Robust adaptive beamforming (RAB) based on interference-plus-noise covariance (INC) matrix reconstruction can experience performance degradation when model mismatch errors exist, particularly when the input signal-to-noise ratio (SNR) is large. In this work, we devise an efficient RAB technique for dealing with covariance matrix reconstruction issues. The proposed method involves INC matrix reconstruction using an idea in which the power and the steering vector of the interferences are estimated based on the power method. Furthermore, spatial match processing is computed to reconstruct the desired signal-plus-noise covariance matrix. Then, the noise components are excluded to retain the desired signal (DS) covariance matrix. A key feature of the proposed technique is to avoid eigenvalue decomposition of the INC matrix to obtain the dominant power of the interference-plus-noise region. Moreover, the INC reconstruction is carried out according to the definition of the theoretical INC matrix. Simulation results are shown and discussed to verify the effectiveness of the proposed method against existing approaches.

Robust adaptive beamforming (RAB) plays a vital role in modern communications by ensuring the reception of high-quality signals. This paper proposes a deep learning approach to robust adaptive beamforming. In particular, we propose a novel RAB approach where the sample covariance matrix (SCM) is used as the input of a deep 1D Complex-Valued Convolutional Neural Network (CVCNN). The network employs complex convolutional and pooling layers, as well as a Cartesian Scaled Exponential Linear Unit activation function to directly compute the nearly-optimum weight vector through the training process and without prior knowledge about the direction of arrival of the desired signal. This means that reconstruction of the interference plus noise (IPN) covariance matrix is not required. The trained CVCNN accurately computes the nearly-optimum weight vector for data not used during training. The computed weight vector is employed to estimate the signal-to-interference plus noise ratio. Simulations show that the proposed RAB can provide performance close to that of the optimal beamformer.

In this work, we present an energy-efficient distributed learning framework using low-resolution ADCs and coarsely quantized signals for Internet of Things (IoT) networks. In particular, we develop a distributed quantization-aware least-mean square (DQA-LMS) algorithm that can learn parameters in an energy-efficient fashion using signals quantized with few bits while requiring a low computational cost. We also carry out a statistical analysis of the proposed DQA-LMS algorithm that includes a stability condition. Simulations assess the DQA-LMS algorithm against existing techniques for a distributed parameter estimation task where IoT devices operate in a peer-to-peer mode and demonstrate the effectiveness of the DQA-LMS algorithm.

This work describes a generalization of our previous maximum likelihood (ML) detector to a maximum a posteriori (MAP) detector in distributed networks using the diffusion LMS algorithm. Nodes in the network must decide between two concurrent hypotheses concerning their environment, using local measurements and shared estimates from neighbors. The generalization is provided by a new approximation concerning the network connectivity, whose accuracy is shown by simulations. The new MAP detector inherits from our ML formulation an exponential decay rate in probability of error independent of the LMS step size, if it is sufficiently small.

To ensure signal receiving quality, robust adaptive beamforming (RAB) is of vital importance in modern communications. In this letter, we propose a new low-complexity RAB approach based on interference-plus-noise covariance matrix (IPNC) reconstruction and steering vector (SV) estimation. In this method, the IPNC and desired signal covariance matrices are reconstructed by estimating all interference powers as well as the desired signal power using the principle of maximum entropy power spectrum (MEPS). Numerical simulations demonstrate that the proposed method can provide superior performance to several previously proposed beamformers.

This letter describes a maximum likelihood distributed detector algorithm with a special initialization that yields an exponential decay rate on error probabilities in transient, using the adaptive diffusion least mean squares (LMS) algorithm in a distributed sensor network. The nodes that compose the network must decide between two concurrent hypotheses concerning the environment where they are inserted, using local measurements and shared estimates coming from their neighbors. It is shown analytically and by simulations that the exponential performance does not depend on the LMS step size, provided it is sufficiently small.

It is known that the calculation of a matrix-vector product can be accelerated if this matrix can be recast (or approximated) by the Kronecker product of two smaller matrices. In array signal processing, the manifold matrix can be described as the Kronecker product of two other matrices if the sensor array displays a separable geometry. This forms the basis of the Kronecker Array Transform (KAT), which was previously introduced to speed up the calculations of acoustic images with microphone arrays. If, however, the array has a quasi-separable geometry, e.g., an otherwise separable array with a missing sensor, then the KAT acceleration can no longer be applied. In this letter, we review the definition of the KAT and provide a much simpler derivation that relies on an explicit new relation developed between Kronecker and Khatri-Rao matrix products. Additionally, we extend the KAT to deal with quasi-separable arrays, alleviating the restriction on the need of perfectly separable arrays.

