Showing papers on "Complex normal distribution published in 2018"

Phase Retrieval With Random Gaussian Sensing Vectors by Alternating Projections

Abstract: We consider a phase retrieval problem, where we want to reconstruct a $n$ -dimensional vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable initialization procedure, the classical algorithm of alternating projections (Gerchberg–Saxton) succeeds with high probability when $m\geq Cn$ , for some $C>0$ . We conjecture that this result is still true when no special initialization procedure is used, and present numerical experiments that support this conjecture.

Spatially-Stationary Propagating Random Field Model for Massive MIMO Small-Scale Fading

Abstract: The spatially uncorrelated Rayleigh small-scale fading model is a useful stochastic tool for analyzing multiple-antenna wireless communication systems, and, as experiments have shown, it often is a good approximation to physical propagation. However, the assumption that the propagating field is uncorrelated from one point in space to another breaks down when, for example, antenna spacings are smaller than one-half wavelength - a model defect typically addressed by assuming some spatial correlation. Spatial correlation can have huge effects even in the absence of close spacing between antennas. While an ad-hoc correlation versus distance, such as exponential, may add an element of realism to the model, in general it does not capture the peculiar “action at a distance” phenomena associated with the wave equation. The very desirable property of spatial stationarity can be retained, provided the spatial autocorrelation is chosen such that the complex Gaussian small-scale fading random field satisfies the homogeneous wave equation. The fading model that is closest to iid Rayleigh fading, and that is still consistent with the wave equation, has an autocorrelation equal to sinc(2πR/λ, corresponding to planewaves arriving uniformly from all directions, and having independent, equal variance complex Gaussian amplitudes. The contribution of this paper is twofold: first, a Fourier planewave representation that provides a computationally efficient way to generate samples of the random field, second, an inverse representation that enables the efficient computation of the joint likelihood of noisy measurements of the field over continuous segments of lines, planes, and volumes.

Independent Deeply Learned Matrix Analysis for Multichannel Audio Source Separation

Abstract: In this paper, we address a multichannel audio source separation task and propose a new efficient method called independent deeply learned matrix analysis (IDLMA). IDLMA estimates the demixing matrix in a blind manner and updates the time-frequency structures of each source using a pretrained deep neural network (DNN). Also, we introduce a complex Student's t-distribution as a generalized source generative model including both complex Gaussian and Cauchy distributions. Experiments are conducted using music signals with a training dataset, and the results show the validity of the proposed method in terms of separation accuracy and compnutational cost.

Complex Gaussian Processes for Regression

Abstract: In this paper, we propose a novel Bayesian solution for nonlinear regression in complex fields. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. This approach is limited. Based on the results in complex-valued linear theory and Gaussian random processes, we show that a pseudo-kernel must be included. This is the starting point to develop the new complex-valued formulation for Gaussian process for regression (CGPR). We face the design of the covariance and pseudo-covariance based on a convolution approach and for several scenarios. Just in the particular case where the outputs are proper, the pseudo-kernel cancels. Also, the hyperparameters of the covariance can be learned maximizing the marginal likelihood using Wirtinger’s calculus and patterned complex-valued matrix derivatives. In the experiments included, we show how CGPR successfully solves systems where the real and imaginary parts are correlated. Besides, we successfully solve the nonlinear channel equalization problem by developing a recursive solution with basis removal. We report remarkable improvements compared to previous solutions: a 2–4-dB reduction of the mean squared error with just a quarter of the training samples used by previous approaches.

Estimation of Rician K-Factor in the Presence of Nakagami- $m$ Shadowing for the LoS Component

Abstract: The received signal in Rician multipath channels is modeled as a complex constant, accounting for the main line-of-sight (LoS) path, added to a circular complex Gaussian random variable, modeling all the diffusive secondary paths. The Rician $K$ -factor is accordingly defined as the ratio between the powers of LoS and diffusive components. Under this (widely assumed) multipath model, a closed-form expression for the Rician $K$ -factor estimate is commonly obtained exploiting the second and fourth moments of the received signal amplitude. However, in mobile communications, random shadowing effects make the LoS component fluctuating (Rician shadowed model) leading to a performance degradation of the conventional estimator. With focus on land mobile satellite communications, in this letter, a novel estimation procedure of the Rician $K$ -factor, aimed at avoiding the bias error of the conventional estimator when the LoS component is Nakagami- ${m}$ distributed, is proposed and assessed. Specifically, the estimate is computed using the solution of a system of three equations and three unknowns in conjunction with the likelihood function of the received signal. The performance analysis shows that the new estimator can guarantee better performance than the conventional estimator.

