A General Framework for Designing Sparse FIR MIMO Equalizers Based on Sparse Approximation

In broadband communications, the long channel delay spread, defined as the duration in time, or samples, over which the channel impulse response (CIR) has significant energy, is too long and results in a highly-frequency-selective channel frequency response. Hence, a long CIR can spread over tens, or even hundreds, of symbol periods and causes impairments in the signals that have passed through such channels. For instance, a large delay spread causes inter-symbol interference (ISI) and inter-carrier interference (ICI) in multi-carrier modulation (MCM). Therefore, long finite impulse response (FIR) equalizers have to be implemented at high sampling rates to avoid performance degradation. However, the implementation of such equalizers is prohibitively expensive as the design complexity of FIR equalizers grows proportional to the square of the number of nonzero taps in the filter. Sparse equalization, where only few nonzero coefficients are employed, is a widely-used technique to reduce complexity at the cost of a tolerable performance loss. Nevertheless, reliably determining the locations of these nonzero coefficients is often very challenging. In this work, we first propose a general framework that transforms the problem of design of sparse single-input single-output (SISO) and multiple-input multiple-output (MIMO) linear equalizers (LEs) into the problem of sparsest-approximation of a vector in different dictionaries. In addition, we compare several choices of sparsifying dictionaries under this framework. Furthermore, the worst-case coherence of these dictionaries, which determines their sparsifying effectiveness, are analytically and/or numerically evaluated. Second, we extend our framework to accommodate SISO and MIMO non-linear decision-feedback equalizers (DFEs). Similar to the sparse FIR LEs design problem, the design of sparse FIR DFEs can be cast into one of sparse approximation of a vector by a fixed dictionary whose solution can be obtained by using either greedy algorithms, such as Orthogonal Matching Pursuit (OMP), or convex-optimization-based approaches, with the former being more desirable due to its low complexity. Third, we further generalize our sparse design framework to the channel shortening setup. Channel shortening equalizers (CSEs) are used to ensure that the cascade of a long CIR and the CSE is approximately equivalent to a target impulse response (TIR) with much shorter delay spread. Channel shortening is essential for communication systems operating over highly-dispersive broadband channels with large channel delay spread. Fourth, as an application of recent practical interest for power-line communication (PLC) community, we consider channel shortening for the impulse responses of medium-voltage power-lines (MV-PLs) with length of 10 km and 20 km to reduce the cyclic prefix (CP) overhead in orthogonal frequency-division multiplexing (OFDM) and, hence, improves the data rate accordingly. For all design problems, we propose reduced-complexity sparse FIR SISO and MIMO linear and non-linear equalizers by exploiting the asymptotic equivalence of Toeplitz and circulant matrices, where the matrix factorizations involved in our design analysis can be carried out efficiently using the fast Fourier transform (FFT) and inverse FFT with negligible performance loss as the number of filter taps increases. Finally, the simulation results show that allowing for a little performance loss yields a significant reduction in the number of active filter taps, for all proposed LEs and DFEs design filters, which in turn results in substantial complexity reductions. The simulation results also show that the CIRs of MV-PLs with length of 10 km and 20 km can be shortened to fit within the broadband PLC standards. Additionally, our simulations validate that the sparsifying dictionary with the smallest worst-case coherence results in the sparsest FIR filter design. Furthermore, the numerical results demonstrate the superiority of our proposed approach compared to conventional sparse FIR filters in terms of both performance and computational complexity.Qscienc

Sparse Filter Design Under a Quadratic Constraint: Low-Complexity Algorithms

This paper considers three problems in sparse filter design, the first involving a weighted least-squares constraint on the frequency response, the second a constraint on mean squared error in estimation, and the third a constraint on signal-to-noise ratio in detection. The three problems are unified under a single framework based on sparsity maximization under a quadratic performance constraint. Efficient and exact solutions are developed for specific cases in which the matrix in the quadratic constraint is diagonal, block-diagonal, banded, or has low condition number. For the more difficult general case, a low-complexity algorithm based on backward greedy selection is described with emphasis on its efficient implementation. Examples in wireless channel equalization and minimum-variance distortionless-response beamforming show that the backward selection algorithm yields optimally sparse designs in many instances while also highlighting the benefits of sparse design.Texas Instruments Leadership University Consortium Progra

