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High-order networks are weighted hypergraphs collecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high-order networks have been defined but they are difficult to compute when the number of nodes is large. The goal here is to find tractable approximations of these network distances. The paper does so by mapping high-order networks to filtrations of simplicial complexes and showing that the distance between networks can be lower bounded by the difference between the homological features of their respective filtrations. Practical implications are explored by classifying weighted pairwise networks constructed from different generative processes and by comparing the coauthorship networks of engineering and mathematics academic journals. The persistent homology methods succeed in identifying different generative models, in discriminating engineering and mathematics communities, as well as in differentiating engineering communities with different research interests. [ABSTRACT FROM PUBLISHER]

In this paper, the design of non-orthogonal multiple access (NOMA) in a multiple-input-single-output (MISO) downlink scenario is investigated. The impact of the recently developed concept, quasi-degradation, on NOMA downlink transmission is first studied. Then, a Hybrid NOMA (H-NOMA) precoding algorithm, based on this concept, is proposed. By exploiting the properties of H-NOMA precoding, a low-complexity sequential user pairing algorithm is consequently developed, to further improve the overall system performance. Both analytical and numerical results are provided to demonstrate the performance of the H-NOMA precoding through the average power consumption and outage probability, while conventional schemes, as dirty-paper coding and zero-forcing beamforming, are used as benchmarking. [ABSTRACT FROM PUBLISHER]

We study convex optimization problems that feature low-rank matrix solutions. In such scenarios, non-convex methods offer significant advantages over convex methods due to their lower space complexity, as well as practical faster convergence. Under mild assumptions, these methods feature global convergence guarantees. In this paper, we extend the results on this matter by following a different path. We derive a non-Euclidean optimization framework in the non-convex setting that takes nonlinear gradient steps on the factors. Our framework enables the possibility to further exploit the underlying problem structures, such as sparsity or low-rankness on the factorized domain, or better dimensional dependence of the smoothness parameters of the objectives. We prove that the non-Euclidean methods enjoy the same rigorous guarantees as their Euclidean counterparts under appropriate assumptions. Numerical evidence with Fourier ptychography and FastText applications, using real data, shows that our approach can enhance solution quality, as well as convergence speed over the standard non-convex approaches. [ABSTRACT FROM AUTHOR]

This paper proposes a novel linear estimator named stochastic MV-PURE estimator, developed for the stochastic linear model, and designed to provide improved performance over the linear minimum mean square error (MMSE) Wiener estimator in cases prevailing in practical, real-world settings, where at least some of the second-order statistics of the random vectors under consideration are only imperfectly known. The proposed estimator shares its main mathematical idea and terminology with the recently introduced minimum-variance pseudo-unbiased reduced-rank estimator (MV-PURE), developed for the linear regression model. The proposed stochastic MV-PURE estimator minimizes the mean square error (MSE) of its estimates subject to rank constraint and inducing minimium distortion to the target random vector. Therefore, the stochastic MV-PURE combines the techniques of the reduced rank Wiener filter (named in this paper RR-MMSE) and the distortionless-constrained estimator (named in this paper C-MMSE), in order to achieve greater robustness against noise or model errors than RR-MMSE and C-MMSE. Furthermore, to ensure that the stochastic MV-PURE estimator combines the reduced-rank and minimum-distortion approaches in the MSE-optimal way, we propose a rank selection criterion which minimizes the MSE of the estimates obtained by the stochastic MV-PURE. As a numerical example, we employ the stochastic MV-PURE, RR-MMSE, C-MMSE, and MMSE estimators as linear receivers in a MIMO wireless communication system. This example is chosen as a typical signal processing scenario, where the statistical information on the data, on which the estimates are built, is only imperfectly known. We verify that the stochastic MV-PURE achieves the lowest MSE and symbol error rate (SER) in such settings by employing the proposed rank selection criterion. [ABSTRACT FROM AUTHOR]

This paper presents a rapid and accurate time-synchronization technique using an LED and an off-the-shelf camera. We discuss optimally modulated illumination as a function of camera exposure time through theoretical analysis and derive its mathematical representation. Experiments in real environments with a single LED and 60 fps camera show that the proposed technique can achieve time synchronization within a 17.4 $\mu$ s error at the 90th percentile using only four frames (a measurement time of 0.067 s). Comparative experiments with computer simulations prove that the proposed technique using optimally modulated illumination to camera exposure time needs less CPU time than that using nonoptimally modulated illumination. The performance of the proposed technique derived from the noise distribution of intensities in captured images agrees closely with the experimental results, enabling time synchronization to be estimated within a confidence interval at a given confidence level. Possible applications to be developed by utilizing features of the proposed technique are described. [ABSTRACT FROM PUBLISHER]

This paper considers phase retrieval from the magnitude of one-dimensional over-sampled Fourier measurements, a classical problem that has challenged researchers in various fields of science and engineering. We show that an optimal vector in a least-squares sense can be found by solving a convex problem, thus establishing a hidden convexity in Fourier phase retrieval. We then show that the standard semidefinite relaxation approach yields the optimal cost function value (albeit not necessarily an optimal solution). A method is then derived to retrieve an optimal minimum phase solution in polynomial time. Using these results, a new measuring technique is proposed which guarantees uniqueness of the solution, along with an efficient algorithm that can solve large-scale Fourier phase retrieval problems with uniqueness and optimality guarantees. [ABSTRACT FROM PUBLISHER]

