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66
Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit
, 2006
"... Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our pr ..."
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Cited by 274 (22 self)
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Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our proposal, Stagewise Orthogonal Matching Pursuit (StOMP), successively transforms the signal into a negligible residual. Starting with initial residual r0 = y, at the sth stage it forms the ‘matched filter ’ Φ T rs−1, identifies all coordinates with amplitudes exceeding a speciallychosen threshold, solves a leastsquares problem using the selected coordinates, and subtracts the leastsquares fit, producing a new residual. After a fixed number of stages (e.g. 10), it stops. In contrast to Orthogonal Matching Pursuit (OMP), many coefficients can enter the model at each stage in StOMP while only one enters per stage in OMP; and StOMP takes a fixed number of stages (e.g. 10), while OMP can take many (e.g. n). StOMP runs much faster than competing proposals for sparse solutions, such as ℓ1 minimization and OMP, and so is attractive for solving largescale problems. We use phase diagrams to compare algorithm performance. The problem of recovering a ksparse vector x0 from (y, Φ) where Φ is random n × N and y = Φx0 is represented by a point (n/N, k/n)
Iterative multiuser joint decoding: unified framework and asymptotic analysis
 IEEE TRANS. INFORM. THEORY
, 2002
"... We present a framework for iterative multiuser joint decoding of codedivision multipleaccess (CDMA) signals, based on the factorgraph representation and on the sumproduct algorithm. In this framework, known parallel and serial, hard and soft interference cancellation algorithms are derived in a ..."
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Cited by 116 (3 self)
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We present a framework for iterative multiuser joint decoding of codedivision multipleaccess (CDMA) signals, based on the factorgraph representation and on the sumproduct algorithm. In this framework, known parallel and serial, hard and soft interference cancellation algorithms are derived in a unified way. The asymptotic performance of these algorithms in the limit of large code block length can be rigorously analyzed by using density evolution. We show that, for random spreading in the largesystem limit, density evolution is considerably simplified. Moreover, by making a Gaussian approximation of the decoder soft output, we show that the behavior of iterative multiuser joint decoding is approximately characterized by the stable fixed points of a simple onedimensional nonlinear dynamical system.
Power Control and Capacity of Spread Spectrum Wireless Networks
 Automatica
, 1999
"... Transmit power control is a central technique for resource allocation and interference management in spreadspectrum wireless networks. With the increasing popularity of spreadspectrum as a multiple access technique, there has been significant research in the area in recent years. While power contr ..."
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Cited by 77 (5 self)
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Transmit power control is a central technique for resource allocation and interference management in spreadspectrum wireless networks. With the increasing popularity of spreadspectrum as a multiple access technique, there has been significant research in the area in recent years. While power control has been considered traditionally as a means to counteract the harmful effect of channel fading, the more general emerging view is that it is a flexible mechanism to provide QualityofService to individual users. In this paper, we will review the main threads of ideas and results in the recent development of this area, with a bias towards issues that have been the focus of our own research. For different receivers of varying complexity, we study both questions about optimal power control as well as the problem of characterizing the resulting network capacity. Although spreadspectrum communications has been traditionally viewed as a physicallayer subject, we argue that by suitable abstr...
Iterative Multiuser Joint Decoding: Optimal Power Allocation and LowComplexity Implementation
, 2002
"... We consider a canonical model for coded CDMA with random spreading, where the receiver makes use of iterative BeliefPropagation (BP) joint decoding. We provide simple DensityEvolution analysis in the largesystem limit (large number of users) of the performance of the exact BP decoder and of so ..."
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Cited by 62 (11 self)
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We consider a canonical model for coded CDMA with random spreading, where the receiver makes use of iterative BeliefPropagation (BP) joint decoding. We provide simple DensityEvolution analysis in the largesystem limit (large number of users) of the performance of the exact BP decoder and of some suboptimal approximations based on Interference Cancellation (IC). Based on this analysis, we optimize the received user SNR distribution in order to maximize the system spectral efficiency for given user channel codes, channel load (users per chip) and target user biterror rate. The optimization of the received SNR distribution is obtained by solving a simple linear program and can be easily incorporated into practical power control algorithms. Remarkably, under the optimized SNR assignment the suboptimal Minimum MeanSquare Error (MMSE) ICbased decoder performs almost as well as the more complex exact BP decoder. Moreover, for a large class of commonly used convolutional codes we observe that the optimized SNR distribution consists of a finite number of discrete SNR levels. Based on this observation, we provide a lowcomplexity approximation of the MMSEIC decoder that suffers from very small performance degradation while attaining considerable savings in complexity. As
Downlink capacity of interferencelimited MIMO systems with joint detection
 IEEE Trans. Wireless Commun
, 2004
"... The capacity of downlink cellular multipleinput multipleoutput (MIMO) cellular systems, where cochannel interference is the dominant channel impairment, is investigated in this paper, mainly from a signalprocessing perspective. Turbo spacetime multiuser detection (ST MUD) is employed for intrac ..."
