Análisis Sobre el Municipio de Aguascalientes Respecto

contours but circularity is not sufficient to make them spherical. In order to obtain spherical contours, x must be spherically distributed, which means that x is elliptical with␮x= 0 and augmented generating matrix Hx x = kI for some positive constant k. Hence, only spherical circular random vectors indeed have spherical contours.

2.6

Complex random processes

In this section, we extend some of the ideas introduced so far to a continuous-time complex random process x(t )= u(t) + jv(t), which is built from two real-valued ran- dom processes u(t ) andv(t) defined on IR. We restrict our attention to a second-order description of wide-sense stationary processes. Higher-order statistical characterizations and concepts such as circularity of random processes are more difficult to treat than in the vector case, and we postpone a discussion of more advanced topics to Chapters8

and9.6

To simplify notation in this section, we will assume that x(t ) has zero mean. The

covariance function of x(t ) is denoted by

2.6 Complex random processes 55

and the complementary covariance function of x(t ) is

˜rx x(t, τ) = E[x(t + τ)x(t)]. (2.124)

We also introduce the augmented signal x(t )=x(t ) x∗(t )T, whose covariance matrix

Rx x(t, τ) = E[x(t + τ)xH(t )]=  rx x(t, τ) ˜rx x(t, τ) ˜r_{x x}∗ (t, τ) rx x∗ (t, τ)  (2.125)

is called the augmented covariance function of x(t ).

2.6.1

Wide-sense stationary processes

Definition 2.6. A signal x(t ) is wide-sense stationary (WSS) if and only if Rx x(t, τ) is

independent of t . That is, both the covariance function rx x(t, τ) and the complementary

covariance function ˜rx x(t, τ) are independent of t.

This definition, in keeping with our general philosophy outlined in Section2.2.1, calls

x(t ) WSS if and only if its real and imaginary parts u(t ) andv(t) are jointly WSS. We note

that some researchers call a complex signal x(t ) WSS if rx x(t, τ) alone is independent of t , and second-order stationary if both rx x(t, τ) and ˜rx x(t, τ) are independent of t. If

x(t ) is WSS, we drop the t -argument from the covariance functions. The covariance and

complementary covariance functions then have the symmetries

rx x(τ) = rx x∗ (−τ) and ˜rx x(τ) = ˜rx x(−τ). (2.126) The Fourier transform of Rx x(τ) is the augmented power spectral density (PSD) matrix

Px x( f )=  Px x( f ) P
x x( f )
Px x∗(− f ) Px x∗(− f )  . (2.127)

The augmented PSD matrix contains the PSD Px x( f ), which is the Fourier transform of rx x(τ), and the complementary power spectral density (C-PSD)
Px x( f ), which is the Fourier transform of ˜rx x(τ). The augmented PSD matrix is positive semidefinite, which implies the following result.

Result 2.14. There exists a WSS random process x(t ) with PSD Px x( f ) and C-PSD

Px x( f ) if and only if

(1) the PSD is real and nonnegative (but not necessarily even),

Px x( f )≥ 0; (2.128)

(2) the C-PSD is even (but generally complex),

Px x( f )=
Px x(− f ); (2.129)

(3) the PSD provides a bound on the magnitude of the C-PSD,

56 Complex random vectors and processes

Condition (3) is due to det Px x( f )≥ 0. The three conditions in this result correspond to the three conditions in Result2.1for a complex vector x. Because of (2.128) and (2.129) the augmented PSD matrix simplifies to

Px x( f )=  Px x( f ) P
x x( f )
Px x∗( f ) Px x(− f )  . (2.131)

The time-invariant power of x(t ) is

Px= rx x(0)=  _∞

−∞Px x( f )d f, (2.132)

regardless of whether or not x(t ) is proper.

We now connect the complex description of x(t )= u(t) + jv(t) to the description in terms of its real and imaginary parts u(t ) andv(t). Let ruv(τ) = E[ u(t + τ)v(t)] denote the cross-covariance function between u(t ) andv(t), and Puv( f ) the Fourier transform of

ruv(τ), which is the cross-PSD between u(t) and v(t). Analogously to (2.20) for n= 1, there is the connection

Px x( f )= T  Puu( f ) Puv( f ) Pu∗v( f ) Pvv( f )  TH. (2.133)

From it, we find that

Px x( f )= Puu( f )+ Pvv( f )+ 2 Im Puv( f ), (2.134)

Px x( f )= Puu( f )− Pvv( f )+ 2j Re Puv( f ), (2.135) which are the analogs of (2.21) and (2.22). Note that, unlike the PSD of x(t ), the PSDs of u(t ) andv(t) are even: Puu( f )= Puu(− f ) and Pvv( f )= Pvv(− f ). Propriety is now defined as the obvious extension from vectors to processes.

Definition 2.7. A complex WSS random process x (t ) is called proper if ˜rx x(τ) = 0 for

allτ or, equivalently,
Px x( f )= 0 for all f .

