Showing papers on "Complex normal distribution published in 2016"

A Geometric Analysis of Phase Retrieval

Abstract: Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of $m$ measurements, $y_k = |\mathbf a_k^* \mathbf x|$ for $k = 1, \dots, m$, is it possible to recover $\mathbf x \in \mathbb{C}^n$ (i.e., length-$n$ complex vector)? This **generalized phase retrieval** (GPR) problem is a fundamental task in various disciplines, and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretical explanations. In this paper, we take a step towards bridging this gap. We prove that when the measurement vectors $\mathbf a_k$'s are generic (i.i.d. complex Gaussian) and the number of measurements is large enough ($m \ge C n \log^3 n$), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) there are no spurious local minimizers, and all global minimizers are equal to the target signal $\mathbf x$, up to a global phase; and (2) the objective function has a negative curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.

Multisnapshot Sparse Bayesian Learning for DOA

Abstract: The directions of arrival (DOA) of plane waves are estimated from multisnapshot sensor array data using sparse Bayesian learning (SBL). The prior for the source amplitudes is assumed independent zero-mean complex Gaussian distributed with hyperparameters, the unknown variances (i.e., the source powers). For a complex Gaussian likelihood with hyperparameter, the unknown noise variance, the corresponding Gaussian posterior distribution is derived. The hyperparameters are automatically selected by maximizing the evidence and promoting sparse DOA estimates. The SBL scheme for DOA estimation is discussed and evaluated competitively against LASSO (l 1 -regularization), conventional beamforming, and MUSIC.

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. We assume the sensing vectors to be independently sampled from complex normal distributions. We propose to solve this problem with the classical non-convex method of alternating projections. We show that, when $m\geq Cn$ for $C$ large enough, alternating projections succeed with high probability, provided that they are carefully initialized. We also show that there is a regime in which the stagnation points of the alternating projections method disappear, and the initialization procedure becomes useless. However, in this regime, $m$ has to be of the order of $n^2$. Finally, we conjecture from our numerical experiments that, in the regime $m=O(n)$, there are stagnation points, but the size of their attraction basin is small if $m/n$ is large enough, so alternating projections can succeed with probability close to $1$ even with no special initialization.

Multi Snapshot Sparse Bayesian Learning for DOA Estimation

Abstract: The directions of arrival (DOA) of plane waves are estimated from multi-snapshot sensor array data using Sparse Bayesian Learning (SBL). The prior source amplitudes is assumed independent zero-mean complex Gaussian distributed with hyperparameters the unknown variances (i.e. the source powers). For a complex Gaussian likelihood with hyperparameter the unknown noise variance, the corresponding Gaussian posterior distribution is derived. For a given number of DOAs, the hyperparameters are automatically selected by maximizing the evidence and promote sparse DOA estimates. The SBL scheme for DOA estimation is discussed and evaluated competitively against LASSO ($\ell_1$-regularization), conventional beamforming, and MUSIC

Student's T nonnegative matrix factorization and positive semidefinite tensor factorization for single-channel audio source separation

Abstract: This paper presents a robust variant of nonnegative matrix factorization (NMF) based on complex Student's t distributions (t-NMF) for source separation of single-channel audio signals. The Itakura-Saito divergence NMF (Gaussian NMF) is justified for this purpose under an assumption that the complex spectra of source signals and those of the mixture signal are complex Gaussian distributed (the additiv-ity of power spectra holds). In fact, however, the source spectra are often heavy-tailed distributed. When the source spectra are complex Cauchy distributed, for example, the mixture spectra are also complex Cauchy distributed (the additivity of amplitude spectra holds). Using the complex t distribution that includes the complex Gaussian and Cauchy distributions as its special cases, we propose t-NMF as a unified extension of Gaussian NMF and Cauchy NMF. Furthermore, we propose the corresponding variant of positive semidefinite tensor factorization based on multivariate complex t distributions (t-PSDTF). The experimental results showed that while t-NMF and t-PSDTF were comparative to Gaussian counterparts in terms of peak performance, they worked much better on average because they are insensitive to initialization and tend to avoid local optima.

The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line

Abstract: We prove that if $\omega$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto \zeta(i\omega T+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta_T$ approach zero slowly enough as $T\to\infty$, then $t\mapsto \zeta(1/2+i\delta_T t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This essentially says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

Abstract: New bounds for the \(k\)th-order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature. We apply these bounds and local approach couplings to obtain an order \(n^{-(p-1)/2}\) bound, for smooth test functions, for the distance between the distribution of a standardised sum of independent and identically distributed random variables and the standard normal distribution when the first \(p\) moments of these distributions agree. We also obtain a bound on the convergence rate of a sequence of distributions to the normal distribution when the moment sequence converges to normal moments.

