Tracking a target from a video stream (or a sequence of image frames) involves nonlinear measurements in Cartesian coordinates. However, the target dynamics, modeled in Cartesian coordinates, result in a linear system. We present a robust linear filter based on an analytical nonlinear to linear measurement conversion algorithm. Using ideas from robust control theory, a rigorous theoretical analysis is given which guarantees that the state estimation error for the filter is bounded, i.e., a measure against filter divergence is obtained. In fact, an ellipsoidal set-valued estimate is obtained which is guaranteed to contain the true target location with an arbitrarily high probability. The algorithm is particularly suited to visual surveillance and tracking applications involving targets moving on a plane.

We consider a class of weighted nuclear norm optimization problems with important applications in signal processing, system identification, and model order reduction. The nuclear norm is commonly used as a convex heuristic for matrix rank constraints. Our objective is to minimize a quadratic cost subject to a nuclear norm constraint on a linear function of the decision variables, where the trade-off between the fit and the constraint is governed by a regularization parameter. The main contribution is an algorithm to determine the so-called approximate regularization path, which is the optimal solution up to a given error tolerance as a function of the regularization parameter. The advantage is that we only have to solve the optimization problem for a fixed number of values of the regularization parameter, with guaranteed error tolerance. The algorithm is exemplified on a weighted Hankel matrix model order reduction problem.

Coordinated precoding based on interference alignment is a promising technique for improving the throughputs in future wireless multicell networks. In small networks, all base stations can typically jointly coordinate their precoding. In large networks however, base station clustering is necessary due to the otherwise overwhelmingly high channel state information (CSI) acquisition overhead. In this work, we provide a branch and bound algorithm for finding the globally optimal base station clustering. The algorithm is mainly intended for benchmarking existing suboptimal clustering schemes. We propose a general model for the user throughputs, which only depends on the long-term CSI statistics. The model assumes intracluster interference alignment and is able to account for the CSI acquisition overhead. By enumerating a search tree using a best-first search and pruning sub-trees in which the optimal solution provably cannot be, the proposed method converges to the optimal solution. The pruning is done using specifically derived bounds, which exploit some assumed structure in the throughput model. It is empirically shown that the proposed method has an average complexity which is orders of magnitude lower than that of exhaustive search.

In this letter, we investigate optimal and relaxed constructive interference regions (CIR) for the symbol-level pre-coding (SLP) problem in the downlink of a multiuser multiple-input single-output (MISO) channel. We define two types of CIRs, namely, distance preserving CIR (DPCIR) and union bound CIR (UBCIR) for any given constellation shape and size. We then provide a systematic way to describe these regions as convex sets. Using the definitions of DPCIR and UBCIR, we show that the SLP power minimization problem, minimizing either sum or peak (perantenna) transmit power, can always be formulated as a convex optimization problem. Our results indicate that these regions allow further reduction of the transmit power compared to the current state of the art without increasing the computational complexity at the transmitter or receiver.

In this letter, we study the optimal solution of multiuser symbol-level precoding (SLP) for minimization of the total transmit power under given signal-to-interference-plus-noise ratio constraints. Adopting the distance preserving constructive interference regions (DPCIR), we first derive a simplified reformulation of the problem. Then, we analyze the structure of the optimal solution using the Karush-Kuhn-Tucker optimality conditions. This leads us to obtain a closed-form suboptimal SLP solution (CF-SLP) for the original problem. Meanwhile, we obtain the necessary and sufficient condition under which the power minimizer SLP is equivalent to the conventional zero-forcing beamforming (ZFBF). Simulation results show that CF-SLP provides significant gains over ZFBF, while performing quite close to the optimal SLP in scenarios with rather small number of users. The results further indicate that the CF-SLP method has a reduction of order 10³ in computational time compared to the optimal solution. © 2018 IEEE.

In this letter, we propose a segment-based matching-pursuit algorithm where the psychoacoustical properties of the human auditory system are taken into account. Rather than scaling the dictionary elements according to auditory perception, we define a psychoacoustic-adaptive norm on the signal space that can be used for assigning the dictionary elements to the individual segments in a rate-distortion optimal way. The new algorithm is asymptotically equal to signal-to-mask-ratio-based algorithms in the limit of infinite-analysis window length. However, the new algorithm provides a significantly improved selection of the dictionary elements for finite window length.