Equivalent conditions on real and imaginary parts for propriety are

ruu(τ) = r_vv(τ) and ru_v(τ) = −ru_v(−τ) for all τ (2.136)

or

Puu( f )= P_vv( f ) and Re Pu_v( f )= 0 for all f. (2.137) Therefore, if WSS x(t ) is proper, its PSD is

Px x( f )= 2[Puu( f )− jPu_v( f )]= 2[P_vv( f )+ jP_vu( f )]. (2.138) The PSD of a proper x(t ) is even if and only if Puv( f )= 0 because Puv( f ) is purely imaginary and odd.

From (2.130), we obtain the following important result for WSS analytic signals, which have Px x( f )= 0 for f < 0, and WSS anti-analytic signals, which have Px x( f )= 0 for

Notes 57

Result 2.15. A WSS analytic (or anti-analytic) signal without a DC component, i.e.,

Px x(0)= 0, is proper.

Because the equivalent complex baseband signal of a real bandpass signal is a down- modulated analytic signal, and modulation keeps propriety intact, we also have the following result.

Result 2.16. The equivalent complex baseband signal of a WSS real bandpass signal is

proper.

2.6.2

Widely linear shift-invariant filtering

Widely linear (linear–conjugate-linear) shift-invariant filtering is described in the time domain as

y(t )=

 _∞

−∞[h1(t− τ)x(τ) + h2(t− τ)x

∗_{(τ)]dτ. (2.139)}

There is a slight complication with a corresponding frequency-domain expression: as we will discuss in Section8.1, the Fourier transform of a WSS random process x(t ) does not exist. The way to deal with WSS processes in the frequency domain is to utilize the Cram´er spectral representation for x(t) and y(t),

x(t )=  _∞ −∞dξ( f )e j2π f t_, (2.140) y(t )=  _∞ −∞dυ( f )e j2π f t_, (2.141) where ξ( f ) and υ( f ) are spectral processes with orthogonal increments dξ( f ) and dυ( f ), respectively. This will be discussed in detail in Section8.1. For now, we content ourselves with stating that

dυ( f ) = H1( f )dξ( f ) + H2( f )dξ∗(− f ), (2.142)

which may be written in augmented notation as  dυ( f ) dυ∗(− f )  =  H1( f ) H2( f ) H2∗(− f ) H1∗(− f )    H( f )  dξ( f ) dξ∗(− f )  . (2.143)

The relationship between the PSDs of x(t ) and y(t ) is

Pyy( f )= H( f )Px x( f )HH( f ). (2.144) Strictly linear filters have h2(t )= 0 for all t or, equivalently, H2( f )= 0 for all f . It

is clear that propriety is preserved under linear filtering, and it is also preserved under modulation.

Notes

1 The first account of widely linear filtering for complex signals seems to have been that byBrown and Crane (1969), who used the term “conjugate linear filtering.” Gardner and his co-authors

58 Complex random vectors and processes

have made extensive use of widely linear filtering in the context of cyclostationary signals, in particular for communications. See, for instance,Gardner (1993). The term “widely linear” was introduced byPicinbono and Chevalier (1995), who presented the widely linear minimum mean-squared error (WLMMSE) estimator for complex random vectors.Schreier and Scharf (2003a) revisited the WLMMSE problem using the augmented complex matrix algebra. 2 The terms “proper” and “pseudo-covariance” (for complementary covariance) were coined by

Neeser and Massey (1993), who looked at applications of proper random vectors in commu- nications and information theory. The term “complementary covariance” is used byLee and Messerschmitt (1994), “relation matrix” byPicinbono and Bondon (1997), and “conjugate covariance” by Gardner. Bothvan den Bos (1995) andPicinbono (1996) utilize what we have called the augmented covariance matrix.

3 Conditions 1–3 in Result2.1for the nonsingular case were proved byPicinbono (1996). A rather technical proof of conditions 3a and 3b was presented byWahlberg and Schreier (2008). 4 The complex proper multivariate Gaussian distribution was introduced byWooding (1956).

Goodman (1963) provided a more in-depth study, and also derived the proper complex Wishart distribution. The assumption of propriety when studying Gaussian random vectors was com- monplace untilvan den Bos (1995) introduced the improper multivariate Gaussian distribution, which he called “generalized complex normal.”Picinbono (1996) explicitly connected this distribution with the Hermitian and complementary covariance matrices.

5 There are other distributions thrown off by the scalar improper complex Gaussian pdf. For example, the conditional distribution Pθ|r(θ|r) = prθ(r, θ)/Pr(r ), with prθ(r, θ) given by (2.71) and pr(r ) by (2.72) is the von Mises distribution. The distribution of the radius-squared r2_could be called the improperχ2_{-distribution with one degree of freedom, and the sum of the radii-} squared of k independent improper Gaussian random variables could be called the improper

χ2_{-distribution with k degrees of freedom. One could also derive improper extensions of theβ-} and exponential distributions.

6 Section2.6builds on results byPicinbono and Bondon (1997), who discussed second-order properties of complex signals, both stationary and nonstationary, andAmblard et al. (1996b), who developed higher-order properties of complex stationary signals.Rubin-Delanchy and Walden (2007) present an algorithm for the simulation of improper WSS processes having specified covariance and complementary covariance functions.

Part II