Molecular communication (MC) will enable the exchange of information among nanoscale devices. In this novel bio-inspired communication paradigm, molecules are employed to encode, transmit and receive information. In the most general case, these molecules are propagated in the medium by means of free diffusion. An information theoretical analysis of diffusion-based MC is required to better understand the potential of this novel communication mechanism. The study and the modeling of the noise sources is of utmost importance for this analysis. The objective of this paper is to provide a mathematical study of the noise at the reception of the molecular information in a diffusion-based MC system when the ligand-binding reception is employed. The reference diffusion-based MC system for this analysis is the physical end-to-end model introduced in a previous work by the same authors, where the reception process is realized through ligand-binding chemical receptors. The reception noise is modeled in this paper by following two different approaches, namely, through the ligand-receptor kinetics and through the stochastic chemical kinetics. The ligand-receptor kinetics allows to simulate the random perturbations in the chemical processes of the reception, while the stochastic chemical kinetics provides the tools to derive a closed-form solution to the modeling of the reception noise. The ligand-receptor kinetics model is expressed through a block scheme, while the stochastic chemical kinetics results in the characterization of the reception noise using stochastic differential equations. Numerical results are provided to demonstrate that the analytical formulation of the reception noise in terms of stochastic chemical kinetics is compliant with the reception noise behavior resulting from the ligand-receptor kinetics simulations. [ABSTRACT FROM AUTHOR]

Due to their linear-phase property, symmetric filters are an interesting class of finite-impulse-response (FIR) filters. Moreover, symmetric FIR filters allow an efficient implementation. In this paper we extend the classical definition of Hermitian symmetry to a more general symmetry that is also applicable to complex filters. This symmetry is called generalized-Hermitian symmetry. We show the usefulness of this definition as it allows for a unified treatment of even and odd-length filters. Central in this paper is a theorem on the reduction of generalized-Hermitian-symmetric filters to Hermitian-symmetric filters, both with finite precision coefficients. A constructive proof of this theorem is presented and an associated procedure for reducing generalized-Hermitian-symmetric filters is derived. Two of the examples show the application of the reduction procedure and the achieved savings on arithmetic costs. Finally, all three examples show that a special instance of the generalized-Hermitian-symmetric filters with finite precision coefficients, may have lower arithmetic costs than the Hermitian-symmetric filter from which it is derived. [ABSTRACT FROM AUTHOR]

In order to find sparse approximations of signals, an appropriate generative model for the signal class has to be known. If the model is unknown, it can be adapted using a set of training samples. This paper presents a novel method for dictionary learning and extends the learning problem by introducing different constraints on the dictionary. The convergence of the proposed method to a fixed point is guaranteed, unless the accumulation points form a continuum. This holds for different sparsity measures. The majorization method is an optimization method that substitutes the original objective function with a surrogate function that is updated in each optimization step. This method has been used successfully in sparse approximation and statistical estimation [e.g., expectation-maximization (EM)] problems. This paper shows that the majorization method can be used for the dictionary learning problem too. The proposed method is compared with other methods on both synthetic and real data and different constraints on the dictionary are compared. Simulations show the advantages of the proposed method over other currently available dictionary learning methods not only in terms of average performance but also in terms of computation time. [ABSTRACT FROM AUTHOR]

In this paper, we systematically derive a large class of fast general-radix algorithms for various types of real discrete Fourier transforms (real DFTs) including the discrete Hartley transform (DHT) based on the algebraic signal processing theory. This means that instead of manipulating the transform definition, we derive algorithms by manipulating the polynomial algebras underlying the transforms using one general method. The same method yields the well-known Cooley-Tukey fast Fourier transform (FFT) as well as general radix discrete cosine and sine transform algorithms. The algebraic approach makes the derivation concise, unifies and classifies many existing algorithms, yields new variants, enables structural optimization, and naturally produces a human-readable structural algorithm representation based on the Kronecker product formalism. We show, for the first time, that the general-radix Cooley-Tukey and the lesser known Bruun algorithms are instances of the same generic algorithm. Further, we show that this generic algorithm can be instantiated for all four types of the real DFT and the DHT. [ABSTRACT FROM AUTHOR]

This paper considers localization of a source or a sensor from distance measurements. We argue that linear algorithms proposed for this purpose are susceptible to poor noise performance. Instead given a set of sensors/anchors of known positions and measured distances of the source/sensor to be localized from them we propose a potentially nonconvex weighted cost function whose global minimum estimates the location of the source/sensor one seeks. The contribution of this paper is to provide nontrivial ellipsoidal and polytopic regions surrounding these sensors/anchors of known positions, such that if the object to be localized is in this region, localization occurs by globally exponentially convergent gradient descent in the noise free case. Exponential convergence in the noise free case represents practical convergence as it ensures graceful performance degradation in the presence of noise. These results guide the deployment of sensors/anchors so that small subsets can be made responsible for practical localization in geographical areas determined by our approach. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of adaptively allocating the bits and the power among a set of parallel subchannels. A frame-oriented transmission with convolutional coding, hard Viterbi decoding, and automatic repeat request (ARQ) retransmission protocol is considered. The objective is to maximize the number of information bits delivered without error to the user by unit of time, or goodput. This criterion is a tradeoff between the bit rate and bit error probability criteria. First, a mathematical expression for the goodput of the system is presented. Then, using that expression, different bit and power allocation strategies are derived and compared. It turns out that having an equal bit error rate (BER) on each subchannel is near-optimal in terms of goodput. Finally, we prove that the waterfilling solution with adequate signal to noise ratio (SNR) gap value (or equivalently adequate target BER value) is near-optimal. We analytically show how this SNR gap value depends on the convolutional code used, the frame length, the type of ARQ protocol used, the available transmit power, and the subchannel gains. [ABSTRACT FROM AUTHOR]

