Digital Signal Processing Reference
In-Depth Information
With
T
representing the period of the random signal
x
(
t
),
this can be broken down
into a Fourier series with
random coefficients
:
t
T
1
+
T
/2
−
j
2
π
k
=
∫
()
C
x t e
dt
[1.33]
k
T
−
T
/2
The random variables {
C
k
} have the following striking properties:
( )
( )
( )
EC
=
0for
k
≠
0
k
*
ECC
=
0for
k
≠
m
[1.34]
km
2
EC
=
A
k
k
where the power
A
k
is associated with the line at frequency
f
=
kT
appearing in
/
k
equation [1.31].
Note that there exists a line for
f
= 0 (by convention, we note it as
f
0
= 0) which
is linked to what is commonly known as the “DC component”, as it is easy to show
that
(
( )
2
A
=
Ext
[1.35]
where the average
E
(
x
(
t
))
is identical to the continuous component of electronics
engineers, if the signal is ergodic (see equation [1.27]).
The term
S
3
(
f
) is known as singular and is, generally, absent in physical signals.
The previous properties are very important to understand what kind of object we
will be confronted with when we will perform a 2
nd
order spectral analysis on
stationary signals. Figure 1.4 summarizes the properties of these PSD.
The main role of the autocorrelation function and its Fourier transform, the PSD,
is largely justified by the regularity of their behavior in the invariant linear
transformations. We must briefly recall here that if two random signals are linked by
an invariant linear filtering operation of kernel
h
(
t
)
(continuous time) or
h
(
k
)
(discrete time), a kernel that is also its impulse response, we can write it in the
convoluted form:
∫
()
()
() ( )
yt
=
x ht
=
xuht udu
−
⊗
ℜ
+∞
[1.36]
∑
()
()
()( )
yk
=
x hk
=
xmhk m
−
⊗
m
=−∞
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