Sparse Equalizers for OFDM Signals with Insufficient Cyclic Prefix

The cyclic prefix (CP) is appended in orthogonal frequency division multiplexing (OFDM) signals to combat inter-symbol interference (ISI) and inter-carrier interference (ICI) induced by the communication channel, which limits its spectral efficiency. Therefore, inserting an insufficient CP and equalizing the resulting ICI and ISI is a method that has been circulating the literature for a while, aiming at increasing the efficiency of OFDM systems. In this paper, we propose a reduced-complexity sparse linear equalizer and a decision-feedback equalizer for OFDM signals with insufficient CP. A performance-complexity trade-off is highlighted, where we show that it is possible to equalize the received signal with a reduced complexity equalizer while having a limited performance loss. Our proposed equalizer designs are not only less complex to realize, but are shown to provide a higher data rate. The proposed equalizers are further evaluated in terms of the worst-case coherence, a metric determining the effectiveness of our used approach. Numerical results show that we can significantly and reliably reduce the order of the design complexity while performing very close to the conventional complex optimal equalizers. 2013 IEEE.This work was supported by GSRA from the Qatar National Research Fund (a member of Qatar Foundation) under Grant 2-1-0601-14011. The statements made herein are solely the responsibility of the authors.Scopu

Advanced DSP Algorithms For Modern Wireless Communication Transceivers

A higher network throughput, a minimized delay and reliable communications are some of many goals that wireless communication standards, such as the fifthgeneration (5G) standard and beyond, intend to guarantee for its customers. Hence, many key innovations are currently being proposed and investigated by researchers in the academic and industry circles to fulfill these goals. This dissertation investigates some of the proposed techniques that aim at increasing the spectral efficiency, enhancing the energy efficiency, and enabling low latency wireless communications systems. The contributions lay in the evaluation of the performance of several proposed receiver architectures as well as proposing novel digital signal processing (DSP) algorithms to enhance the performance of radio transceivers. Particularly, the effects of several radio frequency (RF) impairments on the functionality of a new class of wireless transceivers, the full-duplex transceivers, are thoroughly investigated. These transceivers are then designed to operate in a relaying scenario, where relay selection and beamforming are applied in a relaying network to increase its spectral efficiency. The dissertation then investigates the use of greedy algorithms in recovering orthogonal frequency division multiplexing (OFDM) signals by using sparse equalizers, which carry out the equalization in a more efficient manner when the low-complexity single tap OFDM equalizer can no longer recover the received signal due to severe interferences. The proposed sparse equalizers are shown to perform close to conventional optimal and dense equalizers when the OFDM signals are impaired by interferences caused by the insertion of an insufficient cyclic prefix and RF impairments

Sparse equalizer filter design for multi-path channels

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 81-82).In this thesis, sparse Finite Impulse Response (FIR) equalizers are designed for sparse multi-path channels under a pre-defined Mean Squared Error (MSE) constraint. We start by examining the intrinsic sparsity of the Zero Forcing equalizers and the FIR Minimum MSE (MMSE) equalizers. Next the equalization MSE is formulated as a quadratic function of the equalizer coefficients. Both the Linear Equalizer (LE) and the Decision Feedback Equalizer (DFE) are analyzed. Utilizing the quadratic form, designing a sparse equalizer under a single MSE constraint becomes an 10-norm minimization problem under a quadratic constraint, as described in [2]. Three previously developed methods for solving this problem are applied, namely the successive thinning algorithm, the branch-and-bound algorithm, and the simple linear programming algorithm. Simulations under various channel specifications, equalizer specifications and algorithm specifications are conducted to show the dependency of the sparsity on these factors. The channels include the ideal discrete multipath channels and the Vehicular A multi-path channels in both the Single-Input-Single- Output (SISO) and the Multiple-Input-Multiple-Output scenarios. Additionally, the sparse FIR equalizer is designed for MIMO channels under two MSE constraints. This is formulated as an 10-norm minimization problem under two quadratic constraints. A sub-optimal solution by decoupling the two constraints is proposed.by Xue Feng.S.M