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 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

Design and Implementation of Belief Propagation Symbol Detectors for Wireless Intersymbol Interference Channels

In modern wireless communication systems, intersymbol interference (ISI) introduced by frequency selective fading is one of the major impairments to reliable data communication. In ISI channels, the receiver observes the superposition of multiple delayed reflections of the transmitted signal, which will result errors in the decision device. As the data rate increases, the effect of ISI becomes severe. To combat ISI, equalization is usually required for symbol detectors. The optimal maximum-likelihood sequence estimation (MLSE) based on the Viterbi algorithm (VA) may be used to estimate the transmitted sequence in the presence of the ISI. However, the computational complexity of the MLSE increases exponentially with the length of the channel impulse response (CIR). Even in channels which do not exhibit significant time dispersion, the length of the CIR will effectively increase as the sampling rate goes higher. Thus the optimal MLSE is impractical to implement in the majority of practical wireless applications. This dissertation is devoted to exploring practically implementable symbol detectors with near-optimal performance in wireless ISI channels. Particularly, we focus on the design and implementation of an iterative detector based on the belief propagation (BP) algorithm. The advantage of the BP detector is that its complexity is solely dependent on the number of nonzero coefficients in the CIR, instead of the length of the CIR. We also extend the work of BP detector design for various wireless applications. Firstly, we present a partial response BP (PRBP) symbol detector with near-optimal performance for channels which have long spanning durations but sparse multipath structure. We implement the architecture by cascading an adaptive linear equalizer (LE) with a BP detector. The channel is first partially equalized by the LE to a target impulse response (TIR) with only a few nonzero coefficients remaining. The residual ISI is then canceled by a more sophisticated BP detector. With the cascaded LE-BP structure, the symbol detector is capable to achieve a near-optimal error rate performance with acceptable implementation complexity. Moreover, we present a pipeline high-throughput implementation of the detector for channel length 30 with quadrature phase-shift keying (QPSK) modulation. The detector can achieve a maximum throughput of 206 Mb/s with an estimated core area of 3.162 mm^{2} using 90-nm technology node. At a target frequency of 515 MHz, the dynamic power is about 1.096 W. Secondly, we investigate the performance of aforementioned PRBP detector under a more generic 3G channel rather than the sparse channel. Another suboptimal partial response maximum-likelihood (PRML) detector is considered for comparison. Similar to the PRBP detector, the PRML detector also employs a hybrid two-stage scheme, in order to allow a tradeoff between performance and complexity. In simulations, we consider a slow fading environment and use the ITU-R 3G channel models. From the numerical results, it is shown that in frequency-selective fading wireless channels, the PRBP detector provides superior performance over both the traditional minimum mean squared error linear equalizer (MMSE-LE) and the PRML detector. Due to the effect of colored noise, the PRML detector in fading wireless channels is not as effective as it is in magnetic recording applications. Thirdly, we extend our work to accommodate the application of Advanced Television Systems Committee (ATSC) digital television (DTV) systems. In order to reduce error propagation caused by the traditional decision feedback equalizer (DFE) in DTV receiver, we present an adaptive decision feedback sparsening filter BP (DFSF-BP) detector, which is another form of PRBP detector. Different from the aforementioned LE-BP structure, in the DFSF-BP scheme, the BP detector is followed by a nonlinear filter called DFSF as the partial response equalizer. In the first stage, the DFSF employs a modified feedback filter which leaves the strongest post-cursor ISI taps uncorrected. As a result, a long ISI channel is equalized to a sparse channel having only a small number of nonzero taps. In the second stage, the BP detector is applied to mitigate the residual ISI. Since the channel is typically time-varying and suffers from Doppler fading, the DFSF is adapted using the least mean square (LMS) algorithm, such that the amplitude and the locations of the nonzero taps of the equalized sparse channel appear to be fixed. As such, the channel appears to be static during the second stage of equalization which consists of the BP detector. Simulation results demonstrate that the proposed scheme outperforms the traditional DFE in symbol error rate, under both static channels and dynamic ATSC channels. Finally, we study the symbol detector design for cooperative communications, which have attracted a lot of attention recently for its ability to exploit increased spatial diversity available at distributed antennas on other nodes. A system framework employing non-orthogonal amplify-and-forward half-duplex relays through ISI channels is developed. Based on the system model, we first design and implement an optimal maximum-likelihood detector based on the Viterbi algorithm. As the relay period increases, the effective CIR between the source and the destination becomes long and sparse, which makes the optimal detector impractical to implement. In order to achieve a balance between the computational complexity and performance, several sub-optimal detectors are proposed. We first present a multitrellis Viterbi algorithm (MVA) based detector which decomposes the original trellis into multiple parallel irregular sub-trellises by investigating the dependencies between the received symbols. Although MVA provides near-optimal performance, it is not straightforward to decompose the trellis for arbitrary ISI channels. Next, the decision feedback sequence estimation (DFSE) based detector and BP-based detector are proposed for cooperative ISI channels. Traditionally these two detectors are used with fixed, static channels. In our model, however, the effective channel is periodically time-varying, even when the component channels themselves are static. Consequently, we modify these two detector to account for cooperative ISI channels. Through simulations in frequency selective fading channels, we demonstrate the uncoded performance of the DFSE detector and the BP detector when compared to the optimal MLSE detector. In addition to quantifying the performance of these detectors, we also include an analysis of the implementation complexity as well as a discussion on complexity/performance tradeoffs

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

Shortening of paraunitary matrices obtained by polynomial eigenvalue decomposition algorithms

