Independent Complex Gaussian Random Variables Research Articles

Entire Gaussian Functions: Probability of Zeros Absence

In this paper, we consider a random entire function of the form f(z,ω)=∑n=0+∞εn(ω1)×ξn(ω2)fnzn, where (εn) is a sequence of independent Steinhaus random variables, (ξn) is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers fn∈C is such that lim¯n→+∞|fn|n=0 and #{n:fn≠0}=+∞. We investigate asymptotic estimates of the probability P0(r)=P{ω:f(z,ω) has no zeros inside rD} as r→+∞ outside of some set E of finite logarithmic measure, i.e., ∫E∩[1,+∞)dlnr<+∞. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions.

Delocalization and Quantum Diffusion of Random Band Matrices in High Dimensions II: T-expansion

We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}=\bar h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any $n\in \mathbb N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, with an error $O(W^{-nd/2})$, and use it to prove a local law on the Green's function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $d\ge 8$ in part I of this series.

Cross Term Decay in Multiplicative Processors

Multiplicative processors combine the beamformed outputs of undersampled subarrays to estimate the spatial power spectrum. While the multiplicative processor requires fewer sensors to achieve the same resolution as a conventional linear processor, it also produces cross terms that can degrade the spectral estimate. Cross terms are false peaks formed when a signal passing through the grating lobe of one subarray interacts with a different signal passing through the other subarray. Snapshot averaging reduces cross terms when the signals are uncorrelated. This letter quantifies the number of snapshots required for effective cross term mitigation, accounting for the use of subarray tapers to control sidelobe levels. To facilitate the analysis, the letter derives closed form expressions for the probability density function and cumulative distribution function of the sum of products of independent complex Gaussian random variables. The analysis demonstrates that cross terms decay at a rate of 5 dB per decade of snapshots averaged. While subarray tapering reduces sidelobe leakage, it also increases the number of snapshots required for cross term mitigation.

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

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 the ratio of independent complex Gaussian random variables

In this paper, we derive a closed form equation for the joint probability distribution $${{f_{{R}_{z}}},{\varTheta _{z}}}({r_{z}},{\theta _{z}})$$ of the amplitude $${R_{z}}$$ and phase $${\varTheta _{z}}$$ of the ratio $${Z=\frac{X}{Y}}$$ of two independent non-zero mean Complex Gaussian random variables $$X\sim CN(\nu _{x} \mathrm {e}^{j\phi _{x}},{\sigma ^{2}_{x}})$$ and $$Y\sim CN(\nu _{y} \mathrm {e}^{j\phi _{y}},{\sigma ^{2}_{y}})$$ . The derived joint probability distribution only contains a confluent hypergeometric function of the first kind $${_1F_{1}}$$ without infinite summations resulting in computational efficiency. We further derive the probability distribution for the ratio of two non-zero mean independent real Rician random variables containing an infinite summation generated by the estimation of the Cauchy product of equivalent series of two modified Bessel functions.

On the utility of MIMO multi-relay networks for modulation identification over spatially-correlated channels

The necessity to perfectly monitor the intercepted signals for spatially-correlated multiple-input multiple-output (MIMO) systems, involves modulation identification algorithms. In this paper, we present an algorithm dedicated to the modulation identification for correlated MIMO relaying broadcast channels with direct link using multi-relay nodes. By modeling spatially-correlated MIMO channels as Kronecker-structured and the imperfect channel state information of both the source-to-destination and the relay-to-destination errors as independent complex Gaussian random variables, we firstly derive the ergodic capacity of the proposed transmission system. It turns out that the ergodic capacities improve with the number of relay nodes. Based on a pattern recognition approach using the higher order statistics features and the Bagging classifier, we show that the probability to distinguish among M-ary shift keying linear modulation types without any priori modulation information is enhanced compared to the decision tree (J48), the tree augmented naive Bayes, the naive Bayes using discretization and the multilayer perceptron classifiers. We also study the effect of increasing the number of relay nodes. Numerical simulations show that the proposed algorithm using the cooperation of multi-relay nodes with the source node can avoid the performance deterioration in modulation identification caused by both spatial correlation and imperfect CSI.

