Sinusoidal Steady-State Response of Digital Filters - Digital Signal Processing: Fundamentals and Applications - page 815

D.2 Properties of the Sinusoidal Steady-State Response
From Euler's identity and trigonometric identity, we know that

e jð U þk 2 pÞ ¼ cos ð U þ k 2 pÞþj sin ð U þ k 2 pÞ
¼ cos U þ j sin U ¼ e j U

(D.17)

where k is an integer taking values of k ¼ 0 ; 1 ; 2 ; / . Then

Frequency response: Hðe j U Þ¼Hðe jð U þk 2 pÞ Þ (D.18)

Hðe jð U þk 2 pÞ Þ

Magnitude frequency response: Hðe j U Þ

¼

(D.19)

Phase response: : Hðe j U Þ¼ : Hðe j U þ 2 kp Þ (D.20)

Clearly, the frequency response is periodic, with a period of 2 p . Next, let us develop the symmetric
properties. Since the transfer function is written as

XðzÞ ¼ b 0 þ b 1 z 1

þ / þ b M z M

HðzÞ¼ YðzÞ

(D.21)

1 þ a 1 z 1 þ / þ a N z N

substituting z ¼ e j U into Equation (D.21) yields
Hðe j U Þ¼ b 0 þ b 1 e j U þ / þ b M e jM U
1 þ a 1 e j U þ / þ a N e jN U

(D.22)

Using Euler's identity, e j U ¼ cos U j sin U , we have
Hðe j U Þ¼ ðb 0 þ b 1 cos U þ / þ b M cos M U Þjðb 1 sin U þ / þ b M sin M U Þ
ð 1 þ a 1 cos U þ / þ a N cos N U Þjða 1 sin U þ / þ a N sin N U Þ

(D.23)

Similarly,

Hðe j U Þ¼ ðb 0 þ b 1 cos U þ / þ b M cos M U Þþjðb 1 sin U þ / þ b M sin M U Þ
ð 1 þ a 1 cos U þ / þ a N cos N U Þþjða 1 sin U þ / þ a N sin N U Þ

(D.24)

Then the magnitude response and phase response can be expressed as

q

ðb 0 þ b 1 cos U þ / þ b M cos M U Þ

2 2

Hðe j U Þ ¼ þðb 1 sin U þ / þ b M sin M U Þ

q

ð 1 þ a 1 cos U þ / þ a N cos N U Þ

(D.25)

2 2

þða 1 sin U þ / þ a N sin N U Þ

: Hðe j U Þ¼ tan 1 ðb 1 sin U þ / þ b M sin M U Þ
b 0 þ b 1 cos U þ / þ b M cos M U

tan 1 ða 1 sin U þ / þ a N sin N U Þ
1 þ a 1 cos U þ / þ a N cos N U

(D.26)