Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 509

Example 11.19
Plot the magnitude and phase spectra of the LTID system specified in
Example 11.18.

Solution
From Example 11.18, the transfer function of the LTID system is given by

2

H ( Ω ) = .

1 − 3

− j Ω + 1 − j2 Ω

4 e 8 e

Using Euler's formula exp( j Ω ) = cos Ω + j sin Ω and similarly for exp( j2 Ω ),

yields

2

H ( Ω ) = ,

1 − 3 + 1

3
4 sin Ω

− 1

4 cos Ω 8 cos(2 Ω ) + j 8 sin(2 Ω )

which leads to the following expressions for the magnitude and phase responses:

2

H ( Ω ) =

2 + 2

1 − 3 + 1

3
4 sin Ω

− 1

4 cos Ω 8 cos(2 Ω ) 8 sin(2 Ω )

2

= ;

101
64

− 27 + 1

16 cos Ω 4 cos(2 Ω )

1 − 3 + 1

3
4 sin Ω

− 1 < H ( Ω ) = < 2 − <

4 cos Ω 8 cos(2 Ω ) + j 8 sin(2 Ω )

3 4

3
4 sin Ω

− 1

8 sin(2 Ω ) − 1

=− tan .

1 − 3 + 1

5

4 cos Ω 8 cos(2 Ω )

Figures 11.18(a) and (b) plot the magnitude and phase spectra in the frequency
range Ω

= [ −π, π ]. Because the DTFT is periodic with period Ω 0 = 2 π , the
magnitude and phase spectra at other frequencies can be calculated using the
periodicity property. It is observed that the gain H ( Ω ) of the LTID system

has the maximum value of 16/3 at frequency Ω = 0

is also referred to as the dc component of the impulse response h [ k ], and is
the sum

= 0. The gain H ( Ω ) at Ω

h [ k ] over the duration of the impulse response. As the frequency
increases to π (or decreases to −π ), the gain decreases monotonically and has
a minimum value of 16/15 at Ω =π radians/s. For LTID systems, the fre-

quency Ω =π radians/s corresponds to the maximum frequency. The trans-

fer function H ( Ω ) represents a non-uniform amplifier as the lower-frequency
components are amplified at a relatively higher scale than the high-frequency
components.
The phase response < H ( Ω ) of the LTID system has a value of zero at Ω

= 0.
As the frequency increases from zero, the phase decreases to its minimum