Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 322

Table 8.6 Conversion from Analog Filter Specifications to Lowpass Prototype Specifications
Analog Filter Specifications Lowpass Prototype Specifications

Lowpass: u ap , u as v p ¼ 1 , v s ¼ u as =u ap

Highpass: u ap , u as v p ¼ 1 , v s ¼ u ap =u as

v s ¼ u ash u asl
u aph u apl

p

u apl u aph

Bandpass: u apl , u aph , u asl , u ash , u 0 ¼ , v p ¼ 1 ,

p

u asl u ash u 0 ¼

v s ¼ u aph u apl
u ash u asl

p

u apl u aph Bandstop: u apl , u aph , u asl , u ash , u 0 ¼ , v p ¼ 1 ,

p

u asl u ash u 0 ¼

u ap , passband frequency edge; u as , stopband frequency edge;
u apl , lower cutoff frequency in passband; u aph , upper cutoff frequency in passband;
u asl , lower cutoff frequency in stopband; u ash , upper cutoff frequency in stopband;
u 0 , geometric center frequency.

L f B f

0 apl aph

W

Hj

LP ()

Hj

BP ()

aph apl

A P A P

W

A s

A s

0

0

ap apl aph

as asl 0 ash

FIGURE 8.15

Specifications for analog lowpass and bandpass filters.

8.3.2 Lowpass and Highpass Filter Design Examples
The following examples illustrate various designs for the Butterworth and Chebyshev lowpass and
highpass filters.

EXAMPLE 8.7
a. Design a digital lowpasss Butterworth filter with the following specifications:
1. 3-dB attenuation at the passband frequency of 1.5 kHz
2. 10-dB stopband attenuation at the frequency of 3 kHz
3. Sampling frequency at 8,000 Hz
b. Use MATLAB to plot the magnitude and phase responses.
Solution:
a. First, we obtain the digital frequencies in radians per second:
u dp ¼ 2pf ¼ 2pð1500Þ¼3; 000p rad/sec