the user capacity is 1 when all users are known to the receivers and is reduced to (1-N/L) when N users are unknown (with L the processing gain). The performance of these three linear detectors, with and without unknown users, is compared.     

Resource pooling and effective bandwidths in CDMA networks withmultiuser receivers and spatial diversity

Much of the performance analysis on multiuser receivers for direct-sequence code-division multiple-access (CDMA) systems is focused on worst case near-far scenarios. The user capacity of power-controlled networks with multiuser receivers are less well-understood. Tse and Hanly (see ibid., vol.45, p.541-657, 1999) have shown that under some conditions, the user capacity of an uplink power-controlled CDMA cell for several important linear receivers can be very simply characterized via a notion of effective bandwidth. We show that these results extend to the case of antenna arrays. We consider a CDMA system consisting of users transmitting to an antenna array with a multiuser receiver, and obtain the limiting signal-to-interference (SIR) performance in a large system using random spreading sequences. Using this result, we show that the SIR requirements of all the users can be met if and only if the sum of the effective bandwidths of the users is less than the total number of degrees of freedom in the system. The effective bandwidth of a user depends only on its own requirement. Our results show that the total number of degrees of freedom of the whole system is the product of the spreading gain and the number of antennas. In the case when the fading distributions to the antennas are identical, we show that a curious phenomenon of “resource pooling” arises: the multiantenna system behaves like a system with only one antenna but with the processing gain the product of the processing gain of the original system and the number of antennas, and the received power of each user the sum of the received powers at the individual antennas     

Linear multiuser receivers: effective interference, effectivebandwidth and user capacity

Multiuser receivers improve the performance of spread-spectrum and antenna-array systems by exploiting the structure of the multiaccess interference when demodulating the signal of a user. Much of the previous work on the performance analysis of multiuser receivers has focused on their ability to reject worst case interference. Their performance in a power-controlled network and the resulting user capacity are less well-understood. We show that in a large system with each user using random spreading sequences, the limiting interference effects under several linear multiuser receivers can be decoupled, such that each interferer can be ascribed a level of effective interference that it provides to the user to be demodulated. Applying these results to the uplink of a single power-controlled cell, we derive an effective bandwidth characterization of the user capacity: the signal-to-interference requirements of all the users can be met if and only if the sum of the effective bandwidths of the users is less than the total number of degrees of freedom in the system. The effective bandwidth of a user depends only on its own SIR requirement, and simple expressions are derived for three linear receivers: the conventional matched filter, the decorrelator, and the MMSE receiver. The effective bandwidths under the three receivers serve as a basis for performance comparison     

Large system performance of linear multiuser receivers in multipathfading channels

A linear multiuser receiver for a particular user in a code-division multiple-access (CDMA) network gains potential benefits from knowledge of the channels of all users in the system. In fast multipath fading environments we cannot assume that the channel estimates are perfect and the inevitable channel estimation errors will limit this potential gain. We study the impact of channel estimation errors on the performance of linear multiuser receivers, as well as the channel estimation problem itself. Of particular interest are the scalability properties of the channel and data estimation algorithms: what happens to the performance as the system bandwidth and the number of users (and hence channels to estimate) grows? Our main results involve asymptotic expressions for the signal-to-interference ratio of linear multiuser receivers in the limit of large processing gain, with the number of users divided by the processing gain held constant. We employ a random model for the spreading sequences and the limiting signal-to-interference ratio expressions are independent of the actual signature sequences, depending only on the system loading and the channel statistics: background noise power, energy profile of resolvable multipaths, and channel coherence time. The effect of channel uncertainty on the performance of multiuser receivers is succinctly captured by the notion of effective interference     

Linear receivers for the multiple-input multiple-output multiple-access channel

We consider a multiuser multiple-input multiple-output (MIMO) communication system using code-division multiple access (CDMA) and multiuser detection to discriminate the different users. Our focus is on the CDMA uplink of a frequency-nonselective Rayleigh fading channel. We study two types of receivers: joint receivers, which address simultaneously both spatial and multiple-access interference; and separate receivers, addressing the two types of interference individually. This approach allows assessing the benefits of adding MIMO processing capabilities to existing multiuser single-input single-output systems. For both receiver types, we analyze solutions based on linear (matched filter, decorrelator, minimum mean-square error) and maximum-likelihood receivers. For all the receivers considered, we provide closed-form expressions (as expectations of given functions) of the resulting pairwise error probabilities. Performance results are obtained in terms of frame-error rate versus E/sub b//N/sub 0/, following two different approaches. An analytic approach using large-system asymptotic methods, whereby the system parameters (number of users and antennas, spreading gain) are assumed to grow to infinity with finite limiting ratios. A computer-simulation approach is used to illustrate the differences between asymptotic and simulation results.     