This paper proposes a novel estimator named minimum-variance pseudo-unbiased reduced-rank estimator (MVPURE) for the linear regression model, designed specially for the case where the model matrix is ill-conditioned and the unknown deterministic parameter vector to be estimated is subjected to known linear constraints. As a natural generalization of the Gauss-Markov (BLUE) estimator, the MV-PURE estimator is a solution of the following hierarchical nonconvex constrained optimization problem directly related to the mean square error expression. In the first-stage optimization, under a rank constraint, we minimize simultaneously all unitarily invariant norms of an operator applied to the unknown parameter vector in view of suppressing bias of the proposed estimator. Then, in the second-stage optimization, among all pseudo-unbiased reduced-rank estimators defined as the solutions of the first-stage optimization, we find the one achieving minimum variance. We derive a closed algebraic form of the MV-PURE estimator and show that well-known estimators-the Gauss-Markov (BLUE) estimator, the generalized Marquardt's reduced-rank estimator, and the minimum-variance conditionally unbiased affine estimator subject to linear restrictions-are all special cases of the MV-PURE estimator. We demonstrate the effectiveness of the proposed estimator in a numerical example, where we employ the MV-PURE estimator to the ill-conditioned problem of reconstructing a 2-D image subjected to linear constraints from blurred, noisy observation. This example demonstrates that the MV-PURE estimator outperforms all aforementioned estimators, as it achieves smaller mean square error for all values of signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

The problems addressed by this paper is the following: Given a set of measurements over the range of normalized frequencies (ϴ1, ϴ2) on the magnitude and/or phase of a real FIR but otherwise unknown filter, to estimate the order of the FIR filter. The range (ϴ1, ϴ2) may be partial or it may cover the entire range of frequencies. The purpose of the paper is to propose a new algebraic approach to solve the above collection of problems. Specific new results include FIR order estimation from partial or complete noiseless measurements for a real system from: a) phase alone, from b) magnitude alone (not necessarily piece-wise constant), and from c) joint magnitude and phase. The proposed approach is not only capable of dealing with specifications that go beyond the conventional formulas for the standard piece-wise-constant magnitude FIR filter order estimation, but it also furnishes a nexus for order estimation from phase (or group delay) specifications, areas which have remained hitherto unaddressed. The approach is based on the use of Root Moments. In this context, the novel concept of Fractional Root Moments is used in a key fashion to provide an estimate on the number of zeros inside the unit circle. Open problems and new directions of exploration and research are mentioned in the body of the paper. [ABSTRACT FROM AUTHOR]

This paper presents a computationally simple and accurate method to compute the error probabilities in decentralized detection in sensor networks. The cost of the direct computation of these probabilities—e.g., the probability of false alarm, the probability of a miss, or the average error probability—is combinatorial in the number of sensors and becomes infeasible even with small size networks. The method is based on the theory of large deviations, in particular, the saddlepoint approximation and applies to generic parallel fusion sensor networks, including networks with nonidentical sensors, nonidentical observations, and unreliable communication links. The paper demonstrates with parallel fusion sensor network problems the accuracy of the saddlepoint methodology: 1) computing the detection performance for a variety of small and large sensor network scenarios; and 2) designing the local detection thresholds. Elsewhere, we have used the saddlepoint approximation to study tradeoffs among parameters for networks of arbitrary size. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of blind separation of an instantaneous mixture of sources (BSS), which has been addressed in many ways. When power spectral densities of the sources are different, methods using second-order statistics are sufficient to solve this problem. Otherwise, these methods fail and others (higher order statistics, etc.) must be used. In this paper, we propose an iterative method to process the case of sources with the same power spectral density. This method is based on an evaluation of conditional first and second-order statistics only. Restrictions on characteristics of sources are given to reach a solution, and proofs of convergence of the algorithm are provided for particular cases of probability density functions. Robustness of this algorithm with respect to the number of sources is shown through computer simulations. A particular case of sources that have a probability density function with unbounded domain of definition is described; here, the algorithm does not lead directly to a separation state but to an a priori known mixture state. Finally, prospects of links with contrast functions are mentioned, with a possible generalization of them based on results obtained with particular sources. [ABSTRACT FROM AUTHOR]

Waveform design is a pivotal component of the fully adaptive radar construct. In this paper, we consider waveform design for radar space time adaptive processing (STAP), accounting for the waveform dependence of the clutter correlation matrix. Due to this dependence, in general, the joint problem of receiver filter optimization and radar waveform design becomes an intractable, nonconvex optimization problem, Nevertheless, it is, however, shown to be individually convex either in the filter or in the waveform variables. We derive constrained versions of a) the alternating minimization algorithm, b) proximal alternating minimization, and c) the constant modulus alternating minimization, which, at each step, iteratively optimizes either the STAP filter or the waveform independently. A fast and slow time model permits waveform design in radar STAP, but the primary bottleneck is the computational complexity of the algorithms. [ABSTRACT FROM PUBLISHER]