Two crucial tasks arising in current wireless packet networks are those of link adaptation and power control. It has been recently shown that large performance gains can be attained if proper algorithms for the aforementioned tasks are employed. These algorithms require the estimation of interference power. To this end, an intermediate step is the estimation of the number of active co-channel interferers. This letter aims at determining such an estimate without resorting to traditional training sequences.

Designing the optimal experiment for the recovery of an unknown system with respect to the end performance metric of interest is a recently established practice in the system identification literature. This practice leads to superior end performance to designing the experiment with respect to some generic metric quantifying the distance of the estimated model from the true one. This is usually done by choosing and fixing the estimation method to either a standard maximum likelihood (ML) or a Bayesian estimator. In this paper, we pose the intuitive question: Can we design better estimators than the usual ones with respect to an end performance metric of interest? Based on a simple linear regression example we affirmatively answer this question.

When the squared error of observable signal parameters is below the just noticeable difference (JND), it is not registered by human perception. We modify commonly used distortion criteria to account for this phenomenon and study the implications for quantizer design and performance. Taking the JND into account in the design of the quantizer generally leads to improved performance in terms of mean distortion and the number of outliers. Moreover, the resulting quantizer exhibits better robustness against source mismatch.

The commonly used line spectral frequencies form the roots of symmetric and antisymmetric polynomials constructed from a linear predictor. In this letter, we provide a new, simpler proof that the symmetric and antisymmetric polynomials can be regarded as optimal constrained predictors that correspond to predicting from the low-pass and high-pass filtered signal, respectively.

 In this letter, we study the diversity unfairness associated with the conventional Tomlinson-Harashima precoding (THP) in multiuser multiple-input multiple-output downlink transmission. A single-retransmission scheme that combines two THP signals at each user with a complementary multi-user suppression order is investigated. For a system with M antennas at the transmitter and a single antenna at each user, the proposed scheme provides a diversity order M+1 for all users and ensures diversity fairness. We study two retransmission policies, where the users either consider only the current received codeword or combine both codewords for decoding. An asymptotic analysis of the outage probability for both THP retransmission schemes is provided. In addition, a power allocation policy that minimizes the outage probability and accommodates the same coding gain at each user is discussed by formulating a geometric optimization problem.

In this letter, we employ orthogonal random beamforming (ORBF) for the worst-case multi-user downlink scenario where each user is wiretapped by one eavesdropper. Two opportunistic scheduling techniques that ensure confidentiality by exploiting multi-user diversity are investigated; the first technique (optimal) requires limited feedback of the effective signal-to-interference ratio (SIR) from all the users and the eavesdroppers while the second technique (suboptimal) incorporates SIR knowledge from only the legitimate users. By using extreme value theory, we derive the achievable SIR-based secrecy sum-rate and the associated scaling laws for both scheduling techniques.

In this letter, we investigate the effects of network-layer cooperation in a wireless three-node network with energy-harvesting nodes and bursty data traffic. By modelling energy harvesting in each node as a queue (buffer) that stores the received energy, we study the interaction between data and energy queues when only knowledge of the arrival rates is available. The maximum stable throughput (in packets/slot) of the source as well as the required transmitted power for both a non-cooperative and an orthogonal decode-and-forward cooperative schemes are derived in closed-form. We prove that cooperation achieves a higher maximum stable throughout than direct link for scenarios with poor energy arrival rates.

A new quantization scheme that preserves the probability distribution of the source signal is presented. The distribution preserving quantization (DPQ) achieves the optimal trade-off between mean square error and bit rate asymptotically. It provides a continuum ranging from rate-distortion optimal signal quantization to parametric coding. The method can be used as a core component for scalable coding. Its efficacy is illustrated by applying the scheme to audio coding.

In classic data compression, the optimal transform for energy compaction is the Karhunen-Loeve transform with the eigenvectors of the covariance matrix. In coding applications, neither the covariance matrix nor the eigenvectors can be easily transmitted to the decoder. In this letter, we introduce a covariance matrix model based on graphs determined by hierarchical block motion in image sequences and use its eigenvector matrix for compression. The covariance matrix model is defined using the graph distance matrix, where the graph is determined by block motion. As the proposed covariance matrix is closely related to the graph, the relation between the covariance matrix and the Laplacian matrix is studied and their eigenvector matrices are discussed. From our assumptions, we show that our covariance model can be viewed as a Gaussian graphicalmodel where the signal is described by the second order statistics and the zeros in the precision matrix indicate missing edges in the graph. To assess the compression performance, we relate the coding gain due to the eigenbasis of the covariance model to that of the Laplacian eigenbasis. The experimental results show that the eigenbasis of our covariance model is advantageous for tree-structured blockmotion in image sequences.