CDMA systems in fading channels: admissibility, network capacity,and power control

We study the admissibility and network capacity of imperfect power-controlled code-division multiple access (CDMA) systems with linear receivers in fading environments. In a CDMA system, a set of users is admissible if their simultaneous transmission does not result in violation of any of their quality-of-service (QoS) requirements; the network capacity is the maximum number of admissible users. We consider a single-cell imperfect power-controlled CDMA system, assuming known received power distributions. We identify the network capacities of single-class systems with matched-filter (MF) receivers for both the deterministic and random signature cases. We also characterize the network capacity of single-class systems with linear minimum-mean-square-error (MMSE) receivers for the deterministic signature case. The network capacities can be expressed uniquely in terms of the users' signal-to-interference ratio (SIR) requirements and received power distributions. For multiple-class systems equipped with MF receivers, we find a necessary and sufficient condition on the admissibility for the random signature case, but only a sufficient condition for the deterministic signature case. We also introduce the notions of effective target SIR and effective bandwidth, which are useful in determining the admissibility and hence network capacity of an imperfect power-controlled system     

Asymptotic Performance of Linear Receivers in MIMO Fading Channels

Linear receivers are an attractive low-complexity alternative to optimal processing for multiple-antenna multiple-input multiple-output (MIMO) communications. In this paper, we characterize the information-theoretic performance of MIMO linear receivers in two different asymptotic regimes. For fixed number of antennas, we investigate the limit of error probability in the high-signal-to noise-ratio (SNR) regime in terms of the diversity-multiplexing tradeoff (DMT). Following this, we characterize the error probability for fixed SNR in the regime of large (but finite) number of antennas.As far as the DMT is concerned, we report a negative result: we show that both linear zero-forcing (ZF) and linear minimum mean- square error (MMSE) receivers achieve the same DMT, which is largely suboptimal even in the case where outer coding and deAcircnot coding is performed across the antennas. We also provide an apAcircnot proximate quantitative analysis of the markedly different behavior of the MMSE and ZF receivers at finite rate and nonasymptotic SNR, and show that while the ZF receiver achieves poor diversity at any finite rate, the MMSE receiver error curve slope flattens out progressively, as the coding rate increases. When SNR is fixed and the number of antennas becomes large, we show that the mutual information at the output of an MMSE or ZF linear receiver has fluctuations that converge in distribution to a Gaussian random variable, whose mean and variance can be characterized in closed form. This analysis extends to the linear reAcircnot ceiver case a well-known result previously obtained for the optimal receiver. Simulations reveal that the asymptotic analysis captures accurately the outage behavior of systems even with a moderate number of antennas.     

How fading affects CDMA: an asymptotic analysis with linearreceivers

Using asymptotic analysis, we study the effect of frequency-flat fading on code division multiple access (CDMA) systems with linear receivers and random spreading sequences. Specifically, we let the number of users grow without bound, while the ratio of number of users to spreading sequence length is kept fixed to a value α. We treat separately the cases of slow fading (nonergodic channel) and of fast fading (ergodic channel). For the former channel, we derive the outage probability, while for the latter we compute the channel capacity. In both cases, multiple classes of users with different qualities of service are dealt with. As α→∞, the system throughput tends to the same limit of 1.44 bit/symbol as for the nonfading channel with both single-user matched filter (SUMF) and linear minimum mean-square-error (MMSE) receivers. The outage probability exhibits a floor for all α with the SUMF receivers, while with MMSE receiver the floor is present only for α>1. We also address the tradeoffs involved in the allocation of available bandwidth between spreading and coding     

