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The Properties of the DTFT 67
TABLE 3.1
Signal Type
Some common DTFT pairs
Sequence x(n) DTFT X ( e jω) , −π ≤ ω ≤ π
Unit impulse δ(n) 1
Constant 1 2πδ(ω)
Unit step
u(n)
1
+ πδ(ω)
1 − e−jω Causal exponential α n u(n)
1
1 − αe −jω
Complex exponential e jω0n 2πδ(ω − ω 0 )
Cosine cos(ω 0 n) π[δ(ω − ω 0 )+δ(ω + ω 0 )]
Sine sin(ω 0 n) jπ[δ(ω + ω 0 ) − δ(ω − ω 0 )]
Double exponential
α |n| u(n)
1 − α 2
1 − 2α cos(ω)+α 2
Note: Since X ( e jω) is periodic with period 2π, expressions over only
the primary period of −π ≤ ω ≤ π are given.
of these sequences can be easily obtained using the basic definitions (3.1)
and (3.2). These transform pairs and those of few other pairs are given
in Table 3.1. Note that, even if sequences like unit step u(n) are not
absolutely summable, their discrete-time Fourier transforms exist in the
limiting sense if we allow impulses in the Fourier transform. Such sequences
are said to have finite power, that is, ∑ n |x(n)|2 < ∞. Using
this table and the properties of the Fourier transform (discussed in Section
3.2), it is possible to obtain discrete-time Fourier transform of many
more sequences.
3.2 THE PROPERTIES OF THE DTFT
In the previous section, we discussed two important properties that
we needed for plotting purposes. We now discuss the remaining useful
properties, which are given below without proof. Let X(e jω ) be the
discrete-time Fourier transform of x(n).
1. Linearity: The discrete-time Fourier transform is a linear transformation;
that is,
F [αx 1 (n)+βx 2 (n)] = αF [x 1 (n)] + βF [x 2 (n)] (3.5)
for every α, β, x 1 (n), and x 2 (n).
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