We consider estimating the transition probability matrix of a finite-state finite-observation alphabet hidden Markov model with known observation probabilities. We propose a two-step algorithm: a method of moments estimator (formulated as a convex optimization problem) followed by a single iteration of a Newton-Raphson maximum-likelihood estimator. The two-fold contribution of this letter is, first, to theoretically show that the proposed estimator is consistent and asymptotically efficient, and second, to numerically show that the method is computationally less demanding than conventional methods-in particular for large datasets.

The multiple-input multiple-output interference channel is considered with perfect channel information at the transmitters and single-user decoding receivers. With all transmissions restricted to single stream beamforming, we consider the problem of finding all Pareto optimal rate-tuples in the achievable rate region. The problem is cast as a rate profile optimization problem. Due to its nonconvexity, we resort to an alternating approach: For fixed receivers, optimal transmission is known. For fixed transmitters, we show that optimal receive beamforming is a solution to an inverse field of values problem. We prove the solution's stationarity and compare it with existing approaches.

Consider a hidden Markov model describing a system with two types of states: a monitored state and a private state. The two types of states are dependent and evolve jointly according to a Markov process with a stationary transition probability. It is desired to reveal the monitored states to a receiver but hide the private states. For this purpose, a privacy filter is necessary which suitably perturbs the monitored states before communication with the receiver. Our objective is to design the privacy filter to optimize the tradeoff between the monitoring accuracy and privacy, measured through a time-invariant distortion measure and Shannon's equivocation, respectively. As the optimal privacy filter is difficult to compute using the dynamic programming, we adopt a suboptimal greedy approach through which the privacy filter can be computed efficiently. Here, the greedy approach has the additional advantage of not being restricted to the finite time horizon setups. Simulations show the superiority of the approach compared to a privacy filter which only adds independent noise to the observations.

The derivation of MMSE estimators for the DFT coefficients of speech signals, given an observed noisy signal and super-Gaussian prior distributions, has received a lot of interest recently. In this letter, we look at the distribution of the periodogram coefficients of different phonemes, and show that they have a gamma distribution with shape parameters less than one. This verifies that the DFT coefficients for not only the whole speech signal but also for individual phonemes have super-Gaussian distributions. We develop a spectral domain speech enhancement algorithm, and derive hidden Markov model (HMM) based MMSE estimators for speech periodogram coefficients under this gamma assumption in both a high uniform resolution and a reduced-resolution Mel domain. The simulations show that the performance is improved using a gamma distribution compared to the exponential case. Moreover, we show that, even though beneficial in some aspects, the Mel-domain processing does not lead to better results than the algorithms in the high-resolution domain.

It is often desirable to find the underlying trends in time series data. This is a well known signal processing problem that has many applications in areas such as financial data analysis, climatology, biological and medical sciences. Mean filtering finds a piece-wise constant trend in the data while trend filtering finds a piece-wise linear trend. When the signal is noisy, the main difficulty is finding the changing points in the data that mark the transition points when the mean or the trend changes. Previously proposed methods based on l(1) filtering suffer from the occurrence of false changing points in the estimate. This is known as the staircase effect. The main contribution in this paper is incorporating a technique to remove these false changing points to a fast mean filtering algorithm, referred to as the taut-string method, resulting in an efficient procedure with accurate change point detection and thus the removal of the stair-case effect.

We consider the problem of offline change point detection from noisy piecewise constant signals. We propose normalized fused LASSO (FL), an extension of the FL, obtained by normalizing the columns of the sensing matrix of the LASSO equivalent. We analyze the performance of the proposed method, and in particular, we show that it is consistent in detecting change points as the noise variance tends to zero. Numerical experiments support our theoretical findings.