Output MAI distributions of linear MMSE multiuser receivers inDS-CDMA systems

Multiple-access interference (MAI) in a code-division multiple-access (CDMA) system plays an important role in performance analysis and characterization of fundamental system limits. We study the behavior of the output MAI of the minimum mean-square error (MMSE) receiver employed in the uplink of a direct-sequence (DS)-CDMA system. We focus on imperfect power-controlled systems with random spreading, and establish that in a synchronous system (1) the output MAI of the MMSE receiver is asymptotically Gaussian, and (2) for almost every realization of the signatures and received powers, the conditional distribution of the output MAI converges weakly to the same Gaussian distribution as in the unconditional case. We also extend our study to asynchronous systems and establish the Gaussian nature of the output interference. These results indicate that in a large system the output interference is approximately Gaussian, and the performance of the MMSE receiver is robust to the randomness of the signatures and received powers. The Gaussianity justifies the use of single-user Gaussian codes for CDMA systems with linear MMSE receivers, and implies that from the viewpoints of detection and channel capacity, signal-to-interference ratio (SIR) is the key parameter that governs the performance of the MMSE receiver in a CDMA system     

Asymptotic analysis of improved linear receivers for BPSK-CDMAsubject to fading

In this paper, we design and analyze a new class of linear multiuser detectors, which can be applied when the users employ BPSK modulation and the fading coefficients of the active users are known at the receiver (such as base-station demodulation). The tools of asymptotic distribution of the spectrum of large random matrices are used to show that relative to the classical minimum mean-square-error (MMSE) receiver, the output signal-to-noise ratio (SNR) improves by halving the number of effective interferers and adding 3 dB to the input SNR. We also propose sensible approximations to the proposed linear receivers so as to facilitate their use in CDMA systems that employ long codes     

Fast stochastic power control algorithms for nonlinear multiuser receivers

Uplink communication in a cellular radio network is considered where the base station in each cell employs linear or nonlinear (decision feedback) multiuser receivers. For any such receiver, the problem of interest is that of minimizing the total transmit power under the constraint that all the users of the network achieve their quality-of-service objective in terms of signal-to-interference ratio (SIR). When the solution is feasible for the desired SIR requirements, the optimum powers are computed with a distributed iterative power control strategy suitable for implementation at each base station. While the deterministic algorithm requires both in-cell and out-of-cell user information, the stochastic algorithm proposed in this paper can be implemented at the base stations in a truly distributed manner requiring knowledge of only in-cell parameters. Such an algorithm was proposed previously for the case where base stations use linear (single user) matched filter (MF) receivers. However, the feasibility region in terms of attainable SIRs for a well-designed multiuser receiver, particularly for a nonlinear receiver that employs decision feedback, is generally much larger than it is for the linear MF receiver. The stochastic power control algorithm in this paper, for linear or nonlinear multiuser receivers, converges in the mean-square sense to the minimal powers when the target SIRs are feasible. The second major focus of this paper is to improve the convergence properties of the conventional stochastic approximation based power control strategy by using the more recent results on averaging. Convergence issues of both the "nonaveraged" and "averaged" algorithms are investigated, and numerical examples are presented to demonstrate the performance improvement due to averaging.     

Throughput analysis of CDMA systems using multiuser receivers

Throughput bounds are attained for random channel access multichannel code-division multiple-access (CDMA) systems and spread slotted Aloha systems employing multiuser receivers. It is shown that the normalized throughput of these two systems reaches 1.0 exponentially fast in the region r/K<1, where, r is the average number of simultaneous users in each channel in the random channel access multichannel CDMA system and the packet arrival rate in the spread slotted Aloha system, respectively, and K is the maximum number of users which the multiuser receiver can handle at the same time. Therefore, both of the random channel access multichannel CDMA system and the spread slotted Aloha system employing multiuser receivers can achieve perfect throughput while being stable in the region r/K=1-δ, δ>0. The maximum throughput of the random channel access multichannel CDMA systems is found as K-√(1-(1/M))KlogK-O(logK), where M is the number of channels in the system. The maximum throughput is reached when the average number of simultaneous users is r_m=K-√((1-(1/M))KlogK))+O(√(K/logK)). The maximum throughput of the spread slotted Aloha systems is K-√(KlogK)-O(log K). The maximum throughput is reached when the packet arrival of Poisson distribution has the arrival rate λ_m=K-√(KlogK)+O(√(K/logK))     

