Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 488

Fig. 11.10. DTFT of the periodic
sequence x 2 [ k ], with
fundamental period K 0 = 4.
(a) Magnitude spectrum; (b)
phase spectrum.

X 2 ( W )

5 p 5 5 5 p 5 5 5 p

p p p p

2 2 2 2

W

−2 p −1.5 p − p −0.5 p 0 0.5 p p 1.5 p 2 p

(a)

< X 2 ( W )

p

p

−

4

4

−1.5 p 0.5 p

W

−0.5 p −2 p − p 0 p 1.5 p 2 p

p p

−

4

4

(b)

functions in the magnitude spectrum is given by 2 π D n and is indicated at the
top of each impulse in Fig. 11.9(a).
(ii) Using Eq. (11.36a), the DTFT of x 2 [ k ]isgivenby

Ω −

∞ ∞

2 n π
4

n π
2

X 2 ( Ω ) = 2 π D n δ = 2 π D n δ

Ω −

.

n =−∞ n =−∞

Substituting Ω 0 = 2 π/ K 0 = π/ 2 in Eq. (11.36b), the DTFS coefficients D n are
as follows:

1

1
4

1
4 [5 + 5e − j π n / 2 ] =

5
2 e − j π n / 4 cos

π n
4

5e − j n π k / 2 D n = = .

k = 0

For 0 ≤ n ≤ 3, the values of the DTFS coefficients are as follows:

= 5
2

with D 0 = 5

n = 0( Ω = 0) D 0

2 ,< D 0

= 0;

5 5

=− π

− j π/ 4

n = 1( Ω = 0 . 5 π ) D 1 = √

2 e

with D 0 = √

2 ,< D 0

4 ;

2 2

n = 2( Ω = π ) D 2 = 0 with D 0 = 0 ,< D 0 = 0;

=− 5
2

5

− j3 π/ 4

n = 3( Ω = 1 . 5 π ) D 3 √

2 e

with D 0 = √

2 , 2

− 3 π
4

= π

4 .
The magnitude and phase spectra are plotted separately in Figs. 11.10(a) and
(b), where the values of the DTFS coefficients lying outside 0 ≤ n

< D 0 = π

≤ 3 are

obtained using the periodicity property D n + 4 = D n .
(iii) Using Eq. (11.36a), the DTFT of x 3 [ k ]isgivenby

Ω

.

n =−∞ D n δ

∞

− 2 n π
15

X 3 ( Ω ) = 2 π