Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 513

The DTFT Y ( Ω ) is obtained by substituting the values of the transfer function
H ( Ω ) at frequencies Ω 1

=π/ 4 .
(i) For the transfer function in Eq. (11.65), the values of H 1 ( Ω ) are given by

=π/ 8 and Ω 2

Ω = π/ 8

H 1 ( π/ 8) = 4 . 04 − j2 . 03 , H 1 ( π/ 8) = 4 . 52 , < H 1 ( π/ 8) =− 0 . 465 radians;

Ω =−π/ 8

H 1 ( −π/ 8) = 4 . 04 + j2 . 03 , H 1 ( −π/ 8) = 4 . 52 ,< H 1 ( −π/ 8) = 0 . 465 radians;

Ω

= π/ 4 H 1 ( π/ 4) = 2 . 44 − j2 . 11 , H 1 ( π/ 4) = 3 . 22 , < H 1 ( π/ 4) =− 0 . 71 radians;

Ω

=−π/ 4 H 1 ( −π/ 4) = 2 . 44 + j2 . 11 , H 1 ( −π/ 4) = 3 . 22 ,< H 1 ( −π/ 4) = 0 . 71 radians.

The DTFT Y 1 ( Ω ) of the output sequence is therefore given by

− π
8

+ π
8

− j0 . 465 + δ

4 . 52e j0 . 465 Y 1 ( Ω ) = 3 π δ 4 . 52e

Ω Ω

− π
4

+ π
4

− j0 . 71 + δ

3 . 22e j0 . 71

+ π 5 δ 3 . 22e

Ω Ω

δ

− π
8

+ π
8

− j0 . 465 + δ

e j0 . 465 = 13 . 56 π e

Ω Ω

− π
4

+ π
4

− j0 . 71 + δ

e j0 . 71

+ 16 . 10 π δ e .

Ω Ω

Calculating the inverse DTFT, the output sequence is obtained as

π
8 k − 0 . 465

π
4 k − 0 . 71

y 1 [ k ] = 13 . 56 cos + 16 . 10 cos ,

where we have expressed the constant phase in radians. Expressing the constant
phase in degrees yields

π
8 k − 26 . 67 ◦

π
4 k − 40 . 80 ◦

y 1 [ k ] = 13 . 56 cos + 16 . 10 cos .

The LTID system H 1 ( Ω ) acts like an amplifier as the sinusoidal component
3 cos( π k / 8) with fundamental frequency Ω 1 = π/ 8 is amplified by a factor
of 4.52, while the sinusoidal component 3 cos( π k / 4) with fundamental fre-
quency Ω 1 = π/ 4 is amplified by a factor of 3.22. The difference in the gains
is also apparent in the magnitude spectrum plotted in Fig. 11.18, where the
low-frequency components have a higher amplification factor than that of the
higher-frequency components.
(ii) For the transfer function in Eq. (11.65), the values of the transfer function
H 2 ( Ω ) at frequencies Ω 1 =π/ 8 and Ω 2 =π/ 4 are given by

Ω = π/ 8

H 2 ( π/ 8) = 1 , H 2 ( π/ 8) = 1 ,< H 2 ( π/ 8) = 0 radians;

Ω =−π/ 8

H 2 ( −π/ 8) = 1 , H 2 ( −π/ 8) = 1 ,< H 2 ( −π/ 8) = 0 radians;

Ω = π/ 4

H 2 ( π/ 4) = 0 , H 2 ( π/ 4) = 0 ,< H 2 ( π/ 4) = 0 radians;

Ω =−π/ 4

H 2 ( −π/ 4) = 0 , H 2 ( −π/ 4) = 0 ,< H 2 ( −π/ 4) = 0 radians.