Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 136

e − jω
e − jω

1

a − n u( x 4 (n) = − n) ⇔

ae jω =

when | a | < 1 (3.35)

1 − − a

ae jω
1 −

a − n u(

x 3 (n) = − n − 1 ) ⇔ when | a | < 1 (3.36)

ae jω

a 2
1 − 2 a cos ω

1 −

x 13 (n) = x 1 (n) + x 3 (n) ⇔ when | a | < 1 (3.37)

a 2

+

α n u [ −

For the sequence x 5 (n) = (n + 1 ) ], note that the transform pair is given

by (3.38), which is valid when | α | > 1:

e jω

1
αe − jω

α n u [ −

x 5 (n) = (n + 1 ) ] ⇔

1 =

when | α | > 1 (3.38)

− α − e jω

Example 3.9

A few examples are given below to help explain these differences. From the
results given above, we see that

1. If the DTFT X 1 (e jω )

= 1 /( 1 − 0 . 8 e − jω ) , its IDTFT is x 1 (n) ( 0 . 8 ) n u(n) .

=

2. The IDTFT of X 3 (e jω ) 0 . 8 e jω /( 1 0 . 8 e jω ) is given by x 3 (n)

= − =

( 0 . 8 ) − n [ u( 1 ) ].

3. The IDTFT of X 4 (e jω )

− n −

0 . 8 e jω ) is x 4 (n) ( 0 . 8 ) − n u(

= 1 /( 1 − = − n) .But

4. The IDTFT of X 5 (e jω ) e jω /( 2 e jω ) is x 5 (n) ( 2 ) n u( = − = − n − 1 ) .

Note the differences in the examples above, particularly the DTFT-IDTFT pair
for x 5 (n) .
The magnitude and phase responses of X 1 (e jω ), X 3 (e jω ) ,and X 13 (e jω ) are
shown in Figures 3.17, 3.18, and 3.19, respectively. The magnitude responses of
X 1 (e jω ), X 4 (e jω ) ,and X 3 (e jω ) given below appear the same except for a scale

5 4.5

4

4

3

3.5

2 3

1 2.5

0 2

0 1 2 − 2 − 1 0 1 2

− 2 − 1

Figure 3.17 The magnitude and phase responses of x 1 (n) .