Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 498

Adding the DTFT pairs for p k u [ k ] and kp k u [ k ] yields

− j Ω

1 p e 1

DTFT
←−−→ ( k + 1) p k u [ k ] + =

p e − j Ω ) 2 . 1 − p e − j Ω (1 − p e − j Ω ) 2 (1 −

11.5.9 Time summation

The time summation in the DT domain is the counterpart of integration in the
CT domain. The time-summation property is defined as follows.
If

DTFT
←−−→

x [ k ] X ( Ω )

then

n =−∞ x [ n ]

k

m =−∞ δ ( Ω

∞

1
(1 − e − j Ω ) X ( Ω ) + π X (0)

DTFT
←−−→

− 2 π m ) . (11.46)

Example 11.14
Based on the DTFT of the unit impulse sequence and the time-summation
property, calculate the DTFT of the unit step sequence.

Solution
Using Table 11.2, the DTFT of the unit impulse sequence is given by

DTFT
←−−→

δ [ k ] 1 .

Using the time-summation property, we obtain
k

n =−∞ δ [ n ] m =−∞ δ ( Ω

∞

1
1 − e − j Ω

DTFT
←−−→ 1 + π 1 − 2 π m ) ,

which yields

m =−∞ δ ( Ω

∞

1
1 − e

DTFT
←−−→ u [ k ] + π − 2 π m ) .

− j Ω

11.5.10 Time convolution

In Section 10.5, we showed that the output response of an LTID system is
obtained by convolving the input sequence with the impulse response of the
system. Sometimes the resulting convolution sum is difficult to solve analyti-
cally in the time domain. The convolution property provides us with an alter-
native approach, based on the DTFT, of calculating the output response. Below
we state the convolution property and explain its application in calculating the
output response of an LTID system using an example.
If x 1 [ k ] and x 2 [ k ] are two DT sequences with the following DTFT pairs:

DTFT
←−−→

DTFT
←−−→

x 1 [ k ] X 1 ( Ω ) and x 2 [ k ] X 2 ( Ω ) ,