Digital Signal Processing Reference
In-Depth Information
These representations do not allow the signal to be reconstructed, as the phase of
the transform is lost. We will call them ESD (energy spectral densities).
But the
main reason for interest in these partial representations is that they show the energy
distribution along a non-time axis (here the frequency). In fact, Parseval's theorem
states:
2
2
∫
∫
()
()
[1.19]
E
=
x t
dt
=
x
f
df
R
R
x f
represents the energy distribution in
f
, just as |
x
(
t
)|
2
therefore represents
its distribution over
t.
and
()
2
For discrete time signals, Parseval's theorem is expressed a little differently:
N
−
1
N
−
1
1
∑∑
2
2
()
()
E
=
x k
x l
N
k
=
0
l
=
0
but the interpretation is the same.
The quantities
S
x
(
f
) or
S
x
(
l
) defined above have an inverse Fourier transform,
which can easily be shown as being expressed by:
(
)
∫
()
() ( )
()
IFT
S
f
=
x t
x
∗ −
t
τ
dt
γ τ
[1.20]
x
xx
R
This function is known as the autocorrellation function of
x
(
t
).
We see that the
ESD of
x
(
t
)
could have axiomatically been defined as follows:
(
( )
(
)
()
T
γ τ
()
Sf
=
SDxt
[1.21]
x
xx
When we are interested in
finite power
signals, their Fourier transform exists in
terms of distributions. This generally prohibits the definition of the square of the
modulus, and thus the PSD as by equation [1.17] or [1.18]. It is, however, always
possible to define the autocorrelation function by:
+
T
/2
1
∫
()
() ( )
γτ
lim
x tx t
*
−
τ
dt
[1.22]
xx
T
−
T
/2
T
→∞
Its Fourier transform is used to define the PSD by:
(
( )
(
)
()
()
Sf
=
SDxt
T
γ τ
[1.23]
x
xx
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