An algorithm for multiple description coding (MDC) based on Gaussian mixture models (GMMs) is presented. Based on the parameters of the GMM, the algorithm combines MDC scalar quantizers, yielding a source-optimized vector MDC system. The performance is evaluated on a speech spectrum source in terms of mean-squared error and log spectral distortion. It is demonstrated experimentally that the proposed system outperforms single description coding and repetition coding over a wide range of channel failure probabilities. The proposed algorithm has a complexity that is linear in rate and dimension while retaining a near optimal vector quantizer point density.

In this letter, we study the problem of fault detection and mitigation in networks where the measurements satisfy Kirchhoff's voltage law. First, we characterise the class of faults appearing as an additive fault vector (injected by a malicious adversary or due to equipment failures) that can be detected by taking into account the topology of the network. Second, we consider the problem of estimating the fault vector via tools from compressive sensing. Moreover, we comment on the applicability of the developed methods to the case where the measurements satisfy Kirchhoff's current law. The proposed methods are validated via numerical examples with application to time synchronization networks.

We present a parametric model-based multichannel approach for speech enhancement. By employing an autoregressive model for the speech signal and using a trained codebook of speech linear predictive coefficients, minimum mean square error estimation of the speech signal is performed. By explicitly accounting for steering errors in the signal model, robust estimates are obtained. Experiments show that the proposed method results in significant performance gains.

We consider an interference network with independent links, whose multi-Antenna transmitters have access to an imperfect analog estimate of their local channels. Assuming that the receivers treat the interference as noise, we define the outage rate region as the set of rate-Tuples that are achievable with a given probability and we characterize the boundary of the region for transmit beamforming. Our study shows that the Pareto-optimal beamforming vectors judiciously balance the desired signal power and the interference power based on the quality of the estimated channel state. Our analysis further reveals that, in contrast to the well-known results by Jorswieck , for the perfect channel side-information case, transmission at full power is not necessarily Pareto-optimal.

In this letter the focus is on linear filtering of speech before degradation due to additive background noise. The goal is to design the filter such that the speech intelligibility index (SII) is maximized when the speech is played back in a known noisy environment. Moreover, a power constraint is taken into account to prevent uncomfortable playback levels and deal with loudspeaker constraints. Previous methods use linear approximations of the SII in order to find a closed-form solution. However, as we show, these linear approximations introduce errors in low SNR regions and are therefore suboptimal. In this work we propose a nonlinear approximation of the SII which is accurate for all SNRs. Experiments show large intelligibility improvements with the proposed method over the unprocessed noisy speech and better performance than one state-of-the art method.

We address the problem of blind source separation in acoustic applications where there is no prior knowledge about the number of mixing sources. The presented method employs a mixture of complex Watson distributions in its generative model with a sparse Dirichlet distribution over the mixture weights. The problem is formulated in a fully Bayesian inference with assuming prior distributions over all model parameters. The presented model can regulate its own complexity by pruning unnecessary components by which we can possibly relax the assumption of prior knowledge on the number of sources.

We consider the problem of estimating a deterministic unknown vector which depends linearly on noisy measurements, additionally contaminated with (possibly unbounded) additive outliers. The measurement matrix of the model (i.e., the matrix involved in the linear transformation of the sought vector) is assumed known, and comprised of standard Gaussian i.i.d. entries. The outlier variables are assumed independent of the measurement matrix, deterministic or random with possibly unknown distribution. Under these assumptions we provide a simple proof that the minimizer of the Huber penalty function of the residuals converges to the true parameter vector with a root n-rate, even when outliers are dense, in the sense that there is a constant linear fraction of contaminated measurements which can be arbitrarily close to one. The constants influencing the rate of convergence are shown to explicitly depend on the outlier contamination level.

In this letter, we consider a network of single-antenna sensors that aim at the estimation of an unknown deterministic parameter. The sensors collect the observations and forward them to a fusion center via applying a phase-only beamforming weight. The derivation of the optimal beamforming weights requires the solution of a unit-modulus quadratic program (UQP), which in the relevant literature is solved via the semidefinite relaxation (SDR) technique or via a variation of the analytic constant modulus algorithm. The former achieves better performance, though it exhibits high computational cost that increases drastically with the number of sensors. The latter requires much less complexity, though it achieves worse performance. To that end, we propose an efficient algorithm for the solution of UQPs based on the alternating direction method of multipliers. The new approach achieves almost identical performance to that of the SDR-based approach while exhibiting significantly reduced computational complexity. The convergence of the proposed algorithm to a Karush-Kuhn-Tucker point is theoretically studied, and its effectiveness is verified via numerical results.

