Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 135

x 5 (n)

0.5

0.25

0.125

0.0315 0.0625

− 7 − 6 − 5 − 4 − 3 − 2 − 1 01 2 n

Figure 3.16 The discrete-time sequence x 5 (n) .

α n u [ − Now let us consider the case of x 5 (n) = (n + 1 ) ], where | α | > 1. A plot

of this sequence is shown in Figure 3.16 for α = 2.ItsDTFTisderivedbelow:

∞

X 5 (e jω ) α n u [ 1 ) ] e − jωn = − (n +

n =−∞

1

α e jω − n

−∞ −∞

αe − jω

n

= =

n =− 1 n =− 1

By a change of variable n

=− m ,weget

1

α e jω m

1

α e jω m

∞ ∞

X 5 (e jω ) = =− 1 +

m = 1 m = 0

1

=− 1 +

− α e jω

1

1
αe − jω

=

− 1

So we have the transform pair

e jω 1

αe − jω

α n u [ − x 5 (n) = (n + 1 ) ] ⇔

− 1 =

when | α | > 1 (3.33)

α − e jω

It is important to exercise caution in determining the differences in this pair
(3.33), which is valid for

< 1.
All of them are given below (again, the differences between the different DTFT-
IDTFT pairs and the corresponding plots should be studied carefully and clearly
understood):

| α | > 1, and the earlier pairs, which are valid for | a |

e jω
e jω

1

a n u(n) x 1 (n) = ⇔

ae − jω =

| a | < 1

when (3.34)

1 − − a