Filter-bank-based waveform processing has been suggested as an alternative for the plain cyclic-prefix (CP) orthogonal frequency-division multiplexing (OFDM) based schemes in fifth-generation and future wireless communication systems. This is because of the new requirements, such as asynchronous and mixed numerology scenarios supporting multi-service operation in a common framework, including enhanced mobile broadband, low-latency high-reliability communications, and low-rate machine-type communications (MTC). Nevertheless, advanced multicarrier waveforms impose significantly increased computational complexity compared to the CP-OFDM scheme. Multirate fast-convolution (FC) processing has recently been proposed as an effective implementation for advanced waveforms, such as filtered OFDM (F-OFDM) and filter-bank multicarrier (FBMC) schemes, providing extreme flexibility in the subband spectral control. In this paper, we investigate the computational complexity of FC-based waveform processing and propose two computationally efficient schemes using the idea of circular convolution decomposition. The first scheme targets at narrow bandwidth scenarios, such as MTC. The second scheme considers dense spectral use of non-overlapping subbands. Both schemes achieve significant reduction in the computational complexity compared to direct FC and polyphase filter-bank-based implementations. This reduction in the complexity is achieved without performance loss with respect to direct FC processing. Mathematical analyses are provided for both schemes, along with evaluation and comparison of the computational complexities considering F-OFDM and FBMC waveforms in long-term-evolution-like scenarios. [ABSTRACT FROM AUTHOR]

Signal processing community has recently shown interest in Ramanujan sums, which were defined by S. Ramanujan in 1918. In this paper, we propose a family of sequences, termed here as orthogonal Ramanujan sums, $c^{k}_q(n)$ and also derive their various properties. We also prove two new properties of the Ramanujan Sums, namely, shifted multiplication property and interpolation property. These properties are used to define higher order orthogonal Ramanujan sums. These sums are periodic in nature and are orthogonal for different values of $k$ and $q$. Using orthogonal Ramanujan sums, a new representation of finite length signal is given, termed here as orthogonal Ramanujan periodic transform. The coefficients obtained using this transform can be divided into two parts, first coefficient gives smoothing component and the rest give the detail components of the signal. Further, application of the orthogonal Ramanujan sums in discrete wavelet transform is developed. Construction of Ramanujan wavelet analysis matrix for applying discrete wavelet transform on a signal is described. The application of this matrix on an image of size $N\times N$ , where $N$ is divisible by integer $q$ will result in $q^2-1$ detail components and one average component in transform domain. Some of the applications of this transform are also demonstrated, especially in the field of image processing. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel generalization of the graph Fourier transform (GFT). Our approach is based on separately considering the definitions of signal energy and signal variation, leading to several possible orthonormal GFTs. Our approach includes traditional definitions of the GFT as special cases, while also leading to new GFT designs that are better at taking into account the irregular nature of the graph. As an illustration, in the context of sensor networks we use the Voronoi cell area of vertices in our GFT definition, showing that it leads to a more sensible definition of graph signal energy even when sampling is highly irregular. [ABSTRACT FROM AUTHOR]

In this paper, we provide a performance analysis of cognitive radio systems employing energy detection based on power spectral density estimation. Specifically, we derive mathematical expressions for the probabilities of false alarm and missed detection when the periodogram estimate is used as a decision statistic. First, using the characteristic function of the quadratic form representation of the periodogram, we study the characteristics of the product of the sample covariance matrix of the observed vector and the symmetric matrix of the quadratic form. We have found that this product is Rank-1 with one non-zero eigenvalue and therefore a simplified expression for the cumulative density function (CDF) is obtained. Second, the CDF is exploited to derive accurate mathematical expressions for the probabilities of false alarm and missed detection in both Rayleigh and Rician fading channels. Also, for each case, we have derived the probability distribution function (PDF) of the corresponding eigenvalues. The results demonstrate that the probabilities of false alarm and missed detection are independent of the length of the observations when all the other parameters remain fixed. The results also reveal that the considered frequency-domain based energy detector is superior to its conventional time-domain counterpart as it provides fewer false alarms. [ABSTRACT FROM PUBLISHER]

The Chinese remainder theorem (CRT) allows to reconstruct a large integer from its remainders modulo several moduli. In this paper, we propose a robust reconstruction algorithm called robust CRT when the remainders have errors. We show that, using the proposed robust CRT, the reconstruction error is upper bounded by the maximal remainder error range named remainder error bound, if the remainder error bound is less than one quarter of the greatest common divisor (gcd) of all the moduli. We then apply the robust CRT to estimate frequencies when the signal waveforms are undersampled multiple times. It shows that with the robust CRT, the sampling frequencies can be significantly reduced. [ABSTRACT FROM AUTHOR]

The PARAFAC decomposition of a higher-order tensor is a powerful multilinear algebra tool that becomes more and more popular in a number of disciplines. Existing PARAFAC algorithms are computationally demanding and operate in batch mode-both serious drawbacks for on-line applications. When the data are serially acquired, or the underlying model changes with time, adaptive PARAFAC algorithms that can track the sought decomposition at low complexity would be highly desirable. This is a challenging task that has not been addressed in the literature, and the topic of this paper. Given an estimate of the PARAFAC decomposition of a tensor at instant t, we propose two adaptive algorithms to update the decomposition at instant t + 1, the new tensor being obtained from the old one after appending' a new slice in the 'time' dimension. The proposed algorithms can yield estimation performance that is very close to that obtained via repeated application of state-of-art batch algorithms, at orders of magnitude lower complexity. The effectiveness of the proposed algorithms is illustrated using a MIMO radar application (tracking of directions of arrival and directions of departure) as an example. [ABSTRACT FROM AUTHOR]

One of the tasks for which empirical mode decomposition (EMD) is potentially useful is nonparametric signal denoising, an area for which wavelet thresholding has been the dominant technique for many years. In this paper, the wavelet thresholding principle is used in the decomposition modes resulting from applying EMD to a signal. We show that although a direct application of this principle is not feasible in the EMD case, it can be appropriately adapted by exploiting the special characteristics of the EMD decomposition modes. In the same manner, inspired by the translation invariant wavelet thresholding, a similar technique adapted to EMD is developed, leading to enhanced denoising performance. [ABSTRACT FROM AUTHOR]