In this work, we develop a new method for jointly optimal quantization of sinusoidal frequencies, amplitudes, and phases and apply the method to sinusoidal audio coding. This is an extension of an earlier work on quantization of sinusoidal amplitudes and phases to frequencies. The optimization is performed for a set of sinusoids that models a short segment of an audio signal. For a given bit-rate constraint, the optimal quantizers minimize a single-letter weighted distortion measure that accounts for perceptual importance of sinusoids. The quantizers are derived analytically using high-rate theory. The method yields high performance and has a number of practical advantages over conventional sinusoidal quantization methods.

Gabor’s analytic signal (AS) is a unique complexsignal corresponding to a real signal, but in general, it admitsinfinitely-many combinations of amplitude and frequency modu-lations (AM and FM, respectively). The standard approach is toenforce a non-negativity constraint on the AM, but this results indiscontinuities in the corresponding phase modulation (PM), andhence, an FM with discontinuities particularly when the under-lying AM-FM signal is over-modulated. In this letter, we analyzethe phase discontinuities and propose a technique to computesmooth AM and FM from the AS, by relaxing the non-negativityconstraint on the AM. The proposed technique iseffective athandling over-modulated signals. We present simulation results tosupport the theoretical calculations.

The amplitude-modulation (AM) and phase-modulation (PM) of an amplitude-modulated frequency-modulated (AM-FM) signal are defined as the modulus and phase angle, respectively, of the analytic signal (AS). The FM is defined as the derivative of the PM. However, this standard definition results in a PM with jump discontinuities in cases when the AM index exceeds unity, resulting in an FM that contains impulses. We propose a new approach to define smooth AM, PM, and FM for the AS, where the PM is computed as the solution to an optimization problem based on a vector interpretation of the AS. Our approach is directly linked to the fractional Hilbert transform (FrHT) and leads to an eigenvalue problem. The resulting PM and AM are shown to be smooth, and in particular, the AM turns out to be bipolar. We show an equivalence of the eigenvalue formulation to the square of the AS, and arrive at a simple method to compute the smooth PM. Some examples on synthesized and real signals are provided to validate the theoretical calculations.

We address the problem of temporal envelope modeling for transient audio signals. We propose the Gamma distribution function (GDF) as a suitable candidate for modeling the envelope keeping in view some of its interesting properties such as asymmetry, causality, near-optimal time-bandwidth product, controllability of rise and decay, etc. The problem of finding the parameters of the GDF becomes a nonlinear regression problem. We overcome the hurdle by using a logarithmic envelope fit, which reduces the problem to one of linear regression. The logarithmic transformation also has the feature of dynamic range compression. Since temporal envelopes of audio signals are not uniformly distributed, in order to compute the amplitude, we investigate the importance of various loss functions for regression. Based on synthesized data experiments, wherein we have a ground truth, and real-world signals, we observe that the least-squares technique gives reasonably accurate amplitude estimates compared with other loss functions.

We address the problem of prediction of multivariate data process using an underlying graph model. We develop a method that learns a sparse partial correlation graph in a tuning-free and computationally efficient manner. Specifically, the graph structure is learned recursively without the need for cross validation or parameter tuning by building upon a hyperparameter-free framework. Our approach does not require the graph to be undirected and also accommodates varying noise levels across different nodes. Experiments using real-world datasets show that the proposed method offers significant performance gains in prediction, in comparison with the graphs frequently associated with these datasets.

Square-root (in particular, Cholesky) factorization of Toeplitz matrices and of their inverses is a classical area of research. The Schur algorithm yields directly the Cholesky factorization of a symmetric Toeplitz matrix, whereas the Levinson algorithm does the same for the inverse matrix. The objective of this letter is to use results from the theory of rational orthonormal functions to derive square-root factorizations of the inverse of an positive definite Toeplitz matrix. The main result is a new factorization based on the Takenaka-Malmquist functions, that is parameterized by the roots of the corresponding auto-regressive polynomial of order. We will also discuss briefly the connection between our analysis and some classical results such as Schur polynomials and the Gohberg-Semencul inversion formula.