Signaling Over Two-User Parallel Gaussian Interference Channels: Outage Analysis

This paper presents an outage analysis for a two-user parallel Gaussian interference channel consisting of two sub-channels. Each sub-channel is modeled as a two-user Gaussian interference channel with quasi-static and flat fading. Both users employ single-layer Gaussian code-books and maintain a statistical correlation $\rho $ between the signals transmitted over the underlying sub-channels. When joint decoding (JD) is performed at the receivers, setting $\rho =0$ minimizes the outage probability, regardless of the value of the signal-to-noise ratio (SNR). It is shown, however, that if the receivers treat interference as noise (TIN) or cancel interference (CI), the value of optimum $\rho $ approaches 1 as SNR goes to infinity. Motivated by these observations, we let $\rho =0$ under JD and $\rho =1$ under TIN and CI and compute the outage probability in finite SNR, assuming that the direct and crossover channel coefficients are independent zero-mean complex Gaussian random variables with possibly different variances. In the asymptote of large SNR and assuming the transmission rate per user is $r\log \mathrm {snr}$ , it is shown that the outage probability scales like $\mathrm {snr}^{-(1-r)}$ under both TIN and CI, while it vanishes at least as fast as $\mathrm {snr}^{-\min \{2-r,4(1-r)\}}\log \mathrm {snr}$ under JD. This paper is concluded by extending some of the results to a two-user parallel Gaussian interference channel with an arbitrary number of sub-channels.

Detection performance of coprime sensor arrays

Coprime sensor arrays (CSAs) achieve the resolution of a fully populated uniform linear array (ULA) with the same aperture using fewer sensors. The CSA's reduced number of sensors diminishes its ability to attenuate white noise, and consequently also reduces its array gain. Assuming narrowband independent signal and white noise, when the signal and the noise are modeled by circular complex zero mean Gaussian distributions, the detection statistic distribution for the ULA simplifies to an exponential distribution with its parameter dependent on the input SNR. The detection statistic for the CSA is the product of the output of one subarray with the complex conjugate of the output of the second subarray. Being the product of two independent complex Gaussian random variables, the CSA detection statistic has a distribution proportional to a modified zeroth order Bessel function of the second kind [O'Donoughue and Moura, IEEE Signal Process. (2012)]. Manipulating the ULA and CSA detection statistic distributions provides analytical expressions for probabilities of detection and false alarm and also mean discriminating information. Evaluating these expressions confirms that a CSA pays a detection gain penalty of about 5 dB over a wide range of SNRs and array sizes. [Work supported by ONR.]

Capacity Performance of Relay Beamformings for MIMO Multirelay Networks With Imperfect ${\cal R}$-${\cal D}$ CSI at Relays

In this paper, we consider a dual-hop multiple-input-multiple-output (MIMO) wireless relay network in the presence of imperfect channel state information (CSI), in which a source-destination pair, both equipped with multiple antennas, communicates through a large number of half-duplex amplify-and-forward (AF) relay terminals. We investigate the performance of three linear beamforming schemes when the CSI of the relay-to-destination ( ℜ- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> ) link is not perfect at the relay nodes. The three efficient linear beamforming schemes are based on the matched-filter (MF), zero-forcing (ZF) precoding, and regularized ZF (RZF) precoding techniques, which utilize the CSI of both the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -ℜ channel and the ℜ- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> channel at the relay nodes. By modeling the ℜ- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> CSI error at the relay nodes as independent complex Gaussian random variables, we derive the ergodic capacities of the three beamformers in terms of instantaneous signal-to-noise ratio. Using the Law of Large Number, we obtain the asymptotic capacities, upon which the optimized MF-RZF is derived. Simulation results show that the asymptotic capacities match the respective ergodic capacities very well. Analysis and simulation results demonstrate that the optimized MF-RZF outperforms MF and MF-ZF for any power of the ℜ- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> CSI error.

Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups

Let $M$ be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size $n$, the vector of traces of consecutive powers of $M$ tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method of exchangeable pairs. In this note, we extend Fulman's result to the multivariate central limit theorem for the full vector of traces of powers.

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this direction, we obtain the Tracy–Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko–Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.

Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrödinger Equation Driven by the Square of a Gaussian Field

Solutions to the equation Open image in new window are investigated, where S(x, t) is a complex Gaussian field with zero mean and specified covariance, and m≠0 is a complex mass with Im(m) ≥ 0. For real m this equation describes the backscattering of a smoothed laser beam by an optically active medium. Assuming that S(x, t) is the sum of a finite number of independent complex Gaussian random variables, we obtain an expression for the value of λ at which the qth moment of Open image in new window w.r.t. the Gaussian field S diverges. This value is found to be less or equal for all m ≠ 0, Im(m) ≥ 0 and |m|<+∞ than for |m| = +∞, i.e. when the Open image in new window term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.

Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

We provide an elementary proof for a theorem due to Petz and Reffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor $$\sqrt{n}$$ and as $$n\to\infty$$ , to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.