On the asymptotic distribution of the correlation receiver output for time-hopped UWB signals

In ultra-wideband (UWB) communications based on time-hopping (TH) impulse radio, one of the most frequently studied receivers is the correlation receiver. The multiuser interference (MUI) at the output of this receiver is sometimes modeled as a Gaussian random variable. In order to justify this assumption, the conditions of validity of the Central Limit Theorem (CLT) have to be studied in an asymptotic regime where the number of interferers and the processing gain grow toward infinity at the same rate, with the channel degree being kept constant. An asymptotic study is made in this paper based on the so-called Lindeberg's condition for the CLT for martingales. Nonsynchronized users sending their signals over independent multipath channels are considered. These users may also have different powers. It is shown that when the frame length grows and the repetition factor is kept constant, then the MUI does not converge in distribution toward a Gaussian random variable. On the other hand, this convergence can be established if the repetition factor grows at the rate of the frame length. In this last situation, closed-form expressions for the signal-to-interference-plus-noise ratio (SINR) are given for TH pulse amplitude modulation (PAM) and pulse position modulation (PPM) UWB transmissions.     

Design of reduced-rank MMSE multiuser detectors using random matrix methods

Reduced-rank minimum mean-squared error (MMSE) multiuser detectors using asymptotic weights have been shown to reduce receiver complexity while maintaining good performance in long-sequence code-division multiple-access (CDMA) systems. In this paper, we consider the design of reduced-rank MMSE receivers in a general framework which includes fading, single and multiantenna receivers, as well as direct-sequence CDMA (DS-CDMA) and multicarrier CDMA (both uplink and downlink). In all these cases, random matrix results are used to obtain explicit expressions for the asymptotic eigenvalue moments of the interference autocorrelation matrix and for the asymptotic weights used in the reduced-rank receiver.     

Asymptotic Performance Analysis of Blind Minimum Output Energy Receivers for Large DS-CDMA Systems

The random matrix theory is used to analyze the asymptotic performance of the blind minimum output energy (MOE) receiver in direct-sequence code division multiple-access (DS-CDMA) systems in the presence of unknown multipath channel under the condition that the spreading factor and the number of users go to infinity with the same rate. As a special case, the asymptotic properties of the blind Capon receiver are also studied and the conditions of convergence of the signal-to-interference-plus-noise ratio (SINR) of this receiver to that of the optimal minimum-mean-square error (MMSE) receiver are discussed. In particular, it is shown that the SINR performances of the Capon and MMSE receivers are nearly identical in the uplink scenario, while the performance of the Capon receiver may be considerably inferior to that of the MMSE receiver in the downlink transmission case. As the performance of the Capon receiver is closely related to the performance of the Capon channel estimator, the asymptotic properties of the latter estimator are also studied and the conditions of convergence of the Capon channel estimate to a scaled version of the channel vector of the user-of-interest are obtained.     

Asymptotic analysis of optimum and suboptimum CDMA downlink MMSE receivers

In this paper, we investigate the performance of two linear receivers for code-division multiple-access (CDMA) downlink transmissions over frequency-selective channels, the users having possibly different powers. The optimum minimum mean-square error (MMSE) receiver is first considered. Because this receiver requires the knowledge of the code vectors attributed to all the users within the cell when these vectors are time varying, its use may be unrealistic in the forward link. A classical suboptimum receiver, consisting in a chip rate equalizer followed by a despreading with the code of the user of interest, is therefore studied and compared to the optimum MMSE receiver. Performance of both receivers is assessed through the signal-to-interference-plus-noise ratio (SINR) at their outputs. The analytical expressions of these SINRs depend in a rather nonexplicit way on the codes allocated to the users of the cell, and are therefore not informative. This difficulty is dealt with by modeling the users code matrix by a random matrix. Because the code matrices used in the forward link are usually isometric, the code matrix is assumed to be extracted from a Haar-distributed random unitary matrix. The behavior of the SINRs is studied when the spreading factor and the number of users converge to /spl infin/ at the same rate. Using certain results of the free probability theory, we establish the fact that the SINRs converge almost surely toward quantities that depend only on the complex amplitudes of propagation channel paths. We then use the expressions of these SINR limits to discuss the influence of the various parameters on the performance of the receivers.