This paper extends the analysis of the recently introduced row-shift corrected truncation method for paraunitary matrices to those produced by the state-of-the-art sequential matrix diagonalisation (SMD) family of polynomial eigenvalue decomposition (PEVD) algorithms. The row-shift corrected truncation method utilises the ambiguity in the paraunitary matrices to reduce their order. The results presented in this paper compare the effect a simple change in PEVD method can have on the performance of the paraunitary truncation. In the case of the SMD algorithm the benefits of the new approach are reduced compared to what has been seen before however there is still a reduction in both reconstruction error and paraunitary matrix order

On the sparse beamformer design

In designing acoustic broadband beamformers, the complexity can grow significantly when the number of microphones and the filter length increase. It is advantageous if many of the filter coefficients are zeroes so that the implementation can be executed with less computation. Moreover, the size of the array can also be pruned to reduce complexity. These problems are addressed in this paper. A suitable optimization model is proposed. Both array pruning and filter thinning can be solved together as a two-stage optimization problem to yield the final sparse designs. Numerical results show that the complexity of the designed beamformers can be reduced significantly with minimal effect on performance

Revisiting Efficient Multi-Step Nonlinearity Compensation with Machine Learning: An Experimental Demonstration

Efficient nonlinearity compensation in fiber-optic communication systems is considered a key element to go beyond the "capacity crunch''. One guiding principle for previous work on the design of practical nonlinearity compensation schemes is that fewer steps lead to better systems. In this paper, we challenge this assumption and show how to carefully design multi-step approaches that provide better performance--complexity trade-offs than their few-step counterparts. We consider the recently proposed learned digital backpropagation (LDBP) approach, where the linear steps in the split-step method are re-interpreted as general linear functions, similar to the weight matrices in a deep neural network. Our main contribution lies in an experimental demonstration of this approach for a 25 Gbaud single-channel optical transmission system. It is shown how LDBP can be integrated into a coherent receiver DSP chain and successfully trained in the presence of various hardware impairments. Our results show that LDBP with limited complexity can achieve better performance than standard DBP by using very short, but jointly optimized, finite-impulse response filters in each step. This paper also provides an overview of recently proposed extensions of LDBP and we comment on potentially interesting avenues for future work.Comment: 10 pages, 5 figures. Author version of a paper published in the Journal of Lightwave Technology. OSA/IEEE copyright may appl

Blind MultiChannel Identification and Equalization for Dereverberation and Noise Reduction based on Convolutive Transfer Function

This paper addresses the problems of blind channel identification and multichannel equalization for speech dereverberation and noise reduction. The time-domain cross-relation method is not suitable for blind room impulse response identification, due to the near-common zeros of the long impulse responses. We extend the cross-relation method to the short-time Fourier transform (STFT) domain, in which the time-domain impulse responses are approximately represented by the convolutive transfer functions (CTFs) with much less coefficients. The CTFs suffer from the common zeros caused by the oversampled STFT. We propose to identify CTFs based on the STFT with the oversampled signals and the critical sampled CTFs, which is a good compromise between the frequency aliasing of the signals and the common zeros problem of CTFs. In addition, a normalization of the CTFs is proposed to remove the gain ambiguity across sub-bands. In the STFT domain, the identified CTFs is used for multichannel equalization, in which the sparsity of speech signals is exploited. We propose to perform inverse filtering by minimizing the $\ell_1$-norm of the source signal with the relaxed $\ell_2$-norm fitting error between the micophone signals and the convolution of the estimated source signal and the CTFs used as a constraint. This method is advantageous in that the noise can be reduced by relaxing the $\ell_2$-norm to a tolerance corresponding to the noise power, and the tolerance can be automatically set. The experiments confirm the efficiency of the proposed method even under conditions with high reverberation levels and intense noise.Comment: 13 pages, 5 figures, 5 table

Complexity and search space reduction in cyclic-by-row PEVD algorithms

In recent years, several algorithms for the iterative calculation of a polynomial matrix eigenvalue decomposition (PEVD) have been introduced. The PEVD is a generalisation of the ordinary EVD and uses paraunitary operations to diagonalise a parahermitian matrix. This paper addresses potential computational savings that can be applied to existing cyclic-by-row approaches for the PEVD. These savings are found during the search and rotation stages, and do not significantly impact on algorithm accuracy. We demonstrate that with the proposed techniques, computations can be significantly reduced. The benefits of this are important for a number of broadband multichannel problems

Investigation of mode filtering as a preprocessing method for shallow-water acoustic communications

Author Posting. © IEEE, 2010. This article is posted here by permission of IEEE for personal use, not for redistribution. The definitive version was published in IEEE Journal of Oceanic Engineering 35 (2010): 744-755, doi:10.1109/JOE.2010.2045444.Acoustical array data from the 2006 Shallow Water Experiment (SW06) was analyzed to show the feasibility of broadband mode decomposition as a preprocessing method to reduce the effective channel delay spread and concentrate received signal energy in a small number of independent channels. The data were collected by a vertical array, which spans the water column from 12-m depth to the bottom in shallow water 80 m in depth. Binary-sequence data were used to phase-shift-keyed (PSK) modulate signals with different carrier frequencies. No error correction coding was used. The received signals were processed by a system that does not use training or pilot signals. Signals received both during periods of ordinary internal wave activity and during a period with unusually strong internal wave solitons were processed and analyzed. Different broadband mode-filtering methods were analyzed and tested. Broadband mode filtering decomposed the received signal into a number of independent signals with a reduced delay spread. The analysis of signals from the output of mode filters shows that even a simple demodulator can achieve a low bit error rate (BER) at a distance 19.2 km.This work was supported by the U.S. Office of Naval Research (ONR)