Eigenvector correlations in the complex Ginibre ensemble

Abstract: The complex Ginibre ensemble is an $N\times N$ non-Hermitian random matrix over $\mathbb{C}$ with i.i.d. complex Gaussian entries normalized to have mean zero and variance $1/N$. Unlike the Gaussian unitary ensemble, for which the eigenvectors are distributed according to Haar measure on the compact group $U(N)$, independently of the eigenvalues, the geometry of the eigenbases of the Ginibre ensemble are not particularly well understood. In this paper we systematically study properties of eigenvector correlations in this matrix ensemble. In particular, we uncover an extended algebraic structure which describes their asymptotic behavior (as $N$ goes to infinity). Our work extends previous results of Chalker and Mehlig [CM98], in which the correlation for pairs of eigenvectors was computed.

Approximating the Permanent of a Random Matrix with Vanishing Mean

Abstract: The permanent is #P-hard to compute exactly on average for natural random matrices including matrices over finite fields or Gaussian ensembles. Should we expect that it remains #P-hard to compute on average if we only care about approximation instead of exact computation? In this work we take a first step towards resolving this question: We present a quasi-polynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean µ = O(ln ln n)^-1/8 to within inverse polynomial multiplicative error. (alternatively, one can achieve permanent approximation for matrices with mean µ = 1/polylog(n) in time 2^n^e, for arbitrarily small e>0). The proposed algorithm significantly extends the regime of matrices for which efficient approximation of the permanent is known. This is because unlike previous algorithms which require a stringent correlation between the signs of the entries of the matrix [1], [2] it can tolerate random ensembles in which this correlation is negligible (albeit non-zero). Among important special cases we note: 1) Biased Gaussian: each entry is a complex Gaussian with unit variance 1 and mean µ. 2) Biased Bernoulli: each entry is -1 + µ with probability 1/2, and 1 with probability 1/2. These results counter the common intuition that the difficulty of computing the permanent, even approximately, stems merely from our inability to treat matrices with many opposing signs. The Gaussian ensemble approaches the threshold of a conjectured hardness [3] of computing the permanent of a zero mean Gaussian matrix. This conjecture is one of the baseline assumptions of the BosonSampling paradigm that has received vast attention in recent years in the context of quantum supremacy experiments. We furthermore show that the permanent of the biased Gaussian ensemble is #P-hard to compute exactly on average. To our knowledge, this is the first natural example of a counting problem that becomes easy only when average case analysis and approximation are combined. On a technical level, our approach stems from a recent approach taken by Barvinok [1], [4], [5], [6] who used Taylor series approximation of the logarithm of a certain univariate polynomial related to the permanent. Our main contribution is to introduce an average-case analysis of such related polynomials. We complement our approach with a new technique for iteratively computing a Taylor series approximation of a function that is analytical in the vicinity of a curve in the complex plane. This method can be viewed as a computational version of analytic continuation in complex analysis.

Generalized Proper Complex Gaussian Ratio Distribution and Its Application to Statistical Inference for Frequency Response Functions

Abstract: The frequency response function governs many important processes. Defined as the quotients of the fast Fourier transform coefficients, frequency response functions can be modeled as ratio r...

Approximate Asymptotic Distribution of Locally Most Powerful Invariant Test for Independence: Complex Case

Abstract: Usually, it is very difficult to determine the exact distribution for a test statistic. In this paper, asymptotic distributions of locally most powerful invariant test for independence of complex Gaussian vectors are developed. In particular, its cumulative distribution function (CDF) under the null hypothesis is approximated by a function of chi-squared CDFs. Moreover, the CDF corresponding to the non-null distribution is expressed in terms of non-central chi-squared CDFs for close hypothesis, and Gaussian CDF as well as its derivatives for far hypothesis. The results turn out to be very accurate in terms of fitting their empirical counterparts. Closed-form expression for the detection threshold is also provided. Numerical results are presented to validate our theoretical findings.

