Yuriy V. Zakharov

Low-Complexity RLS Algorithms Using Dichotomous Coordinate Descent Iterations

In this paper, we derive low-complexity recursive least squares (RLS) adaptive filtering algorithms. We express the RLS problem in terms of auxiliary normal equations with respect to increments of the filter weights and apply this approach to the exponentially weighted and sliding window cases to derive new RLS techniques. For solving the auxiliary equations, line search methods are used. We first consider conjugate gradient iterations with a complexity of operations per sample; being the number of the filter weights. To reduce the complexity and make the algorithms more suitable for finite precision implementation, we propose a new dichotomous coordinate descent (DCD) algorithm and apply it to the auxiliary equations. This results in a transversal RLS adaptive filter with complexity as low as multiplications per sample, which is only slightly higher than the complexity of the least mean squares (LMS) algorithm ( multiplications). Simulations are used to compare the performance of the proposed algorithms against the classical RLS and known advanced adaptive algorithms. Fixed-point FPGA implementation of the proposed DCD-based RLS algorithm is also discussed and results of such implementation are presented.

Architecture and FPGA Design of Dichotomous Coordinate Descent Algorithms

In the areas of signal processing and communications, such as antenna-array beamforming, adaptive filtering, multiuser and multiple-input-multiple-output (MIMO) detection, channel estimation and equalization, echo and interference cancellation, and others, solving linear systems of equations often provides an optimal performance. However, this is also a very complicated operation that designers try to avoid by proposing different suboptimal techniques. The dichotomous coordinate descent (DCD) algorithm allows linear systems of equations to be solved with high computational efficiency. In this paper, we present architectures and field-programmable gate-array (FPGA) designs of two variants of the DCD algorithm, which are known as cyclic and leading DCD algorithms. For each of these techniques, we present serial designs, group-2 and group-4 designs, as well as a design with parallel update of the residual vector for the cyclic DCD algorithm. These designs have different degrees of parallelism, thus enabling a tradeoff between FPGA resources and computation time. The serial designs require the smallest FPGA resources; they are well suited for applications where many parallel solvers are required, e.g., for detection in MIMO-orthogonal-frequency-division-multiplexing communication systems. The parallelism introduced in the proposed group-2 and group-4 designs allows faster convergence to the true solution at the expense of an increase in FPGA resources. The design with parallel update of the residual vector provides the fastest convergence speed; however, if the system size is high, it may result in a significant increase in FPGA resources. The proposed fixed-point designs provide an accuracy performance that is very close to the performance of floating-point counterparts and require significantly lower FPGA resources than techniques based on QR decomposition.

Coordinate descent iterations in fast affine projection algorithm

We propose a new approach for real-time implementation of the fast affine projection (FAP) algorithm. This is based on exploiting the recently introduced dichotomous coordinate descent (DCD) algorithm, which is especially efficient for solving systems of linear equations on real-time hardware and software platforms since it is free of multiplication and division. The numerical stability of the DCD algorithm allows the new combined DCD-FAP algorithm also to be stable. The convergence and complexity of the DCD-FAP algorithm is compared with that of the FAP, Gauss-Seidel FAP (GS-FAP), and modified GS-FAP algorithms in the application to acoustic echo cancellation. The DCD-FAP algorithm demonstrates a performance close to that of the FAP algorithm with ideal matrix inversion and the complexity smaller than that of the Gauss-Seidel FAP algorithms.

Polynomial spline-approximation of Clarke's model

We investigate polynomial spline approximation of stationary random processes on a uniform grid applied to Clarkes model of time variations of path amplitudes in multipath fading channels with Doppler scattering. The integral mean square error (MSE) for optimal and interpolation splines is presented as a series of spectral moments. The optimal splines outperform the interpolation splines; however, as the sampling factor increases, the optimal and interpolation splines of even order tend to provide the same accuracy. To build such splines, the process to be approximated needs to be known for all time, which is impractical. Local splines, on the other hand, may be used where the process is known only over a finite interval. We first consider local splines with quasioptimal spline coefficients. Then, we derive optimal spline coefficients and investigate the error for different sets of samples used for calculating the spline coefficients. In practice, approximation with a low processing delay is of interest; we investigate local spline extrapolation with a zero-processing delay. The results of our investigation show that local spline approximation is attractive for implementation from viewpoints of both low processing delay and small approximation error; the error can be very close to the minimum error provided by optimal splines. Thus, local splines can be effectively used for channel estimation in multipath fast fading channels.

Multipath-Doppler diversity of OFDM signals in an underwater acoustic channel

High-speed communications over the underwater acoustic channel are difficult due to time-varying multipath propagation and Doppler scattering. These phenomena, however, can be effectively used for joint multipath-Doppler diversity of received signals similarly to multipath diversity in a conventional RAKE receiver. We present a new signal processing technique for data transmission over fast fading underwater acoustic channel by using OFDM signals. This technique exploits multipath-Doppler diversity which is based on the channel model as a sum of macro-rays characterised by delays, Doppler parameters and transfer functions. Delay and Doppler estimation involves calculation of the cross-ambiguity function for a pilot signal; transfer function estimation exploits frequency domain B-spline approximation. Experimental results demonstrate high BER performance of the proposed algorithm for various propagation scenarios.

