BAPPENDIX

Complex Gaussian RandomVariables and Matrices

B.1 SOME USEFUL PROBABILITY DISTRIBUTIONSConsider that h is a complex Gaussian variable, i.e., a circularly complex variable with zero mean andvariance σ 2. This means that both the real and imaginary parts of h are zero mean Gaussian variablesof variance σ 2. In this case, s � |h| follows a Rayleigh distribution,

ps(s) = s

σ 2 exp

(− s2

2σ 2

), (B.1)

with the following properties:

E{s} = σ

√π

2, (B.2)

E{s2} = 2σ 2. (B.3)

The CDF of s is given by

P[s < S] = 1 − e−S2/2σ 2. (B.4)

The random variable y � |h|2 = s2 follows a χ2 distribution (with two degrees of freedom),

py(y) = 1

2σ 2 exp(− y

2σ 2

). (B.5)

Note that for small ε, we have that P[y < ε

] ≈ ε.Let us now consider n i.i.d. zero mean complex Gaussian variables h1, . . . , hn with variance σ 2.

Defining u = ∑nk=1 |hk |2, the moment generating function of u is given by

Mu(τ ) = E{eτu} =n∏

k=1

1

1 − 2σ 2τ=

[1

1 − 2σ 2τ

]n

, (B.6)

and the distribution of u reads as

pu(u) = 1

σ 2n2n�(n)un−1 exp

(− u

2σ 2

). (B.7)

MIMO Wireless Networks. http://dx.doi.org/10.1016/B978-0-12-385055-3.00017-1© 2013 Elsevier Ltd. All rights reserved.

677

678 Appendix B Complex Gaussian Random Variables and Matrices

This distribution is known as the χ2 distribution with 2n degrees of freedom, also noted as χ22n (the

case n = 1 reduces to (B.5)). The corresponding CDF is given by

P[u < U ] = 1 − e−U/2σ 2n−1∑k=1

1

k!(

U

2σ 2

)k

, (B.8)

= 1 − �(n,U/2σ 2)

(n − 1)! . (B.9)

Furthermore, if u is χ22n distributed, we have that

E {log u} = ψ(n), (B.10)

where ψ(n) is the digamma function [Lee97], which, for integer n, may be expressed as

ψ(n) =n−1∑k=1

1

k− γ , (B.11)

γ ≈ 0.57721566 being Euler’s constant.Finally, let us now analyze the case when h1, . . . , hn are non zero mean. Assume that the real

and imaginary parts of hk are Gaussian variables of mean μk and variance σ 2. In this situation,w = ∑n

k=1 |hk |2 follows a non central χ2 distribution with 2n degrees of freedom,

pw(w) = 1

2σ 2

(w

β

) n−12

exp

(−w + β

2σ 2

)In−1

(√wβ

σ 2

), (B.12)

where β = ∑nk=1 2μ2

k is called the noncentrality parameter of the distribution.

B.2 EIGENVALUES OF WISHART MATRICESIn this section, we formalize a few properties of complex Wishart matrices. Indeed, for i.i.d. Rayleighchannels represented by a nr × nt complex Gaussian matrix Hw,Tw = HwHH

w (for nt > nr ) andTw = HH

wHw (for nt < nr ) are central Wishart matrices with parameters n and N, which we denote asTw(n, N ). In our notations, N = max{nt , nr } and n = min{nt , nr }.Lemma B.1. The joint probability density [Ede89] of the elements of a matrix Tw(n, N ) is

1

2Nn�̃n(N )eTr

{− 1

2 Tw}[

det (Tw)]N−n

, (B.13)

where �̃n(N ) = πn(n−1)/2 ∏nk=1 �(N − k + 1) is the multivariate gamma function.

B.2 Eigenvalues of Wishart Matrices 679

B.2.1 Determinant and Product of Eigenvalues of Wishart MatricesUsing a bi-diagonalization of Hw, it is shown in [Ede89] that Wishart matrices have the followingproperty.

Theorem B.1. If Tw has the central Wishart distribution with parameters n and N, its determinanthas the distribution

∏nk=1 χ

22(N−n+k) (the notation refers to the distribution of the product of n random

variables with the indicated χ2 distribution). As the determinant is also the product of the non-zeroeigenvalues, this product has the same distribution.

Proof. The proof relies on the fact that a complex Gaussian matrix with independent entries is unitarilysimilar to a bi-diagonal matrix whose entries are independent χ2 variables [Ede89]. �

Corollary B.1. The expected value of the determinant of Tw is equal to N !(N−n)! .

It is important to note that the above theorem does not imply that ordered or randomly selectedeigenvalues of Tw are χ2 distributed, as the theorem only concerns their product.

B.2.2 Distribution of Ordered of EigenvaluesThe ordered eigenvalues of Tw, λ1 ≥ λ2 ≥ . . . ≥ λn , have the following distribution:

pλ1,...,λn (λ1, . . . , λn) = 2−Nnπn(n−1)

�̃n(N )�̃n(n)e−∑n

k=1 λk

n∏k=1

λN−nk

∏k<l

(λk − λl)2. (B.14)

Asymptotic results regarding the smallest and the largest eigenvalues of Tw are derived in [Ede89]and summarized as follows.

Proposition B.1. If n and N tend to infinity in a constant ratio n/N ≤ 1, the smallest and the largesteigenvalues of Tw respectively satisfy

1

nλmin

a.s.→ 2(1 − √

n/N)2 (B.15)

1

nλmax

a.s.→ (1 + √

n/N)2 (B.16)

This proposition is particularly helpful as it also provides directly the mean values of λmin and λmax .

B.2.3 Distribution of Non-Ordered of Eigenvalues

Theorem B.2. The distribution pλ(λ) of a randomly selected (non-ordered) eigenvalue of Tw is

pλ(λ) = 1

n

n−1∑k=0

k! λN−n e−λ

(k + N − n)! L N−nk (λ), (B.17)

680 Appendix B Complex Gaussian Random Variables and Matrices

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Eigenvalue of Tw

Prob

abilit

y de

nsity

2 × 2 Monte-Carlo2 × 2 theor.3 × 3 Monte-Carlo3 × 3 theor.

FIGURE B.1

Distribution of a randomly selected eigenvalue of Tw in several scenarios.

and L pk (x) is the Laguerre polynomial of order k,

L pk (x) =

k∑m=0

(− 1)m

(k + pk − m

)xm

m! . (B.18)

As an example, Figure B.1 illustrates the theoretical distribution together with the distributionobtained from Monte-Carlo simulations for 2 × 2 and 3 × 3 i.i.d. Rayleigh channels.