23-01-2013, 10:53 AM
Discrete-time Fourier transform
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Fourier transform
Recall from the last lecture that any sufficiently regular (e.g.,
finite-energy) continuous-time signal x(t) can be represented in frequency
domain via its Fourier transform
Spectral content of discrete-time signals
In this lecture, we will look at one way of describing discrete-time signals
through their frequency content: the discrete-time Fourier transform
(DTFT).
Properties of the DTFT
Like its continuous-time counterpart, the DTFT has several very useful
properties. These are listed in any text on signals and systems. We will
take a look at a couple of them.
Summary of the DTFT
The discrete-time Fourier transform (DTFT) gives us a way of
representing frequency content of discrete-time signals.
The DTFT X(
) of a discrete-time signal x[n] is a function of a
continuous frequency
. One way to think about the DTFT is to view
x[n] as a sampled version of a continuous-time signal x(t):
x[n] = x(nT ), n = . . . ,−2,−1, 0, 1, 2, . . . ,
where T is a sufficiently small sampling step. Then X(
) can be thought
of as a discretization of X(ω).
Due to discrete-time nature of the original signal, the DTFT is
2π-periodic. Hence,
= 2π is the highest frequency component a
discrete-time signal can have.
The DTFT possesses several important properties, which can be
exploited both in calculations and in conceptual reasoning about
discrete-time signals and systems.