Dropping the Independence: Singular Values for Products of Two Coupled Random Matrices

Abstract: We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel–Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.

Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices

Abstract: We study the probability density function (PDF) of the smallest eigenvalue of Laguerre–Wishart matrices W = X†X where X is a random M × N (M ≥ N) matrix, with complex Gaussian independent entries. We compute this PDF in terms of semi-classical orthogonal polynomials, which are deformations of Laguerre polynomials. By analyzing these polynomials, and their associated recurrence relations, in the limit of large N, large M with M/N → 1 — i.e. for quasi-square large matrices X — we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painleve III equation, as found by Tracy and Widom, using Fredholm operator techniques. Furthermore, our method allows us to compute explicitly the first 1/N corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and Peche. We also study the soft edge limit, when M − N ∼𝒪(N), for which we conjecture the form of the first correction to the limiting distribution of the small...

Student's t multichannel nonnegative matrix factorization for blind source separation

Abstract: This paper presents a robust generalization of multichannel nonnegative matrix factorization (MNMF) for blind source separation of mixture audio signals recorded by a microphone array. In conventional MNMF, the complex spectra of observed mixture signals are assumed to be complex Gaussian distributed and are decomposed into the product of the power spectra, temporal activations, and spatial correlation matrices of individual sources in such a way that the complex Gaussian likelihood is maximized. Since the mixture spectra usually include outliers, we propose MNMF based on the complex Student's t likelihood, called t-MNMF, including the original MNMF as a special case. The parameters of t-MNMF can be iteratively optimized with an efficient multiplicative updating algorithm. Experiments showed that t-MNMF with a certain range of degrees of freedom tends to be insensitive to parameter initialization and outperform conventional MNMF.

On the Almost Sure Location of the Singular Values of Certain Gaussian Block-Hankel Large Random Matrices

Abstract: This paper studies the almost sure location of the eigenvalues of matrices WNW∗N , where WN=(W(1)TN,…,W(M)TN)T is a ML×N block-line matrix whose block-lines (W(m)N)m=1,…,M are independent identically distributed L×N Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if M→+∞ and MLN→c∗(c∗∈(0,∞)) , then the empirical eigenvalue distribution of WNW∗N converges almost surely towards the Marcenko–Pastur distribution. More importantly, it is established using the Haagerup–Schultz–Thorbjornsen ideas that if L=O(Nα) with α<2/3 , then, almost surely, for N large enough, the eigenvalues of WNW∗N are located in the neighbourhood of the Marcenko–Pastur distribution. It is conjectured that the condition α<2/3 is optimal.

Measurement and empirical modeling of massive MIMO channel matrix in real indoor environment

Abstract: A novel empirical, measurement-based massive MIMO channel matrix model under real indoor environment at 4.1 GHz is proposed. In this model, the channel matrix can be described as the sum of a fixed and a random matrices, furthermore, the random matrix can be decomposed into the product of four parts: the eigenvectors of the transmitting and receiving side correlation matrices, a coupling matrix, whose amplitude and phase are lognormal and uniform distributions, and a complex Gaussian random matrix. By comparing the capacity and singular value spread, the proposed model is validated to be more accurate than the conventional Kronecker and Weichselberger models. The proposed model is promising for the system design and implementation in future 5G systems.

Asymptotic Outage Probability of MIMO-MRC Systems in Double-Correlated Rician Environments

Abstract: We derive an asymptotic outage probability expression for multiple-input multiple-output maximum-ratio-combining (MIMO-MRC) systems considering Rician fading channels with both transmit and receive spatial correlation. This result is facilitated by a new asymptotic expression which we obtain for the cumulative distribution function of the maximum eigenvalue of a noncentral quadratic form in complex Gaussian matrices. Based on our analysis, we demonstrate that for a rank-1 line-of-sight (LoS) path, the interaction with the correlation matrices may yield improved or degraded performance. The relative effect depends largely on the geometric alignment of the LoS path with the eigenvectors of the correlation matrices. Generally speaking, when the LoS path aligns with the weak eigenvectors (i.e., those corresponding to the smallest eigenvalues), both a stronger LoS component, characterized by a higher Rician $K$ factor, and a higher correlation lead to improved performance. In contrast, when the LoS path aligns with the strong eigenvectors, we demonstrate that there exists distinct regions for which increasing the Rician $K$ factor has a beneficial or deleterious effect. Our results, in general, provide novel insight into how the mutual interactions of LoS and spatial correlation components impact the performance of MIMO-MRC systems.