We consider a robust MIMO transceiver design to minimize mean square errors, by taking into account the imperfect channel state information from a worst-case robustness perspective. We show that for a given precoder, the optimal robust equalizer is obtained through channel diagonalization, and vice versa. Thus, a joint but usually suboptimal transceiver design can be readily obtained through alternately optimizing the equalizer and precoder, where in each iteration only a scalar problem is to be solved. We further propose efficient algorithms for the scalar optimization problems.

In this letter, we present a rotation-aided arctangent phase discriminator (RaAPD) with one-bit analog-to-digital conversion. Different from the existing digital phase discriminator (DPD) and noise-balanced digital phase discriminator (NB-DPD), the proposed RaAPD can achieve higher accuracy and better noise robust features through utilizing an extra rotation channel in the arctangent phase discriminator (APD). Experimental results show that RaAPD achieves 98.3%, 79.3%, and 79.4% reduction in terms of the average root-mean-square error of phase estimation with the signal noise ratio range [-20 dB, 20 dB] and a 16.384-MHz sampling frequency comparing to DPD, NB-DPD, and APD, respectively.

In this letter, the linear predictability of discrete-time stationary stochastic processes with 1/vertical bar f vertical bar(alpha)-shaped power spectral density (PSD) is considered. In particular, the spectral flatness measure (SFM)-which yields a lower bound for the normalized mean-squared-error (NMSE) of any linear one-step-ahead (OSA) predictor-is obtained analytically as a function of alpha is an element of [0, 1]. By comparing the SFM bound to the NMSE of the p-tap linear minimum-mean-square error (LMMSE) predictor, it is shown that close to optimal NMSE performance may be achieved for relatively moderate values of. The performance of the LMMSE predictor for the discrete-time fractional Gaussian noise (DFGN), which may be viewed as the conventional discrete-time counterpart of continuous-time processes with 1/vertical bar f vertical bar(alpha)-shaped PSD, shows that the DFGN is more easily predicted than the discrete-time processes considered herein.

We investigate a wireless network localization scenario in which the need for synchronized nodes is avoided. It consists of a set of fixed anchor nodes transmitting according to a given sequence and a self-localizing receiver node. The setup can accommodate additional nodes with unknown positions participating in the sequence. We propose a localization method which is robust with respect to uncertainty of the anchor positions and other system parameters. Further, we investigate the Cramer-Rao bound for the considered problem and show through numerical simulations that the proposed method attains the bound.

For array processing, we consider the problem of estimating signals of interest, and their directions of arrival (DOA), in unknown colored noise fields. We develop an estimator that efficiently utilizes a set of noise-only samples and, further, can incorporate prior knowledge of the DOAs with varying degrees of certainty. The estimator is compared with state of the art estimators that utilize noise-only samples, and the Cramer-Rao bound, exhibiting improved performance for smaller sample sets and in poor signal conditions.

We consider the problem of estimating a random state vector when there is information about the maximum distances between its subvectors. The estimation problem is posed in a Bayesian framework in which the minimum mean square error (MMSE) estimate of the state is given by the conditional mean. Since finding the conditional mean requires multidimensional integration, an approximate MMSE estimator is proposed. The performance of the proposed estimator is evaluated in a positioning problem. Finally, the application of the estimator in inequality constrained recursive filtering is illustrated by applying the estimator to a dead-reckoning problem. The MSE of the estimator is compared with two related posterior Cramer-Rao bounds.

For reconstruction of low-rank matrices from undersampled measurements, we develop an iterative algorithm based on least-squares estimation. While the algorithm can be used for any low-rank matrix, it is also capable of exploiting a-priori knowledge of matrix structure. In particular, we consider linearly structured matrices, such as Hankel and Toeplitz, as well as positive semidefinite matrices. The performance of the algorithm, referred to as alternating least-squares (ALS), is evaluated by simulations and compared to the Cramer-Rao bounds.

This letter studies the collaborative use of amplify-and-forward (AF) relays to form a virtual multiple-input single-output (MISO) beamforming system with the aid of perfect channel state information (CSI) in a flat-fading channel. In particular, we optimize the relay weights jointly to maximize the received signal-to-noise ratio (SNR) at the destination terminal with both individual and total power constraints at the relays. We show that the optimal collaborative-relay beamforming (CRB) solution achieves the full diversity of a MISO antenna system. Another main contribution of this letter is a distributed algorithm that allows each individual relay to learn its own weight, based on the Karush-Kuhn-Tucker (KKT) analysis.