In the context of cooperative wireless networks that convey data on slow fading channels, outage probability Po is the relevant performance index from the point of view of information theory. Derivation and minimization of this probability with respect to the relaying protocol parameters is of central importance. However, it is often hard to derive its expression, let alone to find its exact minimum for all possible values of the signal-to-noise ratio (SNR). This problem can be simplified by studying the behavior of Po in the asymptotic regime where the SNR p converges to infinity. In this regime, usually ρN+1 Po converges to a constant ~ where N is the number of relays. In this paper, a simple and general method for deriving and minimizing ξ with respect to the power distribution between the source and the relays, and with respect to the durations of the slots specified by the relaying protocol, is developed. While the proposed approach is designed for the high SNR regime, simulations show that outage probability is reduced in a similar proportion at moderate SNR. The method applies to a general class of radio channels that includes the Rayleigh and the Rice channels as particular cases. Convexity of ξ with respect to the design parameters is shown. Decode-and-forward, as well as amplify-and-forward protocols are considered in the half duplex mode. [ABSTRACT FROM AUTHOR]

This paper focuses on linear transceiver design for rate optimization in multiuser Gaussian multple-input multiple-output (MIMO) systems. We focus on two design criteria: 1) maximizing the weighted sum rate subject to a total power constraint; 2) maximizing the minimum user rate subject to a total power constraint. For these problems, new power allocation strategies are derived, which can be formulated as geometric programs (GPs) involving mean square errors (MSE5). Based on these solutions, we propose iterative algorithms, where each iteration contains the optimization of the uplink power, uplink receive filters, and downlink receive filters. Monotonic convergence of the algorithms is proved. Simulations show that the algorithms outperform existing linear schemes. Additionally, we extend the results to other variations of the problems, such as, the problem of sum-rate constrained or user-rate constrained power minimization, and the problem of sum-rate maximization subject to user-rate constraints and a total power constraint. [ABSTRACT FROM AUTHOR]

In the multiple description coding (MDC), the source is decomposed into two or more descriptions, and then transmitted over on/off channels. At the receiver end, each description can be reproduced with acceptable side distortion; reconstructed signals with lower central distortion can be accomplished by exploiting more descriptions. In order to apply the classical MDC over error-prone channels, we propose error-resilient multiple description coding (ERMDC) to utilize the inherent dependency and redundancy between descriptions to combat bit errors. In this paper, we develop a pair of the ERMDC encoder and central decoder based on the multiple description scalar quantizer (MDSQ). After deriving optimal and suboptimal ERMDC decoders, a novel index assignment algorithm is provided for the ERMDC encoder to improve the error detection capability of the ERMDC decoders. Analytical and experimental results show that the ERMDC out- performs the conventional MDSQ encoder-decoder pair, and achieves lower reconstruction distortions than the side distortion within a large range of bit error rates (BERs). [ABSTRACT FROM AUTHOR]

In our paper titled "Algebraic Signal Processing Theory: Foundation and 1-D Time" appearing in this issue of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, we presented the algebraic signal processing theory, an axiomatic and general framework for linear signal processing. The basic concept in this theory is the signal model defined as the triple (A, M, Φ), where A is a chosen algebra of filters, M an associated A-module of signals, and Φ is a generalization of the z-transform. Each signal model has its own associated set of basic SP concepts, including filtering, spectrum, and Fourier transform. Examples include infinite and finite discrete time where these notions take their well-known forms. In this paper, we use the algebraic theory to develop infinite and finite space signal models. These models are based on a symmetric space shift operator, which is distinct from the standard time shift. We present the space signal processing concepts of filtering or convolution, "z-transform," spectrum, and Fourier transform. For finite length space signals, we obtain 16 variants of space models, which have the 16 discrete cosine and sine transforms (DCTs/DSTs) as Fourier transforms. Using this novel derivation, we provide missing signal processing concepts associated with the DCTs/DSTs, establish them as precise analogs to the DFT, get deep insight into their origin, and enable the easy derivation of many of their properties including their fast algorithms. [ABSTRACT FROM AUTHOR]

The dual-tree complex wavelet transform (CWT) has recently received significant interest in the wavelet community, owing primarily to its directional selective and near-shift invariant properties. It has been shown that with two separate maximally decimated and dyadic decompositions where filters are offset by a half sample, the resulting CWT wavelet bases form an approximate Hilbert transform pair. In this paper, we present the design, implementation and applications of several families of orthogonal as well as biorthogonal rational-coefficient wavelet filters that satisfy the Hubert transform pair condition and meet other desirable properties such as high coding gain, good directional sensitivity, and sufficient degree of regularity. The wavelet filters presented here, which confirm to Selesnick's and Kingsbury's design schemes, are designed and implemented directly in the lattice and lifting domain using VLSI-friendly dyadic coefficients. We confirm the fact that rational-coefficient constraint does not impose a significant loss in terms of energy compaction, wavelet smoothness, time-invariance, or directionality. We also propose the time-reversal relationships between the two CWT filter pairs in lattice and lifting domain, thereby facilitating both the design and implementation process. In the end, we present several applications and evaluations to illustrate the performance of the proposed designs. [ABSTRACT FROM AUTHOR]

