Digital Signal Processing Reference
In-Depth Information
Energy Conservation.
Parseval's theorem for theDFT is (obviously) iden-
tical to (4.70):
N
−
1
N
−
1
x
[
n
]
X
[
k
]
1
N
l
g
r
,
y
i
d
.
,
©
,
L
s
2
2
=
(4.85)
n
=
0
k
=
0
4.7 Fourier Analysis in Practice
In the previous Sections, we have developed three frequency representa-
tions for the three main types of discrete-time signals; the derivation was
eminently theoretical and concentrated mostly upon the mathematical
properties of the transforms seen as a change of basis in Hilbert space. In
the following Sections we will see how to put the Fourier machinery to prac-
tical use.
We have seen two fundamental ways to look at a signal: its time-domain
representation, in which we consider the values of the signal as a function
of discrete time, and its frequency-domain representation, in which we con-
sider its energy and phase content as a function of digital frequency. The in-
formation contained in each of the two representations is exactly the same,
as guaranteed by the invertibility of the Fourier transform; yet, from an an-
alytical point of view, we can choose to concentrate on one domain or the
other according to what we are specifically seeking. Consider for instance
a piece of music; such a signal contains two coexisting perceptual features,
meter
and
key
. Meter can be determined by looking at the duration patterns
of the played notes: its “natural” domain is therefore the time domain. The
key, on the other hand, can be determined by looking at the pitch patterns
of the played notes: since pitch is related to the frequency content of the
sound, the natural domain of this feature is the frequency domain.
We can recall that the DTFT is mostly a theoretical analysis tool; the
DTFTs which can be computed exactly (i.e. those in which the sum in (4.13)
can be solved in closed form) represent only a small set of sequences; yet,
these sequences are highly representative and they will be used over and
over to illustrate a prototypical behavior. The DFT,
(8)
on the other hand,
is fundamentally a
numerical
tool in that it defines a finite set of operations
which can be computed in a finite amount of time; in fact, a very efficient al-
gorithmic implementation of the DFT exists under the name of Fast Fourier
(8)
This also applies to the DFS, of course, which is formally identical. As a general remark,
whenever we talk about the DFT of a length-
N
signal, the same holds for the DFS of an
N
-periodic signal; for simplicity, from now on we will just concentrate on the DFT.