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Cited by 60 (7 self)
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The capacity of downlink cellular multipleinput multipleoutput (MIMO) cellular systems, where cochannel interference is the dominant channel impairment, is investigated in this paper, mainly from a signalprocessing perspective. Turbo spacetime multiuser detection (ST MUD) is employed for intracell communications, and is shown to closely approach the ultimate capacity limits in Gaussian ambient noise for an isolated cell. Then it is combined with various multiuser detection methods for combating intercell interference. Among various multiuser detection techniques examined, linear minimummeansquareerror (MMSE) MUD and successive interference cancellation are shown to be feasible and effective. Based on these two multiuser detection schemes, one of which may outperform the other for different settings, an adaptive detection scheme is developed, which together with a Turbo ST MUD structure offers substantial performance gain over the well known VBLAST techniques with coding in this interferencelimited cellular environment. The obtained multiuser capacity is excellent in high to medium signaltointerference ratio scenario. Nonetheless, numerical results also indicate that a further increase in system complexity, using basestation cooperation, could lead to further significant increases of the system capacity. The asymptotic multicell MIMO capacity with linear MMSE MUD preprocessing is also derived, and this analysis agrees well with the simulation results.
Asymptotic normality of linear multiuser receiver outputs
 IEEE TRANS. INFORM. THEORY
, 2002
"... This paper proves largesystem 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 singleuser matched filter, the decorrelator, the minimum mean square error (MMSE) ..."
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Cited by 48 (7 self)
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This paper proves largesystem 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 singleuser 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 signaltointerference ratio (SIR) for large linear systems.
On the distribution of SINR for the MMSE MIMO receiver and performance analysis
 IEEE Trans. Inform. Theory
, 2006
"... Abstract — This paper studies the statistical distribution of the signaltointerferenceplusnoise ratio (SINR) for the minimum mean square error (MMSE) receiver in multipleinputmultipleoutput (MIMO) wireless communications. The channel model is assumed to be (transmit) correlated Rayleigh flatf ..."
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Cited by 33 (6 self)
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Abstract — This paper studies the statistical distribution of the signaltointerferenceplusnoise ratio (SINR) for the minimum mean square error (MMSE) receiver in multipleinputmultipleoutput (MIMO) wireless communications. The channel model is assumed to be (transmit) correlated Rayleigh flatfading with unequal powers. The SINR can be decomposed into two independent random variables: SINR = SINR ZF + T, where SINR ZF corresponds to the SINR for a zeroforcing (ZF) receiver and has an exact Gamma distribution. This paper focuses on characterizing the statistical properties of T using the results from random matrix theory. First three asymptotic moments of T are derived for uncorrelated channels and channels with equicorrelations. For general correlated channels, some limiting upperbounds for the first three moments are also provided. For uncorrelated channels and correlated channels satisfying certain conditions, it is proved that T converges to a Normal random variable. A Gamma distribution and a generalized Gamma distribution are proposed as approximations to the finite sample distribution of T. Simulations suggest that these approximate distributions can be used to accurately estimate the probability of errors even for very small dimensions (e.g., 2 transmit antennas). Index Terms — Multipleinputmultipleoutput system, minimum mean square error receiver, signaltointerferenceplusnoise ratio, channel correlation, random matrix, asymptotic distributions, Gamma approximation, error probability. I.
Asymptotical Analysis of Optimum and SubOptimum CDMA Downlink MMSE Receivers
 IEEE Trans. on Inf. Th
, 2004
"... In this paper, we investigate the performance of two linear receivers for 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 ..."
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Cited by 28 (2 self)
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In this paper, we investigate the performance of two linear receivers for 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 non explicit 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 ∞ 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 (1)
Performance analysis of ZF and MMSE equalizers for MIMO systems: An indepth study of the high SNR regime
 IEEE Trans. Inf. Theory
, 2011
"... This paper presents an indepth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multiinput multioutput (MIMO) systems with no fewer receive than transmit antennas. In spite of much prior work on this subject, we reveal several new and surprisi ..."
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Cited by 25 (2 self)
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This paper presents an indepth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multiinput multioutput (MIMO) systems with no fewer receive than transmit antennas. In spite of much prior work on this subject, we reveal several new and surprising analytical results in terms of the wellknown performance metrics of output signaltonoise ratio (SNR), uncoded error and outage probabilities, diversitymultiplexing (DM) gain tradeoff, and coding gain. Contrary to the common perception that ZF and MMSE are asymptotically equivalent at high SNR, we show that the output SNR of the MMSE equalizer (conditioned on the channel realization) is ρmmse = ρzf + ηsnr, where ρzf is the output SNR of the ZF equalizer, and that the gap ηsnr is statistically independent of ρzf and is a nondecreasing function of input SNR. Furthermore, as snr → ∞, ηsnr converges with probability one to a scaled F random variable. It is also shown that at the output of the MMSE equalizer, the interferencetonoise ratio (INR) is tightly upper bounded by ηsnr. Using the decomposition of the output SNR of MMSE, we can approximate its uncoded error as well ρzf as outage probabilities through a numerical integral which accurately reflects the respective SNR gains of the MMSE equalizer relative to its ZF counterpart. The ɛoutage capacities of the two equalizers, however, coincide
On the Capacity Loss due to Separation of Detection and Decoding
, 2002
"... The performance loss due to separation of detection and decoding on the binaryinput additive white Gaussian noise channel is quantified in terms of mutual information. Results are reported for both the codedivision multipleaccess (CDMA) channel in the large system limit and the intersymbol interf ..."
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Cited by 24 (10 self)
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The performance loss due to separation of detection and decoding on the binaryinput additive white Gaussian noise channel is quantified in terms of mutual information. Results are reported for both the codedivision multipleaccess (CDMA) channel in the large system limit and the intersymbol interference (ISI) channel. The results for CDMA rely on the replica method developed in statistical mechanics. It is shown that a previous result in [1] found for Gaussian input alphabet holds also for binary input alphabets. For the ISI channel, the performance loss is calculated via the BCJR algorithm. Comparisons are made to the capacity of separate detection and decoding using suboptimum detectors such as a decisionfeedback equalizer.