Given a correlated Gaussian signal, may a chi-squared law of probability always be used to describe a spectrogram coefficient distribution? If not, would a "chi-squared description" lead to an acceptable amount of error when detection problems are to be faced in the time-frequency domain? These two questions prompted the study reported in this paper. After deriving the probability distribution of spectrogram coefficients in the context of a noncentered Gaussian correlated signal, the Kullback-Leibler divergence is first used to evaluate to what extent the nonwhiteness of the signal and the Fourier analysis window impact the probability distribution of the spectrogram. To complete the analysis, a detection task formulated as a binary hypothesis test is considered. We evaluate the error committed on the probability of false alarm when the likelihood ratio test is expressed with chi-squared laws. From these results, a chi-squared description of the spectrogram distribution appears accurate when the analysis window used to construct the spectrogram decreases to zero at its boundaries, regardless of the level of correlation contained in the signal. When other analysis windows are used, the length of the window and the correlation contained in the analyzed signal impact the validity of the chi-squared description. [ABSTRACT FROM AUTHOR]

In this paper, a class of cooperative communication protocols with arbitrary N-relay nodes is proposed for wireless networks, in which each relay coherently combines the signals received from ,m (1 ≤ m ≤ N - 1) previous relays in addition to the signal from the source. Exact symbol-error-rate (SER) expressions for an arbitrary N-node network employing M'ary phase-shift-keying (MPSK) modulation or quadrature-amplitude modulation (QAM) are provided for the proposed class of protocols. Further, approximate expressions for the SER are derived and shown to be tight at high enough signal-to-noise ratio (SNR). Our analysis reveals an interesting result: The class of cooperative protocols shares the same asymptotic performance at high enough SNR and does not depend on m, the number of previous nodes involving in coherent detection, hence, the asymptotic performance of a simple cooperative scenario in which each relay combines the signals from the source and the previous relay is exactly the same as that for a much more complicated scenario in which each relay combines the signals from the source and all the previous relays. The theoretical results also confirm that full diversity equal to the number of cooperating nodes is indeed achievable by the proposed protocols. Finally, we formulate a power-allocation problem in order to minimize the SER of the system. The analysis shows that the optimum power allocation at different nodes follows a certain ordering, and that the power-allocation scheme at high SNR does not depend on the channel quality of the direct link between the source and the destination. Closed-form solutions for the optimal power-allocation problem are provided for some network topologies. Simulation results confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Differential space-time modulation (DSTM) is an encoding technique for multiantenna systems that allows the receiver to detect the transmitted signals without the knowledge of the fading channels. It can be viewed as an extension of the differential phase-shift keying scheme in single antenna systems. In this paper, we investigate the performance of multiantenna systems employing DSTM under spatially correlated Rayleigh fading channels. We present three expressions for calculating the pairwise error probability (PEP) of the DSTM scheme. The first expression allows us to numerically calculate the exact PEP. The remaining two are closed-form expressions of the exact PEP at asymptotically high signal-to-noise ratios and the PEP upper hound. It is found that when the distance matrix is proportional to the identity matrix, the error probability is determined by the channel correlation matrix and the initial transmitted code matrix. Based on the new PEP upper bound, we derive a criterion for optimizing the initial code matrix for this special case. Simulation results show that the performance of the differential space-time coded systems can be significantly improved by designing the initial transmitted code matrix according to the proposed design criterion. [ABSTRACT FROM AUTHOR]

In this paper, we propose a frequency-domain equalization technique for orthogonal frequency division multiplexing (OFDM) transmission over frequency-selective channels. We consider the case where the receiver analog front-end suffers from IQ-imbalance and the local oscillator suffers from carrier-frequency offset (CFO). While the IQ-imbalance results in a mirroring effect, the CFO induces inter-carrier interference (ICI). In addition to ICI, we consider the channel delay spread is larger than the cyclic prefix (CP). This means that inter-block interference (IBI) is present. The frequency-domain equalizer is obtained by transferring a time-domain equalizer (TEQ) to the frequency-domain resulting in a per-tone equalizer (PTEQ). Due to the presence of IQ-imbalance the conventional TEQ (where only one TEQ is applied to the received Sequence) is not sufficient to cope with the mirroring effect. A sufficient TEQ consists of two time-domain filters; one applied to the received sequence and another applied to a conjugated version of the received sequence. For the case of IQ-imbalance and CFO, the TEQs are designed according the basis expansion model (BEM) which showed to be able to cope with the ICI problem. Finally, in addition to the frequency-domain PTEQ design procedure, a training-based RLS type initialization scheme for direct per-tone equalization is also proposed. [ABSTRACT FROM AUTHOR]

Echo cancellation in DMT modems has been long studied as a means to improve transmission performance. Most of the current echo-canceller structures make use of FIR filters that emulate the echo-path adaptively. In frame-asynchronous operation, the misalignment of the TX and RX DFT windows (combined with other modem and channel parameters like: the transmit tone-mask and the echo-path impulse response) determines the echo input statistics and can worsen the conditioning of the coefficient update formula. This effect is analyzed in the paper, within the context of two specific algorithms: the circulant decomposition canceller (CDC) and also the time-domain canceller (TD-EC). Self-orthogonalizing (S-O) update formulas, implemented through compensation matrices, and their respective low-rank approximations are proposed as a method to improve the convergence rate and, more generally, to render the echo-canceller performances less sensitive to the input statistics. The method introduced does not employ additional power on the unsent tones. [ABSTRACT FROM AUTHOR]

This paper proposes a space-time decision feedback equalization (ST-DFE) assisted multiuser detection (MUD) scheme for multiple receiver antenna aided space division multiple access systems. A minimum bit error rate (MBER) design is invoked for the MUD, which is shown to be capable of improving the achievable bit error rate performance and enhancing the attainable system capacity over that of the standard minimum mean square error (MMSE) design. An adaptive implementation of the MBER ST-DFE assisted MUD is proposed using a stochastic gradient-based least bit error rate algorithm, which is demonstrated to consistently outperform the classical least mean square (LMS) algorithm, while achieving a lower computational complexity than the LMS algorithm for the binary signalling scheme. Our simulation results demonstrate that the MBER ST-DFE assisted MUD is more robust to channel estimation errors as well as to potential error propagation imposed by decision feedback errors, compared to the MMSE ST-DFE assisted MUD. [ABSTRACT FROM AUTHOR]

