Digital Signal Processing Reference
In-Depth Information
is a formidable tool for the analysis and design of signal processing struc-
tures, as we will see in much detail in the context of discrete-time filters.
The purpose of the present Chapter is to introduce and analyze some
key results on Fourier series and Fourier transforms in the context of
discrete-time signal processing. It appears that, as we mentioned in the
previous Chapter, the Fourier transform of a signal is a change of basis in
an appropriate Hilbert space. While this notion constitutes an extremely
useful unifying framework, we also point out the peculiarities of its special-
ization within the different classes of signal. In particular, for finite-length
signals we highlight the eminently algebraic nature of the transform, which
leads to efficient computational procedures; for infinite sequences, we will
analyze some of its interesting mathematical subtleties.
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4.1 Preliminaries
The Fourier transformof a signal is an alternative representation of the data
in the signal. While a signal lives in the
time domain
,
(1)
its Fourier repre-
sentation lives in the
frequency domain
. We can move back and forth at will
fromone domain to the other using the direct and inverse Fourier operators,
since these operators are invertible.
In this Chapter we study three types of Fourier transformwhich apply to
the three main classes of signals that we have seen so far:
•
the Discrete Fourier Transform (DFT), which maps length-
N
signals
into a set of
N
discrete frequency components;
•
the Discrete Fourier Series (DFS), which maps
N
- periodic sequences
into a set of
N
discrete frequency components;
•
the Discrete-Time Fourier Transform (DTFT), which maps infinite se-
quences into the space of 2
π
-periodic function of a real-valued argu-
ment.
The frequency representation of a signal (given by a set of coefficients in the
case of the DFT and DFS and by a frequency distribution in the case of the
DTFT) is called the
spectrum
.
(1)
Discrete
-time, of course.