On the Equivalence of Different Methods for Calculating Resonances: From Complex Gaussian Basis Set to Reflection-Free Complex Absorbing Potentials via the Smooth Exterior Scaling Transformation.

Abstract: The work compares different methods to compute metastable states (resonances) using Gaussian functions. The subject is of current interest, as methods of computing resonances using ab initio methodologies have accumulated, and it is relevant to shed light on their differences and similarities. Since Gaussian functions are usually used in quantum chemistry, we focus on their use in the present. The illustrative numerical example is for a single particle problem. However, one can learn much from the latter on a many-particle quantum chemistry study. Our findings show that the calculations of molecular resonances by the use of complex Gaussian basis functions and by adding complex absorbing potentials to the original molecular Hamiltonian are closely equivalent one to another.

Spherical optimization with complex variablesfor computing US-eigenpairs

Abstract: The aim of this paper is to compute unitary symmetric eigenpairs (US-eigenpairs) of high-order symmetric complex tensors, which is closely related to the best complex rank-one approximation of a symmetric complex tensor and quantum entanglement. It is also an optimization problem of real-valued functions with complex variables. We study the spherical optimization problem with complex variables including the first-order and the second-order Taylor polynomials, optimization conditions and convex functions of real-valued functions with complex variables. We propose an algorithm and show that it is guaranteed to approximate a US-eigenpair of a symmetric complex tensor. Moreover, if the number of US-eigenpair is finite, then the algorithm is convergent to a US-eigenpair. Numerical examples are presented to demonstrate the effectiveness of the proposed method in finding US-eigenpairs.

Asymptotic normality of linear statistics of zeros of random polynomials

Abstract: In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted L-2-space of polynomials endowed with varying measures of the form e-(2n phi n(z)) dz under suitable assumptions on the weight functions phi(n).

On Critical Points of Random Polynomials and Spectrum of Certain Products of Random Matrices

Abstract: In the first part of this thesis, we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables. In the second part we deal with the spectrum of products of Ginibre matrices. Exact eigenvalue density is known for a very few matrix ensembles. For the known ones they often lead to determinantal point process. Let X1, X2,..., Xk be i.i.d Ginibre matrices of size n ×n whose entries are standard complex Gaussian random variables. We derive eigenvalue density for matrices of the form X1 e1 X2 e2 ... Xk ek , where ei = ±1 for i =1,2,..., k. We show that the eigenvalues form a determinantal point process. The case where k =2, e1 +e2 =0 was derived earlier by Krishnapur. In the case where ei =1 for i =1,2,...,n was derived by Akemann and Burda. These two known cases can be obtained as special cases of our result.

The complex multinormal distribution, quadratic forms in complex random vectors and an omnibus goodness-of-fit test for the complex normal distribution

Abstract: This paper first reviews some basic properties of the (noncircular) complex multinormal distribution and presents a few characterizations of it. The distribution of linear combinations of complex normally distributed random vectors is then obtained, as well as the behavior of quadratic forms in complex multinormal random vectors. We look into the problem of estimating the complex parameters of the complex normal distribution and give their asymptotic distribution. We then propose a virtually omnibus goodness-of-fit test for the complex normal distribution with unknown parameters, based on the empirical characteristic function. Monte Carlo simulation results show that our test behaves well against various alternative distributions. The test is then applied to an fMRI data set and we show how it can be used to “validate” the usual hypothesis of normality of the outside-brain signal. An R package that contains the functions to perform the test is available from the authors.

A new method of generating multivariate Weibull distributed data

Abstract: In order to fully test detector frameworks, it is important to have representative simulated clutter data readily available. While measured clutter data has often been fit to the Weibull distribution, generation of simulated complex multivariate Weibull data with prescribed covariance structure has been a challenging problem. As the multivariate Weibull distribution is admissible as a spherically invariant random vector for a specific range of shape parameter values, it can be decomposed as the product of a modulating random variable and a complex Gaussian random vector. Here we use this representation to compare the traditional method of generating multivariate Weibull data using the Rejection Method to a new approximation of the modulating random variable that lends itself to efficient computer generation.

Recursive Bayesian beamforming with uncertain projected steering vector and strong interferences

Abstract: A recursive Bayesian beamforming is proposed for the steering vector uncertainty and strong interferences. Signal and noise powers are unknown, and beamforming weight is modeled as a complex Gaussian vector that characterizes the level of projected steering vector uncertainty. By applying the Bayesian model, a recursive algorithm is developed to estimate beamforming weight. Numerical simulations of linear and planar arrays demonstrate the effectiveness and robustness of the proposed beamforming algorithm. After convergence, the proposed algorithm exhibits a performance similar to that of the optimal $$\mathrm {MaxSINR}$$ beamformer.