Performance Analysis of Optimum and Suboptimum Selection Diversity Schemes on Rayleigh Fading Channels With Imperfect Channel Estimates

In this paper, we present a comparative analysis on the effects of channel estimation errors on the performance of optimum and suboptimum selection diversity (SD) receivers on Rayleigh-fading channels. By modeling the estimation errors as independent complex Gaussian random variables, we derive simple closed-form expressions for the average probability of error for both optimum and suboptimum SD schemes with noisy channel estimates. With dual diversity and imperfect estimates, we establish a connection between optimum SD and maximal-ratio combining (MRC), and between suboptimum SD and equal-gain combining diversity schemes. Interestingly, we show that the optimum SD receiver structure and the resulting performance for differential binary coherent phase-shift keying (DBPSK) signaling can be obtained, in a straightforward way, as a special case of the performance of the optimum SD scheme with binary PSK signaling and channel estimation errors. For a fixed average power and bit duration, in conjunction with pilot-assisted minimum mean-square error channel estimation, we show that the optimum coherent SD scheme coincides with that of the optimum noncoherent SD scheme with binary frequency-shift keying (BFSK) signaling, whereas the coherent MRC scheme coincides with the optimum noncoherent receiver (i.e., the square-law combiner) for BFSK. The optimum number of diversity channels, under an energy-sharing mode of operation, is also studied. Finally, we formulate the problem of optimal pilot placement, consider channel estimation with a practical pilot-symbol-assisted modulation technique, and present some numerical results illustrating the comparative performances of various SD receivers

On the Performance of the MIMO Zero-Forcing Receiver in the Presence of Channel Estimation Error

By employing spatial multiplexing, multiple-input multiple-output (MIMO) wireless antenna systems provide increases in capacity without the need for additional spectrum or power. Zero-forcing (ZF) detection is a simple and effective technique for retrieving multiple transmitted data streams at the receiver. However the detection requires knowledge of the channel state information (CSI) and in practice accurate CSI may not be available. In this letter, we investigate the effect of channel estimation error on the performance of MIMO ZF receivers in uncorrelated Rayleigh flat fading channels. By modeling the estimation error as independent complex Gaussian random variables, tight approximations for both the post-processing SNR distribution and bit error rate (BER) for MIMO ZF receivers with M-QAM and M-PSK modulated signals are derived in closed-form. Numerical results demonstrate the tightness of our analysis

Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrödinger Equation Driven by the Square of a Gaussian Field

Solutions to the equation are investigated, where S(x, t) is a complex Gaussian field with zero mean and specified covariance, and m≠0 is a complex mass with Im(m) ≥ 0. For real m this equation describes the backscattering of a smoothed laser beam by an optically active medium. Assuming that S(x, t) is the sum of a finite number of independent complex Gaussian random variables, we obtain an expression for the value of λ at which the q th moment of w.r.t. the Gaussian field S diverges. This value is found to be less or equal for all m ≠ 0, Im(m) ≥ 0 and |m|<+∞ than for |m| = +∞, i.e. when the term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.

On second-order statistics and linear estimation of cepstral coefficients

Explicit expressions for the second-order statistics of cepstral components representing clean and noisy signal waveforms are derived. The noise is assumed additive to the signal, and the spectral components of each process are assumed statistically independent complex Gaussian random variables. The key result developed here is an explicit expression for the cross-covariance between the log-periodograms of the clean and noisy signals. In the absence of noise, this expression is used to show that the covariance matrix of cepstral components representing N signal samples, is a fixed signal independent matrix, which approaches a diagonal matrix at a rate of 1/N. In addition, the cross-covariance expression is used to develop an explicit linear minimum mean square error estimator for the clean cepstral components given noisy cepstral components. Recognition results on the English digits using the fixed covariance and linear estimator are presented.

Gaussian random band matrices and operator-valued free probability theory

1. Gaussian random band matrices. Let g ij , 1 ≤ i, j ≤ n, k = 1, . . . , r be independent centered complex Gaussian random variables of covariance 1 n (1 + δij) (i.e., the expectation E(g ijg k ij) = 1 n (1 + δij)), such that g k ij = g k ji. Let for each i, j, k, σn(i, j; k) be a positive real number, such that σn(i, j; k) = σn(j, i; k). Consider the r-tuple of n × n matrices Gn(k) = (σn(i, j; k)g k ij)ij , k = 1, . . . , r. The family {Gn(k)}k is called a family of independent Gaussian random band matrices. As an example, assume that σn(i, j; k) = 1 if |i − j| < cn, and is zero otherwise. In this case all entries of Gn(k) are zero outside a band around the diagonal; this is the reason such matrices are called band matrices. In the case that σn(i, j; k) is identically 1 one recovers the so-called Gaussian Unitary Ensemble, see e.g. [9].