Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 223

Table 7.2 Frequency Response Calculation in Example 7.2

: Hðe jU Þ
degree

Hðe jU Þ Hðe jU Þj dB

radians f [ U f s /(2 ) Hz 0.2 D 0.3742 cos

U p U

0 0 0.5742 0.5742 4.82 0

p= 4 1000 0.4646 0.4646 6.66 45

p= 2 2000 0.2 0.2 14.0 90

3 p= 4 3000 0.0646 0.0646 23.8 45

p 4000 0.1742 0.1742 15.2 0

Multiplying X(z) leads to

Y ðzÞ¼0:1871X ðzÞþ0:2z 1 X ðzÞþ0:1871z 2 X ðzÞ

Applying the inverse z-transform on both sides, the difference equation is yielded as
yðnÞ¼0:1871xðnÞþ0:2xðn 1Þþ0:1871xðn 2Þ

c. The magnitude frequency response and phase response can be obtained using the technique introduced in
Chapter 6. Substituting z ¼ e jU into HðzÞ, it follows that
Hðe jU Þ¼0:1871 þ 0:2e jU þ 0:1871e j2U

Factoring term e jU and using the Euler formula e jx þ e jx

¼ 2 cos ðxÞ, we achieve

Hðe jU Þ¼e jU ð0:1871e jU þ 0:2 þ 0:1871e jU Þ
¼ e jU ð0:2 þ 0:3742 cos ðUÞÞ

Then the magnitude frequency response and phase response are found to be
Hðe jU Þ

¼ j0:2 þ 0:3472 cos Uj

and (

U if 0:2 þ 0:3472 cos U > 0

: Hðe jU Þ¼

U þ p if 0:2 þ 0:3472 cos U < 0

Details of the magnitude calculations for several typical normalized frequencies are listed in Table 7.2 .

Due to the symmetry of the coefficients, the obtained FIR filter has a linear phase response as
shown in Figure 7.4 . The sawtooth shape is produced by the contribution of the negative sign of the real
magnitude term 0 : 2 þ 0 : 3742 cos U in the 3-tap filter frequency response, that is,
Hðe jU Þ¼e jU ð 0 : 2 þ 0 : 3742 cos UÞ

In general, the FIR filter with symmetric coefficients has a linear phase response (linear function of U )
as follows:

: Hðe jU Þ¼MU þ possible phase of 180

(7.13)