Low-Complexity Implementation of the Affine Projection Algorithm

In this letter, a new low-complexity implementation of the affine projection (AP) adaptive filtering algorithm is proposed and investigated by simulation. The proposed algorithm uses a novel low complexity recursive filtering technique and filter update that is incorporated in dichotomous coordinate descent (DCD) iterations. If the projection order is significantly smaller than the filter length L, the complexity of the proposed DCD-AP algorithm is as small as about L multiplications per sample.

Broadband Underwater Localization of Multiple Sources Using Basis Pursuit De-Noising

Locating multiple underwater acoustic sources is a problem that can be solved using antenna array beamforming based on the matched field (MF) processing. However, known MF beamforming techniques fail to provide good performance for multiple sources, a high noise power, and/or when the sources are close to each other. This paper proposes an MF technique for solving the localization problem. The proposed technique exploits formulation of the localization problem in terms of sparse representation of a small number of source positions among a much larger number of potential positions. The sparse representation is formulated as the basis pursuit de-noising (BPDN) problem for complex-valued variables. The solution is found as a joint solution to a set of BPDN problems corresponding to the set of source frequencies subject to the joint support. The joint BPDN problem is efficiently solved using the Homotopy approach and coordinate descent search. For further reduction in the complexity, a position grid refinement method is applied. Using simulated and real experimental data, it is shown that the technique can provide accurate source localization for multiple sources. The proposed technique outperforms other MF techniques in resolving sources positioned closely to each other, tolerance to the noise and capability of locating multiple sources.

Pseudo-Affine Projection Algorithms for Multichannel Active Noise Control

For feedforward multichannel active noise control (ANC) systems, the use of adaptive finite-impulse response (FIR) filters is a popular solution, and the multichannel filtered-x least-mean-square (FX-LMS) algorithm is the most commonly used algorithm. The drawback of the FX-LMS is the slow convergence speed, especially for broadband multichannel systems. Recently, some fast affine projection algorithms have been introduced for multichannel ANC, as an interesting alternative to the FX-LMS algorithm. They can provide a significantly improved convergence speed at a reasonable additional computational cost. Yet, the additional computational cost or the potential numerical instability in some of the recently proposed algorithms can prevent the use of those algorithms for some applications. In this paper, we propose two pseudo-affine projection algorithms for multichannel ANC: one based on the Gauss-Seidel method and one based on dichotomous coordinate descent (DCD) iterations. It is shown that the proposed algorithms typically have a lower complexity than the previously published fast affine projection algorithms for ANC, with very similar good convergence properties and good numerical stability. Thus, the proposed algorithms are an interesting alternative to the standard FX-LMS algorithm for ANC, providing an improved performance for a computational load of the same order

Iterative Channel Estimation Based on B-splines for Fast Flat Fading Channels

We propose novel low-complexity iterative channel estimators based on B-splines. Local splines are adopted for computational simplicity. Minimum mean square error (MMSE) local splines with integral sampling are derived. The MSE of the proposed estimators depends on signal-to-noise ratio, fading rate, sampling interval, spline order and the number of weighting coefficients; these dependencies are investigated. The linear and cubic local splines with as few as seven weighting coefficients are capable of achieving MSE and BER performance comparable to those of the Wiener filter and the spheroidal basis expansion. However, a significantly lower complexity is achieved using B-splines

Frequency estimation in slowly fading multipath channels

This paper concerns the estimation of a frequency offset of a known (pilot) signal propagated through a slowly fading multipath channel, such that channel parameters are considered to he constant over the observation interval. We derive a maximum-likelihood (ML) frequency estimation algorithm for additive Gaussian noise and path amplitudes having complex Gaussian distribution when covariance matrices of the fading and noise are known; we consider in detail the algorithm for the white noise and Rayleigh fading, in particular, for independent fading of path amplitudes and pilot signals with diagonal autocorrelation matrices. For the latter scenario, we also derive an ML frequency estimator when the power delay profile is unknown, but the noise variance and bounds for the path amplitude variances are specified; in particular, this algorithm can be used when path delays and amplitude variances are unknown. Finally, we consider frequency estimators which do not use a priori information about the noise variance; these algorithms are also operable without timing synchronization. All the frequency estimators exploit the multipath diversity by combining periodograms of multipath signal components and searching for the maximum of the combined statistic. For implementation of the algorithms, we use a fast Fourier transform-based coarse search and fine dichotomous search. We perform simulations to compare the algorithms. The simulation results demonstrate high accuracy performance of the proposed frequency estimators in wide signal-to-noise ratio and frequency acquisition range.