AppendixSome Useful Definitions

Complex Gaussian random variable. A real valued random vector X = [x1, . . . , xn]T

has a Gaussian distribution if the random variables x1, . . . , xn have a joint Gaussiandistribution. Assuming that X has a zero mean and covariance matrix E[XXT] = RXX,its probability density function can be written as:

f (X) = 1√(2π)n det(RXX)

exp[−1

2XTR−1

XXX

]. (A.1)

A complex random variable Z = Re(Z) + jIm(X) has a complex Gaussian distributionif its real and imaginary parts are jointly Gaussian. Let us define the 2n×1 vectorZ = [

Re(X)T Im(X)T]T as the real valued random vector containing the real and

imaginary parts of X. The probability density function if Z ∼ N(0, RZZ) can be writtenas in Equation (A.1). When the complex Gaussian random vector Z is additionallycircularly symmetric, its distribution can be written in a conveniently compact way asdescribed in the following two items.

Complex circularly symmetric random variable. A complex vectorial random variableZ is said to be circularly symmetric if Z and ejφZ have the same distribution for all realvalues of the phase φ.For a scalar complex random variable, circular symmetry means that its real and imag-inary parts are independent and have the same distribution.A complex circularly symmetric random vector has zero-mean, as E [Z] = ejφE [Z] ⇒(1 − ejφ)E [Z] = 0 ⇒ E [Z] = 0.

Distribution of a complex circularly symmetric Gaussian random variable. Let thevector Z = [z1, . . . , zn]T be a complex circularly symmetric Gaussian random vari-able with mean equal to 0 and covariance matrix E[ZZH] = RZZ. Such a circularly

Practical Guide to the MIMO Radio Channel with MATLAB® Examples, First Edition.Tim Brown, Elisabeth De Carvalho and Persefoni Kyritsi.© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

248 Practical Guide to the MIMO Radio Channel with MATLAB® Examples

symmetric Gaussian distribution is denoted as Z ∼ CN(0, RZZ). The probability den-sity function of Z is:

f (Z) = 1

πn det(RZZ)exp

[−ZHR−1

ZZZ

]. (A.2)

This last expression can be easily extended to a random variable with mean thatis non zero. Let us denote as m the mean of Z and its covariance matrix asRZZ = E[(Z − m) (Z − m)H]. If Z − m is circularly symmetric Gaussian, that is,Z − m ∼ CN(0, RZZ), (A.2) can be applied to the random variable Z − m using asimple change of variable.

MIMO Rayleigh fading model. The MIMO Rayleigh fading model assumes that (a) eachsubchannel fading hji has a ZMCCS Gaussian distribution: hji ∼ CN(0, σ2

h) and b) thesubchannel fading processes are independent from each other.

Random or stochastic process. A random or stochastic process is a sequence of randomvariables {x(t), t ∈ T} indexed by time and where T is the set of time indices. It is usedto describe the evolution in time of a given process.For example, a wireless channel can be viewed as a random variable with a certaindistribution which depends on the position of the wireless device, the propagationenvironment, etc. . . . The value of the channel at a given time is a realisation of therandom variable. The evolution in time of the channel distribution is described usinga random process. At a given time instant, the wireless device might be in the line ofsight of a base station, while, at another time instant, obstacles prevent a line of sightcommunication: in such a case, the channel distribution at different time instances mightchange dramatically.

According to the complexity of the channel modelling, the distribution of the ran-dom variables might change in time or remain the same. A correlation might also existbetween random variables at different time instances. Continuing with the channel pro-cess example, a way to model the evolution in time of the channel when the wirelessdevice moves within a local area is as follows. Because the movement is local, thedistribution of the channel is assumed to be invariant in time. However, the randomvariables representing the channel at different time instants are correlated. This corre-lation factor depends on the mobility of the user. If the user moves slowly comparedto the coherence time of the channel, the correlation between two consecutive timeinstants is high. If it moves fast, the correlation is low.

Two processes {x(t), t ∈ T} and {y(t), t ∈ T′} are said to be independent if, for allvalues t ∈ T, t′ ∈ T′, the random variables x(t) and y(t′) are independent.

Stationary stochastic process. A stochastic process is said to be (strictly) stationary if itsdistribution characteristics are invariant when shifted in time. Consider the stochasticprocess {x(t), t ∈ T}. F (x(t1 + τ), . . . , x(tn + τ)) is the joint cumulative distributionfunction of the stochastic process at time t1 + τ, . . . , tn + τ. Then, {x(t), t ∈ T} is said

Appendix 249

to be stationary if, for all t, τ and n

F (x(t1 + τ), . . . , x(tn + τ)) = F (x(t1), . . . , x(tn)). (A.3)

Hence, the joint cumulative distribution function is not a function of the time shift τ.

Ergodic stochastic process. Let the stochastic process {x(t), t ∈ T} and X(t1), . . . , X(tn)be realisations of the stochastic process. Consider the time averaging of those realisa-tions: 1

n

∑nk=1 X(tk). The stochastic process is said to be ergodic if the time averaging

converges to the same limit when n grows to infinity for all realisations X(t1), . . . , X(tn)of the stochastic process. This limit is the ensemble average E(x) and is independentof time:

1

n

n∑k=1

X(tk)n→∞−→ E(x). (A.4)

Orthogonal projection matrix. Let us consider the simple geometric example in the fig-ure below: u⊥ is the orthogonal projection of vector u into the direction of vectorU.

Uu⊥

u

This geometric orthogonal projection can be formalised for a vectorial space of higherdimension with an Hermitian inner product. With this inner product, it is possible todefine orthogonality: two vectors u and v belonging to the vectorial space are orthogonalis uHv = 0. Note that it is possible to define projections that are not orthogonal.

The orthogonal projection of a vectorial space fromCM (in the figure, vector u) intothe column space U (in the figure, line U) can be represented as a matrix operation usingan orthogonal projection matrix.

An orthogonal projection matrix, denoted as P , is a square matrix with the followingproperties. Let P be of dimension M×M and let us define U as the space spanned bythe columns of P . Let u be a M×1 vector belonging to the column space of P . If P isan orthogonal projection matrix, then Pu = u. If u belongs to the space orthogonal toU, then Pu = 0.

A projection matrix is formally defined as a square matrix verifying P2 = P : thisrelationship means that the column space of P remains intact after the projection. Inaddition if PH = P , the projection is orthogonal.P can be written as a function of U. If the columns of U form an orthonormal basis (i.e.UHU = I), an expression for the orthogonal projection matrix into the column space of

250 Practical Guide to the MIMO Radio Channel with MATLAB® Examples

U is P = UUH. In general, an expression for the orthogonal projection matrix into thecolumn space of U is P = U (UHU)−1 UH.

Matrix inversion lemma (MIL). Let A, B, C, D be 4 matrices of size M×M, M×N,N×N, N×M respectively. The matrix inversion lemma states:

(A + BCD)−1 = A−1 − A−1B(C−1 + DA−1B

)−1DA−1. (A.5)

A simple application of the matrix inversion lemma is as follows. Suppose A is anidentity matrix (or a matrix easy to invert), B is a column vector and D is a rowvector. Then, C−1 + DA−1B is a scalar. Hence, using the matrix inversion lemma, thecomputation of the inverse of A + BCD gets simplified.