Bit Error Probability for MMSE Receiver in GFDM Systems

Abstract: In this letter, we consider the minimum mean-square error receiver for the generalized frequency division multiplexing system (GFDM) over frequency selective fading channels. We derive an approximate probability density function for the signal-to-interference-plus-noise ratio. This expression allows us to obtain a new approximate, but rather accurate formulation for the bit error probability for a $\mathcal {M}$ -quadrature amplitude modulation scheme. Our results resort on the pivotal properties exhibited by eigenvalues of a circulant matrix. Since the entries of the channel matrix $\text {H}_{\text {ch}}$ are complex Gaussian distributed, and the eigenvalues are given as a weighted sum of its entries, the joint eigenvalue distribution is also Gaussian. Comparisons of the simulated and analytical results validate our formulation and allow a quick and efficient tool to compute the bit error rate for the GFDM system.

Permutation-Free Cgmm: Complex Gaussian Mixture Model with Inverse Wishart Mixture Model Based Spatial Prior for Permutation-Free Source Separation and Source Counting

Abstract: Here we propose a permutation-free cGMM (PF-cGMM), a new probabilistic model of observed mixtures, which can resolve permutation ambiguity between frequency bins, and is applicable even when the number of sources is unknown. A recently proposed complex Gaussian mixture model (cGMM) is highly effective for frequency bin-wise clustering when the number of sources is known. However, it cannot resolve the permutation ambiguity, and is inapplicable when the number of sources is unknown. The proposed PF-cGMM is an extension of the cGMM, which resolves these issues. The resolution of the permutation ambiguity can be realized by a spatial prior called a complex inverse Wishart mixture model (cIWMM). The absence of the permutation ambiguity facilitates source counting, which is performed by hierarchical clustering in this paper. Experiments showed that the PF-cGMM was able to (1) resolve the permutation ambiguity and (2) realize source separation even when the number of sources was unknown with little performance degradation compared to when it was known.

A note on “On the ratio of independent complex Gaussian random variables”

Abstract: Nadimi et al. (Multidimens Syst Signal Process 2017. https://doi.org/10.1007/s11045-017-0519-3 ) studied the distribution of the ratio of two independent complex Gaussian random variables. The expressions provided for the distribution involved a hypergeometric function and an infinite sum. Here, we derive simpler and more manageable expressions. The practical usefulness of the expressions in terms of computational time is illustrated.

On complex Gaussian random fields, Gaussian quadratic forms and sample distance multivariance

Abstract: The paper contains results in three areas: First we present a general estimate for tail probabilities of Gaussian quadratic forms with known expectation and variance. Thereafter we analyze the distribution of norms of complex Gaussian random fields (with possibly dependent real and complex part) and derive representation results, which allow to find efficient estimators for the moments of the associated Gaussian quadratic form. Finally, we apply these results to sample distance multivariance, which is the test statistic corresponding to distance multivariance -- a recently introduced multivariate dependence measure. The results yield new tests for independence of multiple random vectors. These are less conservative than the classical tests based on a general quadratic form estimate and they are (much) faster than tests based on a resampling approach. As a special case this also improves independence tests based on distance covariance, i.e., tests for independence of two random vectors.

Design of Discrete Constellations for Peak-Power-Limited complex Gaussian Channels

Abstract: The capacity-achieving input distribution of the complex Gaussian channel with both average- and peak-power constraint is known to have a discrete amplitude and a continuous, uniformly-distributed, phase. Practical considerations, however, render the continuous phase inapplicable. This work studies the backoff from capacity induced by discretizing the phase of the input signal. A sufficient condition on the total number of quantization points that guarantees an arbitrarily small backoff is derived, and constellations that attain this guaranteed performance are proposed.

