FOURIER TRANSFORMS OF DISCRETE-TIME SIGNALS - Signals, Systems, and Transforms - page 606

e -j2pn because thus, periodicity is proved. This proper-

ty is very important, and its implications are covered in detail later in this chapter.
We illustrate this property with an example.

= cos 2pn - j sin 2pn = 1;

EXAMPLE 12.4

Demonstration of the periodicity of the DTFT

x[n] = a n u[n]

ƒ a ƒ 6 1,

From Table 12.1, for with

1
1 - ae -jÆ .

X(Æ) =

Then,

1
1 - ae -j(Æ+2p) =

1
1 - ae -jÆ e -j2p .

X(Æ+2p) =

e -j2p = cos 2p - j sin 2p = 1, Because

1
1 - ae -jÆ = X(Æ).

X(Æ+2p) = ■

Linearity

The linearity property of the Fourier transform states that the Fourier transform of
a sum of functions is equal to the sum of the Fourier transforms of the functions,
provided that the sum exists. The property applies directly to the discrete-time
Fourier transform:

[a 1 x 1 [n] + a 2 x 2 [n]] = a 1 X 1 (Æ) + a 2 X 2 (Æ).

This property was demonstrated in Example 12.2.

Time Shift

It is informative to derive the time-shift property from the definition of the discrete-
time Fourier transform. From (12.1),

[x[n - n 0 ]] = q
n=- q

x[n - n 0 ]e -jnÆ .

We make the change of variables (n - n 0 ) = k on the right side of this equation.

Then, because n = (k + n 0 ),

q

x[k]e -jÆ(k+n 0 )

[x[n - n 0 ]] = a

k=- q

q

= e -jÆn 0 a x[k]e -jÆk = e -jÆn 0 X(Æ);

k=- q