Cyclostationary processes - Statistical Signal Processing of Complex-Valued Data - page 252

Fourier transform in t . These Fourier series coefficients are

T

1
T

)e − j2 π nt / T

/ ,τ = ,τ ,

p xx ( n T ) r xx ( t d t (10.3)

0

T

1
T

)e − j2 π nt / T

p xx ( n / T ,τ ) = r xx ( t ,τ d t , (10.4)

0

and they are called the cyclic correlation function and cyclic complementary correlation
function , respectively. There is a subtle difference between the cyclic correlation function
and the ambiguity function, which is defined as the Fourier transform of the correlation
function in t (see Section 9.2.1 ). The cyclic correlation function and the ambiguity
function have different physical units.
The Fourier transform of the cyclic correlation and cyclic complementary correlation
functions in yields the cyclic power spectral density and cyclic complementary power

spectral density :

τ

∞

)e − j2 π f τ d

P xx ( n / T , f ) = p xx ( n / T ,τ τ, (10.5)

−∞

∞

P xx ( n

)e − j2 π f τ d

/ T , f ) = p xx ( n / T ,τ τ. (10.6)

−∞

The appropriate generalization for the cyclic correlation and complementary correla-
tion of almost CS processes is

T / 2

1
T

)e − j2 πν n t p xx ( ν n ,τ ) = lim

T →∞

r xx ( t ,τ d t , (10.7)

− T / 2

T / 2

1
T

)e − j2 π ν n t p xx (˜ ν n ,τ ) = lim

T →∞

r xx ( t ,τ d t . (10.8)

− T / 2

The frequency offset variables ν n and ˜ ν n are called the cycle frequencies .Theset

of cycle frequencies { ν n } for which p xx ( ν n ,τ ) is not identically zero and the set of

cycle frequencies ) is not identically zero are both count-

able (but possibly countably infinite). These two sets do not have to be identical. CS
processes are a subclass of almost CS processes in which the frequencies

{ ν n } ˜ for which p xx (˜ ν n ,τ

{ ν n } and

{ ν n } ˜ are contained in a lattice of the form { n / T } . Relationships ( 10.5 ) and ( 10.6 )

remain valid for almost CS processes if n / T is replaced with the cycle frequencies ν n

and ˜ ν n :

∞

)e − j2 π f τ d P xx ( ν n , f ) = p xx ( ν n ,τ τ, (10.9)

−∞

∞

P xx (˜

)e − j2 π f τ d

ν n , f ) = p xx (˜ ν n ,τ τ. (10.10)

−∞

Harmonizable signals may be expressed as

∞

( f )e j2 π ft x ( t ) = d ξ . (10.11)

−∞