In practical implementations of estimation algorithms, designers usually have information about the range in which the unknown variables must lie either due to physical constraints (such as power always being non-negative) or due to hardware constraints (such as in implementations using fixed-point arithmetic). In this letter, we propose a fast (i.e., whose complexity grows linearly with the filter length) version of the dichotomous coordinate descent recursive least-squares (RLS) adaptive filter which can incorporate constraints on the variables. The constraints can be in the form of lower and upper bounds on each entry of the filter, or norm bounds. We compare the proposed algorithm with the recently proposed normalized non-negative least-mean-squares (N-NLMS) and projected-gradient normalized LMS (PG-NLMS) filters, which also include inequality constraints in the variables.

This paper addresses adaptive filtering for sparse identification. Proportionate affine projection algorithms (PAPAs) are known to be efficient techniques for this purpose. We show that the PAPA performance may improve with an increase in the projection order M (for example, such as M = 512), which, however, also results in an increased complexity; the complexity is in general O(M 2 N) or at least O(MN) operations per sample, where N is the filter length. We show that PAPAs are equivalent to specific sliding-window recursive least squares (SRLS) adaptive algorithms with time-varying and tap-varying diagonal loading (SRLS-VDLs). We then propose an approximation to the SRLS-VDLs based on dichotomous coordinate descent (DCD) iterations with a complexity of O(N u N), which does not depend on M; it depends on the number of DCD iterations N u , which as we show can be significantly smaller than M, thus allowing a low-complexity implementation of PAPA adaptive filters.

Combinations of adaptive filters have attracted attention as a simple way to improve filter performance, avoiding the compromise between fast adaptation and low excess mean-square error. However, the computational cost of the combined scheme is higher than that of a single filter, since two or more filters must be run in parallel. In this paper, we propose a new approach for reducing the computational cost of combinations of adaptive filters in custom and semi-custom hardware implementations. In the proposed scheme, the fast filter is adapted as usual, but instead of updating the slow filter, we adapt the difference between the slow and fast filters. We show that the coefficients of the difference filter have a lower dynamic range compared to the slow filter, and therefore can be updated using a smaller number of bits. The wordlength used for the difference filter can be chosen to make the performance of this low-cost scheme similar to that of an infinite-precision implementation.

In this work a low-complexity strategy for accelerating the convergence of convex combinations of adaptive filters is proposed. The idea is based on an instantaneous transfer of coefficients from a fast adaptive filter to a slow adaptive filter, which is performed according to a pre-defined window length. A theoretical model that is capable of predicting the excess mean squared error (EMSE) of the proposed strategy is also presented. Simulation results illustrate the good performance of the proposed strategy and the effectiveness of the proposed model to predict the EMSE.

Acoustic imaging is a computationally intensive and ill-conditioned inverse problem, which involves estimating high resolution source distributions with large microphone arrays. In this paper we show how to significantly decrease its computational cost with a fast transform designed for separable array geometries. This transform provides a natural and elegant way of accelerating beamforming, deconvolution methods and regularized least-squares solvers. We accelerate image deconvolution by 10x with respect to FFT-based methods, and accelerate other important imaging algorithms (based on explicit matrix multiplication) by even larger factors. Because of these gains in computational speed, one can reconstruct images with higher resolutions than previously possible, and also enable more accurate reconstruction techniques, opening new and exciting possibilities for acoustic imaging.

Sparse recovery techniques have been shown to produce very accurate acoustic images, significantly outperforming traditional deconvolution approaches. However, so far these proposals have been computationally intractable for all but very small images, because they had no means of efficiently transforming back and forth between a hypothetical image under reconstruction and the measured data. In this paper we obtain a fast transform for planar array geometries using the fast non-equispaced Fourier transform (NFFT). We then apply it to accelerate general-purpose solvers by several orders of magnitude, enabling computationally-efficient regularized acoustic imaging. The proposed approach is not only tractable, but faster than competing deconvolution techniques, while delivering reconstructions with unprecedented accuracy.

Subjective tests are generally regarded as the most reliable and definitive methods for assessing image quality. Nevertheless, laboratory studies are time consuming and expensive. Thus, researchers often choose to run informal studies or use objective quality measures, producing results which may not correlate well with human perception. In this paper we propose a cost-effective and convenient subjective quality measure called crowdMOS, obtained by having Internet workers participate in MOS (mean opinion score) subjective quality studies. Since these workers cannot be supervised, we propose methods for detecting and discarding inaccurate or malicious scores. To facilitate this process, we offer an open source set of tools for Amazon Mechanical Turk, which is an Internet marketplace for crowdsourcing. These tools completely automate the test design, score retrieval and statistical analysis, abstracting away the technical details of Mechanical Turk and ensuring a user-friendly, affordable and consistent test methodology. We demonstrate crowdMOS using data from the LIVE subjective quality image dataset, showing that it delivers accurate and repeatable results.