In this paper, we relax the usual assumptions in denoising that the data consist of a ‘true’ signal to which normally distributed noise is added. Instead of regarding noise as the high-frequency part in the data to be removed either by a ‘hard’ or ‘soft’ threshold, we define it as that part in the data which is harder to compress than the rest with the class of models considered. Here, we model the data by two histograms: one for the denoised signal and the other for the noise, both represented by wavelet coefficients. A code length can be calculated for each part, and by the principle of minimum description length the optimal decomposition results by minimization of the sum of the two code lengths. [ABSTRACT FROM AUTHOR]

This paper proposes a numerical-integration perspective on the Gaussian filters. A Gaussian filter is approximation of the Bayesian inference with the Gaussian posterior probability density assumption being valid. There exists a variation of Gaussian filters in the literature that derived themselves from very different backgrounds. From the numerical-integration viewpoint, various versions of Gaussian filters are only distinctive from each other in their specific treatments of approximating the multiple statistical integrations. A common base is provided for the first time to analyze and compare Gaussian filters with respect to accuracy, efficiency and stability factor. This study is expected to facilitate the selection of appropriate Gaussian filters in practice and to help design more efficient filters by employing better numerical integration methods. [ABSTRACT FROM AUTHOR]

This study considers robust filtering problems for uncertain two dimensional (2-D) discrete systems in nonlinear fractional transformation (NFT) representations. Like one-dimensional (1-D) systems, the NFT in this paper serves to complement the linear fractional transformation (LFT). Linear matrix inequality (LMI) formulations are given to analyze as well as synthesize robust filters. Numerical examples show the advantage of the NFT over the LFT in terms of not only computational savings but also substantial performance gains and verify the efficiency of the proposed techniques. [ABSTRACT FROM AUTHOR]

This paper describes a systematic technique for designing prototype filters for generating perfect-reconstruction (PR) orthogonal cosine-modulated and modified discrete Fourier transform filterbanks. In the proposed design scheme, the stophand energy of the prototype filter is minimized, and the basic unknowns are the angles of a special lattice structure used for implementing the prototype filter so that the PR property is automatically satisfied independent of the angle values. This selection of the unknowns makes the overall optimization problem unconstrained. Due to the fact that there are several local optima, the design is performed in multiple steps in order to arrive at least at a very good suboptimal solution. First, for the given number of channels, the length of the channel filters, and the stopband edge of the prototype filter, the corresponding two-channel filterbank is designed based on the preoptimized data. Then, after knowing the angles for the optimized two-channel case, the prototype filter for the desired filterbank is generated by gradually increasing the number of channels and by properly using the result of the previous step as a start-up solution for the present step. The main benefit of the proposed design technique is that it enables one to effectively design prototype filters for filterbanks with very high-order analysis and synthesis filters. [ABSTRACT FROM AUTHOR]

This paper focuses on the extension of the asymptotic covariance of the sample covariance (denoted Bartlett's formula) of linear processes to third- and fourth-order sample cumulant and to noisy linear processes. Closed-form expressions of the asymptotic covariance and cross-covariance of the sample second-, third-, and fourth-order cumulants are derived in a relatively straight forward manner, thanks to a matrix polyspectral representation and a symbolic calculus akin to a high-level language. As an application of these extended formulae, we underscore the sensitivity of the asymptotic performance of estimated ARMA parameters by an arbitrary third- or fourth-order-based algorithm with respect to the signal-to-noise ratio, the spectra of the linear process, and the colored additive noise. [ABSTRACT FROM AUTHOR]

The estimation of the frequency of a complex exponential is a problem that is relevant to a large number of fields. In this paper, we propose and analyze two new frequency estimators that interpolate on the Fourier coefficients of the received signal samples. The estimators are shown to achieve identical asymptotic performances. They are asymptotically unbiased and normally distributed with a variance that is only 1.0147 times the asymptotic Cramér-Rao bound (AC RB) uniformly over the frequency estimation range. [ABSTRACT FROM AUTHOR]

This paper deals with the design of gain-scheduled filters, whose state-space realization depends on real-time parameters of plants. Similar to well-recognized advantages of gain-scheduled controllers in control theory, gain-scheduled filters are expected to provide enhanced performance in comparison with customary nonadjustable filters. Our construction technique is based on nonlinear fractional transformation (NFT) representations of systems that are a generalization of widely used linear fractional transformation (LFT) representations. Both generalized &hemlit;2 and &hamlit;∞ discrete-time filter design problems are investigated together with their extension to mixed designs. This study leads to new linear matrix inequality (LMI) formulations, which in turn provide an effective and reliable design tool. The proposed design technique is finally evaluated in the light of simulation examples. [ABSTRACT FROM AUTHOR]

Recently, many applications for three-dimensional (3-D) image and video compression have been proposed using 3-D discrete cosine transforms (3-D DCTs). Among different types of DCTs, the type-II DCT (DCT-II) is the most used. In order to use the 3-D DCTs in practical applications, fast 3-D algorithms are essential. Therefore, in this paper, the 3-D vector-radix decimation-in-frequency (3-D VR DIF) algorithm that calculates the 3-D DCT-II directly is introduced. The mathematical analysis and the implementation of the developed algorithm are presented, showing that this algorithm possesses a regular structure, can be implemented in-place for efficient use of memory, and is faster than the conventional row-column-frame (RCF) approach. Furthermore, an application of 3-D video compression-based 3-D DCT-II is implemented using the 3-D new algorithm. This has led to a substantial speed improvement for 3-D DCT-II-based compression systems and proved the validity of the developed algorithm. [ABSTRACT FROM AUTHOR]

