Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for the product of two Gaussian matrices

Kumar, Santosh

doi:10.1088/1751-8113/48/44/445206

Cited by 18 publications

(12 citation statements)

References 30 publications

(54 reference statements)

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“…They are given by These formulae allow us to make some straightforward generalisations of the exact expressions presented by Kumar [37] in the m = 2 case. Following [37], we have…”

Section: Rectangular Matricesmentioning

confidence: 93%

“…as a summation over a linear combination of { 2 F 1 (µ + a, µ + b; µ + c; 1 − z)} n µ=0 has been given by Kumar [37], and this was used to show…”

Section: Probability Of K Real Eigenvaluesmentioning

confidence: 99%

“…Given such generalisations the rest of the results presented in previous sections may be extended as well due to the generality of Proposition 5. , the joint PDF for these eigenvalues is given by Table 3: Consider a product of two Gaussian matrices, X 1 X 2 , with dimensions N ×(N +ν) and (N +ν)×N for X 1 and X 2 , respectively. The table provides exact probabilities for a purely real spectrum for various values of N and ν (the ν = 0 column have previously appeared in [37]). These probabilities are all given as a rational number times a power of π depending on N and whether ν is odd or even.…”

Section: Rectangular Matricesmentioning

confidence: 99%

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Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Forrester

Ipsen

2016

Linear Algebra and its Applications

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Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these -the eigenvalue density of the real eigenvalues -gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.

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“…They are given by These formulae allow us to make some straightforward generalisations of the exact expressions presented by Kumar [37] in the m = 2 case. Following [37], we have…”

Section: Rectangular Matricesmentioning

confidence: 93%

“…as a summation over a linear combination of { 2 F 1 (µ + a, µ + b; µ + c; 1 − z)} n µ=0 has been given by Kumar [37], and this was used to show…”

Section: Probability Of K Real Eigenvaluesmentioning

confidence: 99%

Section: Rectangular Matricesmentioning

confidence: 99%

See 1 more Smart Citation

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Forrester

Ipsen

2016

Linear Algebra and its Applications

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“…and Γ(., .) functions to their Meijer's G equivalents [22]. Further, applying the identity [23, 07.34.21.0081.01], we get the closed form expression for these integrals.…”

Section: Ergodic Capacitymentioning

confidence: 99%

Performance Analysis of Dual-Hop THz Wireless Transmission for Backhaul Applications

Pai¹,

Bhardwaj²,

Zafaruddin

2021

Preprint

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THz transmissions suffer from pointing errors due to antenna misalignment and incur higher path loss because of molecular absorption at such a high frequency. In this paper, we employ an amplify-and-forward (AF) dual-hop relay to mitigate the effect of pointing errors and extend the range of a wireless backhaul network. We provide statistical analysis on the performance of the considered system by deriving analytical expressions for the outage probability, average bit-error-rate (BER), average signal-to-noise ratio (SNR), and a lower bound on the ergodic capacity over independent and identical (i.i.d) αµ fading model and statistical pointing errors. Using computer simulations, we validate the derived analysis of the relay-assisted system. We demonstrate the effect of the system parameters on outage probability and average BER with the help of diversity order. We show that data rates up to several Gbps can be achieved using THz transmissions, which is desirable for next-generation wireless systems, especially for backhaul applications.

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“…e relation of Meijer G-functions with elementary functions as exponential, logarithmic, cosine, and Bessel functions is given by Santosh [32] as…”

Section: Introductionmentioning

confidence: 99%

Relation of Some Known Functions in terms of Generalized Meijer $G$ -Functions

Shah

Mubeen

Rahman

et al. 2021

Journal of Mathematics

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The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

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Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for the product of two Gaussian matrices

Cited by 18 publications

References 30 publications

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Performance Analysis of Dual-Hop THz Wireless Transmission for Backhaul Applications

Relation of Some Known Functions in terms of Generalized Meijer $G$ -Functions

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Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for the product of two Gaussian matrices

Cited by 18 publications

References 30 publications

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Performance Analysis of Dual-Hop THz Wireless Transmission for Backhaul Applications

Relation of Some Known Functions in terms of Generalized Meijer G -Functions

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Product

Resources

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Relation of Some Known Functions in terms of Generalized Meijer $G$ -Functions