The classical approach for acoustic imaging consists of beamforming, and produces the source distribution of interest convolved with the array point spread function. This convolution smears the image of interest, significantly reducing its effective resolution. Deconvolution methods have been proposed to enhance acoustic images and have produced significant improvements. Other proposals involve covariance fitting techniques, which avoid deconvolution altogether. However, in their traditional presentation, these enhanced reconstruction methods have very high computational costs, mostly because they have no means of efficiently transforming back and forth between a hypothetical image and the measured data. In this paper, we propose the Kronecker Array Transform (KAT), a fast separable transform for array imaging applications. Under the assumption of a separable array, it enables the acceleration of imaging techniques by several orders of magnitude with respect to the fastest previously available methods, and enables the use of state-of-the-art regularized least-squares solvers. Using the KAT, one can reconstruct images with higher resolutions than was previously possible and use more accurate reconstruction techniques, opening new and exciting possibilities for acoustic imaging.

In Part I ["Fast Transforms for Acoustic Imaging-Part I: Theory," IEEE Transactions on Image Processing], we introduced the Kronecker array transform (KAT), a fast transform for imaging with separable arrays. Given a source distribution, the KAT produces the spectral matrix which would be measured by a separable sensor array. In Part II, we establish connections between the KAT, beamforming and 2-D convolutions, and show how these results can be used to accelerate classical and state of the art array imaging algorithms. We also propose using the KAT to accelerate general purpose regularized least-squares solvers. Using this approach, we avoid ill-conditioned deconvolution steps and obtain more accurate reconstructions than previously possible, while maintaining low computational costs. We also show how the KAT performs when imaging near-field source distributions, and illustrate the trade-off between accuracy and computational complexity. Finally, we show that separable designs can deliver accuracy competitive with multi-arm logarithmic spiral geometries, while having the computational advantages of the KAT.

Knowledge-aided space-time adaptive processing (KA-STAP) algorithms, which incorporate a priori knowledge into radar signal processing methods, have the potential to substantially enhance detection performance while combating heterogeneous clutter effects. In this paper, we develop a KA-STAP algorithm to estimate the inverse interference covariance matrix rather than the covariance matrix itself, by combining the inverse of the covariance known a priori, R0-1, and the inverse sample covariance matrix estimate R̂-1. The computational load is greatly reduced due to the avoidance of the matrix inversion operation. We also develop a cost-effective algorithm based on the minimum variance (MV) criterion for computing the mixing parameter that performs a convex combination of R0-1and R̂-1. Simulations show the potential of our proposed algorithm, which obtain substantial performance improvements over prior art.

Combinations of adaptive filters have attracted attention as a simple solution to improve filter performance, including tracking properties. In this paper, we consider combinations of LMS and RLS filters, and study their performance for tracking time-varying solutions. We show that a combination of two filters from the same family (i.e., two LMS or two RLS filters) cannot improve the performance over that of a single filter of the same type with optimal selection of the step size (or forgetting factor). However, combining LMS and RLS filters it is possible to simultaneously outperform the optimum LMS and RLS filters. In other words, combination schemes can achieve smaller errors than optimally adjusted individual filters. Experimental work in a plant identification setup corroborates the validity of our results.

This paper proposes an improved model for the transient of convex combinations of adaptive filters. A previous model, based on a first-order Taylor series approximation of the nonlinear functions that appear in convex combinations, tended to overestimate the variance of the auxiliary variable used to estimate the mixing parameter. In this paper, we apply a second-order Taylor approximation that improves these estimates, and obtains better agreement with simulations. In addition, we also extend the model to include a simple mechanism for the transfer of coefficients between the constituent filters, a procedure that greatly improves the convergence of the overall filter, and provide an expression to select the free parameter used in such a scheme.

Blind equalization algorithms with good convergence and tracking properties and numerical robustness are desired to ensure the good performance of communications systems. In this paper, we present transient and steady-state analyses for the dual-mode constant modulus algorithm (DM-CMA), a version of CMA that avoids its well-known divergence problem. We show that DM-CMA is able to avoid divergence without degradation of mean-square performance. Good agreement between analytical and simulation results is observed.

We present a novel array RLS algorithm with forgetting factor that circumvents the problem of fading regularization, inherent to the standard exponentially-weighted RLS, by allowing for time-varying regularization matrices with generic structure. Simulations in finite precision show the algorithm's superiority as compared to alternative algorithms in the context of adaptive beamforming.

We extend the affine combination of one fast and one slow least mean-square (LMS) filter to blind equalization, considering the combination of two constant modulus algorithms (CMA). We analyze the proposed combination in stationary and nonstationary environments verifying that there are situations where the absence of the restriction on the mixing parameter can be advantageous for the combination. Furthermore, we propose a combination of two CMAs with different initializations. Preliminary simulations show that this scheme can avoid local minima and eventually can present a faster convergence rate than that of its components.

