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48
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)
Randomly Spread CDMA: Asymptotics via Statistical Physics
, 2005
"... This paper studies randomly spread codedivision multiple access (CDMA) and multiuser detection in the largesystem limit using the replica method developed in statistical physics. Arbitrary input distributions and flat fading are considered. A generic multiuser detector in the form of the posterio ..."
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Cited by 101 (12 self)
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This paper studies randomly spread codedivision multiple access (CDMA) and multiuser detection in the largesystem limit using the replica method developed in statistical physics. Arbitrary input distributions and flat fading are considered. A generic multiuser detector in the form of the posterior mean estimator is applied before singleuser decoding. The generic detector can be particularized to the matched filter, decorrelator, linear MMSE detector, the jointly or the individually optimal detector, and others. It is found that the detection output for each user, although in general asymptotically nonGaussian conditioned on the transmitted symbol, converges as the number of users go to infinity to a deterministic function of a “hidden ” Gaussian statistic independent of the interferers. Thus the multiuser channel can be decoupled: Each user experiences an equivalent singleuser Gaussian channel, whose signaltonoise ratio suffers a degradation due to the multipleaccess interference. The uncoded error performance (e.g., symbolerrorrate) and the mutual information can then be fully characterized using the degradation factor, also known as the multiuser efficiency, which can be obtained by solving a pair of coupled fixedpoint equations identified in this paper. Based on a general linear vector channel model, the results are also applicable to MIMO channels such as in multiantenna systems.
Asymptotic meansquare optimality of belief propagation for sparse linear systems,"
 Proc. in IEEE Inform. Theory Workshop (ITW),
, 2006
"... AbstractThis paper studies the estimation of a highdimensional vector signal where the observation is a known "sparse" linear transformation of the signal corrupted by additive Gaussian noise. A paradigm of such a linear system is codedivision multiple access (CDMA) channel with sparse s ..."
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Cited by 36 (1 self)
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AbstractThis paper studies the estimation of a highdimensional vector signal where the observation is a known "sparse" linear transformation of the signal corrupted by additive Gaussian noise. A paradigm of such a linear system is codedivision multiple access (CDMA) channel with sparse spreading matrix. Assuming a "semiregular" ensemble of sparse matrix linear transformations, where the bipartite graph describing the system is asymptotically cyclefree, it is shown that belief propagation (BP) achieves the minimum meansquare error (MMSE) in estimating the transformation of the input vector in the largesystem limit. The result holds regardless of the the distribution and power of the input symbols. Furthermore, the mean squared error of estimating each symbol of the input vector using BP is proved to be equal to the MMSE of estimating the same symbol through a scalar Gaussian channel with some degradation in the signaltonoise ratio (SNR). The degradation, called the efficiency, is determined from a fixedpoint equation due to Guo and Verdú, which is a generalization of Tanaka's formula to arbitrary prior distributions.
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.
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
Multiuser Detection for Overloaded CDMA Systems
 IEEE TRANS. INFORM. THEORY
, 2002
"... Multiuser detection for overloaded CDMA systems, in which the number of users is larger than the dimension of the signal space, is of particular interest when bandwidth is at a premium. In this paper, certain fundamental questions are answered regarding the asymptotic forms and performance of subopt ..."
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Cited by 22 (2 self)
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Multiuser detection for overloaded CDMA systems, in which the number of users is larger than the dimension of the signal space, is of particular interest when bandwidth is at a premium. In this paper, certain fundamental questions are answered regarding the asymptotic forms and performance of suboptimum multiuser detectors for cases where the desired and/or interfering signal subspaces are of reduced rank and/or have a nontrivial intersection. In the process, two new suboptimum detectors are proposed that are especially well suited to overloaded systems, namely the group pseudodecorrelator and the group MMSE detector, and the former is seen to be the correct extension of the group decorrelator in the sense that it is the limiting form (in the lownoise regime) of the group MMSE detector. Pseudodecorrelation is also used as a feedforward filter in a new decision feedback scheme. For the particular case of realvalued modulation, it is shown that the recent proposals of the socalled "improved" linear (aka "linearconjugate" or "widely linear") detectors were more simply derived earlier using the idea of minimal sufficiency which we also apply to the new detectors of this paper.
Design of block transceivers with decision feedback detection
 in IEEE Trans. Signal Process
, 2006
"... Abstract—This paper presents a method for jointly designing the transmitter–receiver pair in a blockbyblock communication system that employs (intrablock) decision feedback detection. We provide closedform expressions for transmitter–receiver pairs that simultaneously minimize the arithmetic mean ..."