This paper analyzes both the mean-square error (MSE) and the mutual information in block-based transceivers that employ zero-padding. We consider both linear transceivers and decision-feedback equalizers (DFE) that minimize the MSE of symbols. These systems may enjoy the zero-forcing property or not, and may use unitary precoder or not. We demonstrate mathematically that the MSE/mutual information related to these transceivers are: (i) monotone increasing/decreasing functions of the number of transmitted symbols per block; (ii) monotone decreasing/increasing functions of the number of redundant data used in the equalization process of a block; and (iii) increased/decreased whenever non-minimum phase channels are utilized, instead of their minimum phase counterparts, assuming that one does not use the whole received data block to estimate the transmitted signal. As consequence of the former results, we also prove that, for both DFE and minimum error-probability systems, the average error-probability of symbols maintains the same monotonic behavior as the average MSE of symbols. The theoretical results do not include pure minimum MSE-based systems. [ABSTRACT FROM AUTHOR]

The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar/sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar/sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the \ell1 error of the best k-term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit \ell1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar. [ABSTRACT FROM AUTHOR]

In this paper, a new theory of optimal weighted non-linear filtering is presented. Two filter models are considered. The first model is based on a representation of the filter in the polynomial-like form with q terms where each term consists of weighted matrices and the matrix determined from the error minimization problem. The second model extends the first one to the case of the filter concatenation. The filter models are given in terms of pseudo-inverse matrices, i.e., the requirement of invertibility for covariance matrices is omitted. Thus, our filters always exist. We develop methods which allow us to exploit advantages associated with the proposed nonlinear filter models. The methods consist of the orthogonalization procedure and the reduction of the original problem to q individual minimization problems for smaller matrices. This leads to a considerable reduction in the required computational work. The error associated with the first filter model decreases when the number q of terms of filter increases. Its compression ratio can be adjusted by varying a particular value of ranks in each of its q terms. This means that the proposed filer structure provides the two degrees of freedom. The second filter model provides another degree of freedom, a number k of filters in the concatenation. Variations of the degrees of freedom improve the performance of the proposed filters. In particular, the error associated with the filter concatenation decreases as the filter number k in the concatenation increases. [ABSTRACT FROM AUTHOR]

It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang-Fix conditions of order r, this formula also provides an approximation scheme of order r valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support. [ABSTRACT FROM AUTHOR]

This paper introduces a new class of multilinear functions which can be used for analyzing a signal with time-varying frequency. The new class subsumes a number of existing functions, including the higher order ambiguity functions (HAFs), the polynomial Wigner-Ville distributions (PWVDs), and the higher order phase (HP) functions. As well as establishing a link between these existing functions, the new class provides a formalism which allows for the creation of useful new multilinear functions. A number of new functions are derived from the class. [ABSTRACT FROM AUTHOR]

Microphone arrays sample the sound field in both space and time with the major objective being the extraction of the signal propagating from a desired direction-of-arrival (DOA). In order to reconstruct a spatial sinusoid from a set of discrete samples, the spatial sampling must occur at a rate greater than a half of the wavelength of the sinusoid. This principle has long been adapted to the microphone array context: in order to form an unambiguous beampattern, the spacing between elements in a microphone array needs to conform to this spatial Nyquist criterion. The implicit assumption behind the narrowband beampattern is that one may use linearity and Fourier analysis to describe the response of the array to an arbitrary wideband plane wave. In this paper, this assumption is analyzed. A formula for the broadband beampattern is derived. It is shown that in order to quantify the spatial filtering abilities of a broadband array, the incoming signal's bifrequency spectrum must be taken into account, particularly for nonstationary signals such as speech. Multi-dimensional Fourier analysis is then employed to derive the broadband spatial transform, which is shown to be the limiting case of the broadband beampattern as the number of sensors tends to infinity. The conditions for aliasing in broadband arrays are then determined by analyzing the effect of computing the broadband spatial transform with a discrete spatial aperture. It is revealed that the spatial Nyquist criterion has little importance for microphone arrays. Finally, simulation results show that the well-known steered response power (SRP) method is formulated with respect to stationary signals, and that modifications are necessary to properly form steered beams in nonstationary signal environments. [ABSTRACT FROM AUTHOR]

This paper discusses the constrained stochastic gradient (CSG) algorithm used for controlling antenna arrays, aiming to maximize the signal-to-interference-plus-noise ratio (SINR) in mobile communications. Firstly, analytical expressions for the first moment of the weight vector and the SINR characteristic of the standard CSG algorithm are derived for two interferer signals, considering small step-size conditions and assuming Gaussian signal, interference, and noise. From these model expressions, the CSG algorithm performance is assessed, which predicts undesired behavior (termed here unbalanced behavior, pertaining to an unbalance between maximizing signal power and minimizing interference power) when one or more interference angles-of-arrival are close to the signal angle-of-arrival and the angle-of-arrival spreads of the involved signals are small. Finally, by using the model expressions, an improved CSG (ICSG) algorithm is proposed to compensate the unbalanced behavior of the standard CSG algorithm. The accuracy of the proposed model and the effectiveness of the modified algorithm are assessed through numerical simulations. [ABSTRACT FROM AUTHOR]