Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NP-hard 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 s-th stage it forms the ‘matched filter ’ Φ T rs−1, identifies all coordinates with amplitudes exceeding a specially-chosen threshold, solves a least-squares 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 large-scale problems. We use phase diagrams to compare algorithm performance. The problem of recovering a k-sparse vector x0 from (y, Φ) where Φ is random n × N and y = Φx0 is represented by a point (n/N, k/n)

Abstract—This paper studies randomly spread code-division multiple access (CDMA) and multiuser detection in the large-system 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 single-user decoding. The generic detector can be particularized to the matched filter, decorrelator, linear minimum mean-square error (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 non-Gaussian 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 single-user Gaussian channel, whose signal-to-noise ratio (SNR) suffers a degradation due to the multiple-access interference (MAI). The uncoded error performance (e.g., symbol error rate) 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 fixed-point equations identified in this paper. Based on a general linear vector channel model, the results are also applicable to multiple-input multiple-output (MIMO) channels such as in multiantenna systems. Index Terms—Channel capacity, code-division multiple access (CDMA), free energy, multiple-input multiple-output (MIMO) channel, multiuser detection, multiuser efficiency, replica method, statistical mechanics. I.

Abstract — This paper studies the statistical distribution of the signal-to-interference-plus-noise ratio (SINR) for the minimum mean square error (MMSE) receiver in multiple-input-multipleoutput (MIMO) wireless communications. The channel model is assumed to be (transmit) correlated Rayleigh flat-fading 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 zero-forcing (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 — Multiple-input-multiple-output system, minimum mean square error receiver, signal-to-interference-plusnoise ratio, channel correlation, random matrix, asymptotic distributions, Gamma approximation, error probability. I.

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 non-trivial intersection. In the process, two new suboptimum detectors are proposed that are especially well suited to overloaded systems, namely the group pseudo-decorrelator 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 low-noise regime) of the group MMSE detector. Pseudo-decorrelation is also used as a feedforward filter in a new decision feedback scheme. For the particular case of real-valued modulation, it is shown that the recent proposals of the so-called "improved" linear (aka "linear-conjugate" 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.

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 so-called 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 large-system limit, the bit-error-rate 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 large-system uncoded bit-error-rate 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 mean-square error of the conditional mean estimator output in the large-system limit.

Abstract: This paper presents an asymptotic analysis of multi-signature Code-Division Multiple Access (CDMA) in the presence of frequency-selective 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 Signal-to-Interference-Plus 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 ‘incremental-signature ’ 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 coding-spreading 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 finite-size systems of interest. The resulting expressions are used to highlight and infer properties of the multi-signature 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

Abstract—This paper presents a method for jointly designing the transmitter–receiver pair in a block-by-block communication system that employs (intrablock) decision feedback detection. We provide closed-form 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 bit-loaded system at moderate-to-high signal-to-noise ratios. Separate expressions apply for the “zero-forcing ” 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 mean-square error, mutual information, zero-forcing. I.

In this contribution, the performance of a downlink code division multiple access (CDMA) system with orthogonal spreading and multicell interference is analyzed. A useful framework is provided in order to determine the optimal base station coverage for wireless frequency selective channels with dense networks where each user is equipped with a matched filter. Using asymptotic arguments, explicit expressions of the spectral efficiency are obtained and provide a simple expression of the network spectral efficiency based only on a few meaningful parameters. Contrarily to a common misconception which asserts that to increase spectral efficiency in a CDMA network, one has to increase the number of cells, we show that, depending on the path loss and the fading channel statistics, the code orthogonal gain (due to the synchronization of all the users at the base station) can compensate and even compete in some cases with the drawbacks due to intercell interference. The results are especially realistic and useful for the design of dense networks.

Abstract: The system spectral efficiency is considered for the single-cell DS-CDMA uplink with random spreading and flat Rayleigh fading channels. Based on the asymptotic analysis for large systems, the signal-to-interferenceand-noise ratio after the linear minimum mean-squarederror detector is determined analytically for multi-code transmission. This measure is used to obtain the maximum rate and thus efficiency by applying the concepts of coding at capacity and at cut-off rate. The efficiency of multiple codes is compared to multi-level modulation for different signal-to-noise ratios and system loads. It is shown that multi-code transmission improves efficiency for lower loads, but cannot increase its maximum value. In contrast, with multi-level modulation the maximum efficiency can be increased in the low-noise region. The accuracy of these results is investigated by means of simulations. Both simulation and asymptotic theoretical results are very close. Finally, the optimal parameters are compared to the UMTS parameter set. The study suggests increasing both spreading factor and modulation order to achieve better spectral efficiency when using an linear minimum mean-squarederror detector. 1.

The problem of modelling channels is crucial for the efficient design of wireless...