From Integrable to Chaotic Systems: Universal Local Statistics of Lyapunov exponents

Abstract: Systems where time evolution follows a multiplicative process are ubiquitous in physics. We study a toy model for such systems where each time step is given by multiplication with an independent random $N\times N$ matrix with complex Gaussian elements, the complex Ginibre ensemble. This model allows to explicitly compute the Lyapunov exponents and local correlations amongst them, when the number of factors $M$ becomes large. While the smallest eigenvalues always remain deterministic, which is also the case for many chaotic quantum systems, we identify a critical double scaling limit $N\sim M$ for the rest of the spectrum. It interpolates between the known deterministic behaviour of the Lyapunov exponents for $M\gg N$ (or $N$ fixed) and universal random matrix statistics for $M\ll N$ (or $M$ fixed), characterising chaotic behaviour. After unfolding this agrees with Dyson's Brownian Motion starting from equidistant positions in the bulk and at the soft edge of the spectrum. This universality statement is further corroborated by numerical experiments, multiplying different kinds of random matrices. It leads us to conjecture a much wider applicability in complex systems, that display a transition from deterministic to chaotic behaviour.

An Edge Exclusion Test for Complex Gaussian Graphical Model Selection

Abstract: We consider the problem of inferring the conditional independence graph (CIG) of complex-valued multivariate Gaussian vectors. A p-variate complex Gaussian graphical model (CGGM) associated with an undirected graph with p vertices is defined as the family of complex Gaussian distributions that obey the conditional independence restrictions implied by the edge set of the graph. For real random vectors, considerable body of work exists where one first tests for exclusion of each edge from the saturated model, and then infers the CIG. Much less attention has been paid to CGGMs. In this paper, we propose and analyze a generalized likelihood ratio test based edge exclusion test statistic for CGGMs. The test statistic is expressed in an alternative form compared to an existing result, where the alternative expression is in a form usually given and exploited for real GGMs. The computational complexity of the proposed statistic is $\mathcal {O}(p^{3})$ compared to $\mathcal {O}(p^{5})$ for the existing result.

The largest root of random Kac polynomials is heavy tailed

Abstract: We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviations principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviations principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by the behavior at the origin of the distribution of the coefficients. We also prove that the random process of the roots of modulus smaller than one converges towards a limit point process. Finally, in the case of complex Gaussian coefficients, we use the work of Peres and Virag [15] to obtain explicit formulas for the limiting objects.

On The Non-Detectability of Spiked Large Random Tensors

Abstract: This paper addresses the detection of a low rank high-dimensional tensor corrupted by an additive complex Gaussian noise. In the asymptotic regime where all the dimensions of the tensor converge towards $+\infty $ at the same rate, existing results devoted to rank 1 tensors are extended. It is proved that if a certain parameter depending explicitly on the low rank tensor is below a threshold, then the null hypothesis and the presence of the low rank tensor are undistinguishable hypotheses in the sense that no test performs better than a random choice.

Sparse Bayesian Learning for DOA Estimation of Correlated Sources

Abstract: Direction of arrival (DOA) estimation from array observations in a noisy environment is discussed. The source amplitudes are assumed to be correlated zero-mean complex Gaussian distributed with unknown covariance matrix. The DOAs and covariance parameters of plane waves are estimated from multi-snapshot sensor array data using sparse Bayesian learning (SBL). The performance of SBL is evaluated in terms of the fidelity of the reconstructed coherency matrix of the estimated plane waves.

A series representation for multidimensional Rayleigh distributions

Abstract: The Rayleigh distribution is of paramount importance in signal processing and many other areas, yet an expression for random variables of arbitrary dimensions has remained elusive. In this note, we generalise the results of Beard and Tekinay [1] for quadrivariate random variables to cases of unconstrained order and provide a simple algorithm for evaluation. The assumptions of cross-correlation between in-phase and quadrature, as well as non-singularity of the covariance matrix are retained throughout our computations.

Phase Retrieval by Alternating Minimization with Random Initialization

Abstract: We consider a phase retrieval problem, where the goal is to reconstruct a $n$-dimensional complex vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a random initialization, the classical algorithm of alternating minimization succeeds with high probability as $n,m\rightarrow\infty$ when ${m}/{\log^3m}\geq Mn^{3/2}\log^{1/2}n$ for some $M>0$. This is a step toward proving the conjecture in \cite{Waldspurger2016}, which conjectures that the algorithm succeeds when $m=O(n)$. The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.