A Highly Flexible Trajectory Model Based on the Primitives of Brownian Fields—Part II: Analysis of the Statistical Properties

Abstract: In the first part of our paper, we have proposed a highly flexible trajectory model based on the primitives of Brownian fields (BFs). In this second part, we study the statistical properties of that trajectory model in depth. These properties include the autocorrelation function (ACF), mean, and the variance of the path along each axis. We also derive the distribution of the angle-of-motion (AOM) process, the incremental traveling length process, and the overall traveling length. It is shown that the path process is in general non-stationary. We show that the AOM and the incremental traveling length processes can be modeled by the phase and the envelope of a complex Gaussian process with nonidentical means and variances of the quadrature components. In accordance with empirical studies, we prove that the AOM process does not follow the uniform distribution. As special cases, we show that the incremental traveling length process follows the Rice and Nakagami-q distributions, whereas the overall traveling path can be modeled by a random variable following either the Gaussian or the lognormal distribution. The flexibility of the results is demonstrated and discussed extensively. It is shown that the results are in line with those of real-world user tracings. The results can be used in many areas of wireless communications.

Theoretical convergence analysis of complex Gaussian kernel LMS algorithm

Abstract: With the vigorous expansion of nonlinear adaptive filtering with real-valued kernel functions,its counterpart complex kernel adaptive filtering algorithms were also sequentially proposed to solve the complex-valued nonlinear problems arising in almost all real-world applications.This paper firstly presents two schemes of the complex Gaussian kernel-based adaptive filtering algorithms to illustrate their respective characteristics.Then the theoretical convergence behavior of the complex Gaussian kernel least mean square(LMS) algorithm is studied by using the fixed dictionary strategy.The simulation results demonstrate that the theoretical curves predicted by the derived analytical models consistently coincide with the Monte Carlo simulation results in both transient and steady-state stages for two introduced complex Gaussian kernel LMS algonthms using non-circular complex data.The analytical models are able to be regard as a theoretical tool evaluating ability and allow to compare with mean square error(MSE) performance among of complex kernel LMS(KLMS) methods according to the specified kernel bandwidth and the length of dictionary.

Noise-robust radar HRR target recognition based on complex Gaussian statistical models

Abstract: In order to utilize the phase information to improve radar target recognition performance of high range resolution (HRR) target signals, this paper takes adaptive Gaussian classifier (AGC) model and factor analysis (FA) model for instance to generalize Gaussian statistical models to complex domain to build complex HRR models. It is demonstrated that the structures and parameter estimators of complex Gaussian statistical models are invariant to the initial phases of HRR complex data. Furthermore, to enhance the recognition performance under low signal-to-noise ratio (SNR) conditions, a noise-robust modification algorithm is introduced. It solves the mismatch between complex Gaussian models and noisy test signals. Experimental results show that the proposed models can obtain higher average correct recognition rates by utilizing the phase information. Also, the modified models can deal well with noisy test signals.

MIMO Radar DOD/DOA Estimation and Performance Analysis in the Presence of SIRP Clutter

Abstract: This thesis investigates the problem of target parameter estimation and performance analysis of multiple-input multiple-output (MIMO) radar in the presence of non-Gaussian clutter. During the past decades, multiple-input multiple-output (MIMO) radar has become a research subject of growing interest, due to its superior performance in many aspects over the traditional phased-array radar. Conventionally, MIMO radar clutter is modeled as Gaussiandistributed. This modeling, however, becomes unrealistic and inadequate in certain specific scenarios, where the clutter shows distinct non-Gaussianity. In the radar literature, one of the most notable and popular models for such non-Gaussian clutter is the so-called spherically invariant random process (SIRP) model. A SIRP is a complex, compound Gaussian process with random power and can be represented as the product of two components: a complex Gaussian process, called the speckle, and the square root of a positive scalar random process, called the texture. The goal of this thesis is to devise estimation algorithms for target parameters, more specifically, for direction-of-departures/arrivals (DODs/DOAs) of the targets, in a MIMO radar context in the presence of SIRP clutter, and to evaluate the ultimate performance of this estimation problem, in terms of performance bounds and of target resolvability. First, three DOD/DOA estimation algorithms are proposed, which differ from one another in the modeling of the texture, as well as in the respective likelihood functions that they are based on, but have in common that all three algorithms employ the same concept of the stepwise numerical concentration approach and thus have similar iterative procedures. Performance properties like convergence of iterations and computational complexity of the three proposed algorithms are then examined. Next, various Cramer-Rao-type bounds (CRTBs) for the DOD/DOA parameters in this context are derived for performance assessment and their relationships between one another are determined. The respective impacts of the texture parameters on the CRTBs are investigated to illustrate the effect of the clutter spikiness on the same. Then, the estimation performance achievable in the presence of SIRP clutter is studied from another point of view, namely, that of the target resolvability, which is quantified by the concept of the resolution limit (RL). As a result, an analytical, closed-form expression of the RL with respect to (w.r.t.) the angular parameters between twoclosely spaced targets in this context is derived based on Smith’s criterion. For this aim, nonmatrix, closed-form expressions for several of the aforementioned CRTBs w.r.t. the angular spacing between the targets are also obtained as byproducts. Moreover, an alternative, more concrete expression for the RL is propsed for asymptotic scenarios. Like for the CRTBs, the respective impacts of the texture parameters on the RL are also determined. Finally, numerical simulations are provided to assess the performance of the proposed algorithms, to show the validity of the derived RL expressions, as well as to reveal the CRTBs’ and the RL’s insightful properties.

