Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 479

X 3 ( W )

2.5

x 3 [ k ] =0.6 k u [ k ]

1 0.6

0.36

k W

−10 −8 −6 −4 −2

02 4 68 10

−3 p −2 p − p 0 p 2 p 3 p

(a) (b)

< X 3 ( W )

0.2 p

Fig. 11.8. (a) Decaying
exponential sequence x 3 [ k ]
with a decay factor p = 0.6.
(b) Magnitude spectrum and
(c) phase spectrum of x 3 [ k ]as
derived in Example 11.6(iii).

W

−3 p −2 p − p 0 p 2 p 3 p

−0.2 p

(c)

As a special case, we plot the DT sequence x 3 [ k ] and its magnitude and phase
spectra for p = 0 . 6 in Figs. 11.8(a)-(c).

In Example 11.6, we calculated the DTFTs for three different sequences and
observed that all three DTFTs are periodic with period Ω 0

= 2 π . This property
is referred to as the frequency-periodicity property and is satisfied by all DTFTs.
In Section 11.4, we present a mathematical proof verifying the frequency-
periodicity property.

Example 11.7
Calculate the DT sequences for the following DTFTs:

∞

(i) X 1 ( Ω ) = 2 π δ ( Ω − 2 m π );

m =−∞

∞

(ii) X 2 ( Ω ) = 2 π δ ( Ω − Ω 0 − 2 m π ) .

m =−∞

Solution
(i) Using the synthesis equation, Eq. (11.28a), the inverse DTFT of X 1 ( Ω )is
given by

π

m =−∞ δ ( Ω

∞

1
2 π

1
2 π

X 1 ( Ω )e j k Ω d Ω − 2 m π )e j k Ω d Ω

x 1 [ k ] = = 2 π

2 π − 2 π

π

m =−∞ δ ( Ω

∞

− 2 m π )e j k 2 m π = d Ω [ ∵ δ ( Ω − θ ) f ( Ω ) = δ ( Ω − θ ) f ( θ )]

= 1

−π

π

m =−∞ δ ( Ω

∞

= − 2 m π )d Ω .

−π