The Golden Quantizer: The Complex Gaussian Random Variable Case

Abstract: The problem of quantizing a circularly symmetric complex Gaussian random variable is considered. For this purpose, we design two non-uniform quantizers, a high-rate- and a Lloyd-Max-quantizer, that are both based on the (golden angle) spiral-phyllotaxis packing principle. We find that the proposed schemes have lower mean-square error distortion compared to (non)-uniform polar/rectangular-quantizers, and near-identical to the best performing trained vector quantizers. The proposed quantizer scheme offers a structured design, a simple natural index ordering, and allows for any number of centroids.

Weighted graphs and complex Gaussian free fields

Abstract: We prove a combinatorial lemma about the distribution of directed currents in a complex "loop soup" and use it to give a new proof of the isomorphism relating loop measures and complex Gaussian fields.

On the non-detectability of spiked large random tensors.

Abstract: This paper addresses the detection of a low rank high-dimensional tensor corrupted by an additive complex Gaussian noise. In the asymptotic regime where all the dimensions of the tensor converge towards $+\infty$ at the same rate, existing results devoted to rank 1 tensors are extended. It is proved that if a certain parameter depending on the low rank tensor is below a threshold, then the null hypothesis and the presence of the low rank tensor are undistinguishable hypotheses in the sense that no test performs better than a random choice.

Improvement of Greedy Algorithms With Prior Information for Sparse Signal Recovery

Abstract: In this paper, we improve greedy algorithms to recover sparse signals with complex Gaussian distributed nonzero elements, when the probability of sparsity pattern is known a priori. By exploiting this prior probability, we derive a correction function that minimizes the probability of incorrect selection of a support index at each iteration of the orthogonal matching pursuit (OMP). In particular, we employ the order statistics of exponential distribution to create the correction function. Simulation results demonstrate that the correction function significantly improves the recovery performance of OMP and subspace pursuit (SP) for random Gaussian and Bernoulli measurement matrices.

Low-Complexity Complex KLMS based Non-linear Estimators for OFDM Radar System

Abstract: Recently, kernel-based adaptive filtering (KAF) algorithms have found widespread application in numerous nonlinear signal processing problems; one of them being radar signal processing. In particular, considering the inherent non-linearity in a radar system, KAF has been recently applied for estimation of delay and found to achieve lower variance as compared to classical Fourier-Transform based approach. However, as the radar-return is complex-valued in general, using a traditional complex Gaussian kernel in KAF based estimator yields inaccurate estimates. In this work, we explore Wirtinger’s calculus-based complexification of a reproducing kernel Hilbert space (RKHS) for estimation of delay and Doppler-shift, which guarantees lower estimator-variance, and kernel-stability. Furthermore, since the choice of suitable kernel-width is crucial for RKHS-based estimation of delay and Doppler parameters, we derive an adaption for joint-estimation of kernel-width for the proposed normalized complex kernel least mean square (NCKLMS) based estimator from the radar return. Simulations performed over orthogonal frequency division multiplexed (OFDM)-radar system indicate that the proposed NCKLMS based estimator converges to a significantly lower dictionary-size, thereby leading to simpler implementation, receiver-simplicity, and latency whilst maintaining equivalent squared error performance, which makes the proposed estimators suitable for practical OFDM-radar systems.

A New Approach to the Statistical Analysis of Non-Central Complex Gaussian Quadratic Forms with Applications

Abstract: This paper proposes a novel approach to the statistical characterization of non-central complex Gaussian quadratic forms (CGQFs). Its key strategy is the generation of an auxiliary random variable (RV) that converges in distribution to the original CGQF. Since the mean squared error between both is given in a simple closed-form formulation, the auxiliary RV can be particularized to achieve the required accuracy. The technique is valid for both definite and indefinite CGQFs and yields simple expressions of the probability density function (PDF) and the cumulative distribution function (CDF) that involve only elementary functions. This overcomes a major limitation of previous approaches, in which the complexity of the resulting PDF and CDF prevents from using them for subsequent calculations. To illustrate this end, the proposed method is applied to maximal ratio combining systems over correlated Rician channels, for which the outage probability and the average bit error rate are derived.