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Cited by 21 (4 self)
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Abstract—This paper presents a method for jointly designing the transmitter–receiver pair in a blockbyblock communication system that employs (intrablock) decision feedback detection. We provide closedform expressions for transmitter–receiver pairs that simultaneously minimize the arithmetic mean squared error (MSE) at the decision point (assuming perfect feedback), the geometric MSE, and the bit error rate of a uniformly bitloaded system at moderatetohigh signaltonoise ratios. Separate expressions apply for the “zeroforcing ” and “minimum MSE” (MMSE) decision feedback structures. In the MMSE case, the proposed design also maximizes the Gaussian mutual information and suggests that one can approach the capacity of the block transmission system using (independent instances of) the same (Gaussian) code for each element of the block. Our simulation studies indicate that the proposed transceivers perform significantly better than standard transceivers and that they retain their performance advantages in the presence of error propagation. Index Terms—Bit error rate, block precoding, channel capacity, decision feedback detection, minimum meansquare error, mutual information, zeroforcing. I.
Multiuser Detection and Statistical Physics
, 2002
"... We present a framework for analyzing multiuser detectors in the context of statistical physics. A multiuser detector is shown to be equivalent to a conditional mean estimator which finds the mean value of the stochastic output of a socalled Bayes retrochannel. The Bayes retrochannel is equivalent t ..."
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Cited by 19 (6 self)
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We present a framework for analyzing multiuser detectors in the context of statistical physics. A multiuser detector is shown to be equivalent to a conditional mean estimator which finds the mean value of the stochastic output of a socalled Bayes retrochannel. The Bayes retrochannel is equivalent to a spin glass in the sense that the distribution of its stochastic output conditioned on the received signal is exactly the distribution of the spin glass at thermal equilibrium. In the largesystem limit, the biterrorrate of the multiuser detector is simply determined by the magnetization of the spin glass, which can be obtained using powerful tools developed in statistical mechanics. In particular, we derive the largesystem uncoded biterrorrate of the matched filter, the MMSE detector, the decorrelator and the optimal detectors, as well as the spectral efficiency of the Gaussian CDMA channel. It is found that all users with different received energies share the same multiuser efficiency, which uniquely determines the performance of a multiuser detector. A universal interpretation of multiuser detection relates the multiuser efficiency to the meansquare error of the conditional mean estimator output in the largesystem limit.
Achievable Sum Rate of MIMO MMSE Receivers: A General Analytic Framework 1
, 903
"... This paper investigates the achievable sum rate of multipleinput multipleoutput (MIMO) wireless systems employing linear minimum meansquared error (MMSE) receivers. We present a new analytic framework which unveils an interesting connection between the achievable sum rate with MMSE receivers and ..."
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Cited by 12 (0 self)
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This paper investigates the achievable sum rate of multipleinput multipleoutput (MIMO) wireless systems employing linear minimum meansquared error (MMSE) receivers. We present a new analytic framework which unveils an interesting connection between the achievable sum rate with MMSE receivers and the ergodic mutual information achieved with optimal receivers. This simple but powerful result enables the vast prior literature on ergodic MIMO mutual information to be directly applied to the analysis of MMSE receivers. The framework is particularized to various Rayleigh and Rician channel scenarios to yield new exact closedform expressions for the achievable sum rate, as well as simplified expressions in the asymptotic regimes of high and low signal to noise ratios. These expressions lead to the discovery of key insights into the performance of MIMO MMSE receivers under practical channel conditions.
Asymptotic spectral efficiency of multiuser multisignature CDMA in frequencyselective channels
 IEEE Trans. Inform. Theory
"... Abstract: This paper presents an asymptotic analysis of multisignature CodeDivision Multiple Access (CDMA) in the presence of frequencyselective channels. We characterize the sum spectral efficiency and spectral efficiency regions for both the optimal and Linear Minimum Mean Squared Error (LMMSE) ..."
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Cited by 11 (3 self)
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Abstract: This paper presents an asymptotic analysis of multisignature CodeDivision Multiple Access (CDMA) in the presence of frequencyselective channels. We characterize the sum spectral efficiency and spectral efficiency regions for both the optimal and Linear Minimum Mean Squared Error (LMMSE) multiuser receivers. Both i.i.d. signatures and isometric signatures, which are orthogonal at each transmitter, are considered. Our results are asymptotic as the number of signatures per user and processing gain both tend to infinity with fixed ratio. The spectral efficiency of the LMMSE receiver is determined from the asymptotic output SignaltoInterferencePlus Noise Ratio (SINR). Our results rely on approximating certain covariance matrices with unitarily invariant matrices that are asymptotically free. This approximation is shown to be very accurate through comparison with both simulation and an ‘incrementalsignature ’ analysis, which can be used to compute asymptotic moments. Also, a novel proof of the convergence of the empirical spectral distribution of the signal correlation matrix is presented. From these results, we derive the optimal codingspreading tradeoff, which maximizes the LMMSE spectral efficiency, for the case of a single user with multiple i.i.d. signatures. Simulation studies demonstrate that the asymptotic results accurately predict the performance of finitesize systems of interest. The resulting expressions are used to highlight and infer properties of the multisignature CDMA system, including the benefit of orthogonal relative to i.i.d. signatures, and the tradeoff between spectral efficiency and the versatility of providing a variable data rate