Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 521

11.3 Determine if the following DT sequences satisfy the DTFT existence
property:
(i) x [ k ] − 2;

3 − k k < 3
0

(ii) x [ k ] =

otherwise;

− k

(iii) x [ k ] = k 3 ;

(iv) x [ k ] = α k cos( ω 0 k ) u [ k ], α < 1;
(v) x [ k ] = α k sin( ω 0 k + φ ) u [ k ], α < 1;
(vi) x [ k ] = sin( π k / 5) sin( π k / 7)
π 2 k 2

;

∞

(vii) x [ k ] = δ ( k − 5 m − 3);

m =−∞

3 − k k < 3
0

(viii) x [ k ] = and x [ k + 7] = x [ k ];

k = 3

(ix) x [ k ] = e j(0 . 2 π k + 45 ◦ ) ;
(x) x [ k ] = k 3 − k u [ k ] + e j(0 . 2 π k + 45 ◦ ) .

11.4 (a) Calculate the DTFT of the DT sequences specified in Problem 11.3.

(b) Calculate the DTFT of the periodic DT sequences specified in
Problem 11.1.

11.5 Given the following transform pair:

DTFT
←−−→

DTFT
←−−→ X 2 ( Ω ) , x 1 [ k ] X 1 ( Ω ) and x 2 [ k ]

express the DTFT of the following DT sequences in terms of the DTFTs
X 1 ( Ω ) and X 2 ( Ω ):
(i) ( − 1) k x 1 [ k ];
(ii) ( k − 5) 2 x 2 [ k − 4];
(iii) k e

− j4 k x 1 [3 − k ];

m =−∞ [ x 1 [ k − 4 m ] + x 2 [ k − 6 m ]];
(v) x 1 [5 − k ] x 2 [7 − k ].

∞

(iv)

11.6 Calculate the DT sequences with the following DTFT representations
defined over the frequency range −π ≤ ≤ π :

Ω

− j Ω
1 − 5e − j Ω + 6e − j2 Ω ;

4e

(i) X ( Ω ) =

− j2 Ω
(1 − 4e − j Ω ) 2 (1 − 2e − j Ω ) ;
(iii) X ( Ω ) = 8 sin(7 Ω ) cos(9 Ω );

2e

(ii) X ( Ω ) =

− j4 Ω

4e

(iv) X ( Ω ) = ;

10 − 6 cos Ω

1 . 25 π ≤ < 0 . 75 π

Ω

(v) X ( Ω ) =

≤ 0 . 25 π and 0 . 75 π ≤ <π.

0

Ω Ω