Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 323

u ds ¼ 2pf ¼ 2pð3; 000Þ¼6; 000p rad/sec
T ¼ 1=f s ¼ 1=8; 000 sec

We then follow the design procedure steps
1. We apply the warping equation as follows:

u d T
2

3; 000p=8; 000
2

u ap ¼ T tan

¼ 1:0691 10 4 rad=sec:

¼ 16; 000 tan

u d T
2

6; 000p=8; 000
2

u as ¼ T tan ¼ 3:8627 10 4 rad=sec: ¼ 16; 000 tan

We then find the lowpass prototype specifications using Table 8.6 as follows:

1:0691 10 4

v s ¼ u as =u ap ¼ 3:8627 10 4 = ¼ 3:6130 rad=s and A s ¼ 10 dB

The filter order is computed as

ε 2 ¼ 10 0:13 1 ¼ 1

log 10 10 0:110 1
2$ log 10 ð3:6130Þ

n ¼ ¼ 0:8553

2. Rounding n up, we choose n ¼ 1 for the lowpass prototype. From Table 8.3 , we have
H P s ¼

1
s þ 1

Applying the prototype transformation (lowpass to lowpass) yields the analog filter

HðsÞ¼H P ðsÞ s

1:0691 10 4
s þ 1:0691 10 4

1 u ap

s þ u ap ¼

u ap ¼

u ap þ 1 ¼

s

3. Finally, using the BLT, we have

s¼16;000ðz1Þ=ðzþ1Þ

HðzÞ¼ 1:0691 10 4
s þ 1:0691 10 4

Substituting the BLT leads to

1:0691 10 4

16; 000

HðzÞ¼

z 1
z þ 1

þ 1:0691 10 4

To simply the algebra, we both numerator and denominator by 16,000 to get

0:6682
z 1
z þ 1

HðzÞ¼

þ 0:6682