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3.4 The Fourier Transform 63

ferent properties in the frequency domain. Examples of such systems are fre-
quency selective filters. The Fourier transform is particularly suitable for
describing (linear shift-invariant) systems, since complex sinusoidal sequences are
eigenfunctions to such systems. In other words, a sinusoidal input sequence will
lead to a sinusoidal output sequence with the same frequency, but with a phase-
shift and possibly a different magnitude. Analysis of the frequency properties of
such systems can be done using the Fourier transform.
The Fourier transform for a discrete-time signal is defined

and the inverse Fourier transform is

cafr
if the sum in Equation(3.3) exists. Note that the Fourier transform, X(eJ ), is
periodic in coT with period 2n, and that we use the angle 0)T as the independent
variable.
The magnitude
function, correspond-
ing to a real sequence,
is shown in Figure 3.1.
The magnitude and
the phase function as
well as the real and
imaginary parts of the
Fourier transform of a
real sequence are even
and odd functions of
1
off , respectively.
.For real sequences we nave

and

EXAMPLE 3.1
Determine the Fourier transform of the sequence