Benefits of noncircular statistics for nonstationary signals

Abstract: Conventional statistical signal processing of nonstationary signals uses circular complex Gaussian distributions to model the complex-valued short-time Fourier transform. In this paper, we show how noncircular complex Gaussian distributions can provide better statistical models of a variety of nonstationary acoustic signals. The estimators required for this model are computationally efficient, and also have a simple approximate finite-sample distribution. We also show that noncircular Gaussian models provide distinct benefits for statistical signal processing. In particular, we show how noncircular Gaussian models can improve detection of nonstationary acoustic events, and we explore how estimator parameter choices affect performance.

Laplace Autoregressive Model for Cascaded Systems

Abstract: System modeling problems, such as the channel of a single-hop relay communication system in a flat-fading environment, require a cascade of two or more autoregressive (AR) processes to capture the entire system characteristics However, for the purpose of system simulation and parameter estimations, it is more convenient if the entire system is modeled by a single AR model In this paper, we consider a cascade system whose statistical characteristics of its subsystems is represented by two independent ${p}$ th-order complex Gaussian AR processes, and model it by a single ${p}$ th-order Laplace AR process In our analysis, we first demonstrate that the marginal probability distribution functions of the real and imaginary components of a system described by a cascade of the two complex Gaussian AR processes are Laplace distributed Then, to model the cascaded system, we develop a single complex Laplace AR process whose parameters are configured to match other statistical characteristics of the cascaded system Specifically, we show that the autocorrelation of the developed Laplace AR process satisfies Yule–Walker type of equations and derive the steps for the design of its parameters through autocorrelation matching

Radar signal adaptive detection method based on autoregressive model

Abstract: The invention discloses a radar signal adaptive detection method based on an autoregressive model. According to the concept, a radar receives coherent pulse sequences of N pulses, the coherent pulse sequences of the N pulses serve as to-be-detected unit echoes z0 of a target, and then a target detection problem of the radar is represented by a binary hypothesis test, wherein H0 indicates that z0 only has an interference hypothesis, H1 indicates that z0 has target and interference hypotheses, and first-order partial derivatives of a joint probability density function f(z0, ZK/theta) of z0 and ZK on two-dimensional column vectors theta r of a target amplitude, an upper left block matrix of a Fisher information matrix J(theta) inverse of a to-be-estimated parameter theta, the maximum likelihood estimator of variance sigma of white complex Gaussian noise and the maximum likelihood estimator of an autoregressive parameter vector a of the M-order autoregressive model are calculated respectively; the detection threshold eta AR-Rao of the autoregressive model based on the Rao detection method is set, and a target detection expression TR, based on the autoregressive model, in z0 is calculated; if the TR is larger than eta AR-Rao, a target exists in z0; otherwise, the target does not exist in z0.

Revisiting the RIP of Real and Complex Gaussian Sensing Matrices Through RIV Framework

Abstract: In this paper, we aim to revisit the restricted isometry property of real and complex Gaussian sensing matrices. We do this reconsideration via the recently introduced restricted isometry random variable (RIV) framework for the real Gaussian sensing matrices. We first generalize the RIV framework to the complex settings and illustrate that the restricted isometry constants (RICs) of complex Gaussian sensing matrices are smaller than their real-valued counterpart. The reasons behind the better RIC nature of complex sensing matrices over their real-valued counterpart are delineated. We also demonstrate via critical functions, upper bounds on the RICs, that complex Gaussian matrices with prescribed RICs exist for larger number of problem sizes than the real Gaussian matrices.