One of the most popular algorithms for blind equalization is the constant modulus algorithm (CMA), due to its simplicity and low computational cost. However, if the step-size is not properly chosen or if the initialization is distant from the optimal solution, CMA can diverge or converge to undesirable local minima. In order to avoid divergence, we propose a dual-mode algorithm, which works as CMA with a time-variant step-size, but rejects non-consistent estimates of the transmitted signal. We present a deterministic analysis of the stability of the new algorithm for scalar filters. In the vector case, the good performance of the new algorithm is confirmed through numerical simulations.

We present a computationally efficient method for sinusoidal noise cancellation based on the FXLMS algorithm. It features subsampling in order to increase convergence speed and decrease computational requirements, and most importantly, does not require extra noise added to the filter output for secondary path identification. In addition, it is robust to secondary path variations and in low SNR scenarios, which are frequently found in practical active noise control systems, features fast tracking and can be directly generalized to multichannel systems. We illustrate its operation with simulations.

Recently, we proposed a model for the steady-state estimation error of real-valued constant-modulus-based algorithms as a function of the a priori error and of a term that measures the variability in the modulus of the transmitted signal. In this paper, we extend this model to complex-valued data and use it in conjunction with the feedback analysis method to obtain an analytical expression for the steady-state excess mean-square error (EMSE) of the Constant Modulus Algorithm (CMA). Such expression is more accurate for larger step-sizes than the previous ones in the literature, as confirmed by the good agreement between analytical and simulation results. Furthermore, from the EMSE expression, we obtain an estimate for the CMA step-size interval to ensure its convergence and stability, when it is initialized sufficiently close to the zero-forcing solution.

We propose a convex combination of two blind equalizers adapted respectively by CMA (constant modulus algorithm) and SWA (Shalvi-Weinstein algorithm). The performance of the proposed scheme is, in the worst case, as good as that of the best of its components. This behavior provides a good tracking capability, since both CMA or SWA may have a better tracking performance, depending on the kind of nonstationary environment. A steady-state analysis (using energy conservation) is also presented, considering both the proposed scheme, and the convex combination of two CMAs.

Recently, an adaptive convex combination of two LMS (least mean-square) filters was proposed and its tracking performance analyzed. Motivated by the performance of such scheme and by the differences between the tracking capabilities of the RLS (recursive least-squares) and LMS algorithms, we propose a convex combination of one LMS and one RLS filter. The resulting combination should profit of the best tracking behavior of each component filter. A steady-state analysis via energy conservation relation is also presented for stationary and non-stationary environments.

We propose a new model for the behavior of the Least Mean Square (LMS) algorithm when implemented in finite precision. We model the adaptive filter coefficients as a Markov chain and determine its transition probability matrix for the one-dimensional case. We also determine conditions to avoid the so-called stopping phenomenon. The proposed model eliminates the linearizations used in previous models, accounts for saturation effects and leads to accurate estimations of the mean-square error behavior. Monte Carlo simulation results illustrate the quality of the proposed model.

We show that the least-mean fourth (LMF) and the least-mean mixed-norm (LMMN) algorithms are not mean-square stable when the input regressor is Gaussian-distributed. For the LMF algorithm, we propose an upper bound for the algorithm's probability of divergence, given the input and noise statistics, the stepsize and the filter length. We show that the upper bound can also be used for the LMMN algorithm, which is a combination of LMS and LMF.

We propose a new simple model for the input regressor vectors in adaptive filters. This model allows more insight on the effect of normalization on the convergence rate and eigenvalue spread for the normalized least-mean-squares algorithm (NLMS). Using the new model, we show that NLMS works best to reduce eigenvalue spread when the input regressor vector points to all directions with equal probability, but with direction-dependent power.

This paper presents a new analysis for the convergence of the LMF (least mean fourth) adaptive algorithm. The analysis improves previous results because it explicitly shows how the stability of the algorithm depends on the initial conditions of the weights, i.e., the analysis is also valid when the algorithm is initialized far from the optimum weight vector. Analytical expressions are derived relating the limiting values of the adaptation constant and the initial weight error vector. The analysis assumes a white zero-mean Gaussian reference signal and a white measurement noise with any even probability density function (p.d.f.) and finds conditions for convergence in the mean square sense.

We analyze the hierarchical least mean-square (HLMS) algorithm, providing expressions for its steady-state mean-square error (MSE). We find conditions for the hierarchical structure to be equivalent to the optimal (full-length) Wiener solution. When these conditions are not satisfied, we show that HLMS will compute biased estimates. Our analysis also shows that even when these conditions hold, the MSE obtained using HLMS may be much larger than that obtained using LMS, since the potentially large MSEs at the subfilters in the first hierarchical level directly affect the output MSE.