Method For Designing Digital Filters Using BZT

METHOD FOR DESIGNING DIGITAL FILTERS USING BZT

step 1. Find out the value of ωc*,.

step 2. Find out the value of frequency scaled analog transfer function

Normalized analog transfer function is frequency scaled by replacing s by s/ωp^*.

step 3. Convert into digital filter

Apply BZT. i.e Replace s by the ((z-1)/(z+1)). And find out the desired transfer function of digital function.

Example:

Q) Design first order high pass butterworth filter whose cutoff frequency is 1 kHz at sampling frequency of 10⁴ sps. Use BZT Method

Step 1. To find out the cutoff frequency

ωc = 2∏f

= 2000 rad/sec

Step 2. To find the prewarp frequency

ωc* = tan (ωc Ts/2) = tan(∏/10)

Step 3. Scaling of the transfer function

For First order HPF transfer function H(s) = s/(s+1) Scaled transfer function H*(s) = H(s) |s=s/ωc*

H^*(s)= s/(s + 0.325)

Step 4. Find out the digital filter transfer function. Replace s by (z-1)/(z+1)

Q) Design second order low pass butterworth filter whose cutoff frequency is 1 kHz at sampling frequency of 10⁴ sps.

Q) First order low pass butterworth filter whose bandwidth is known to be 1 rad/sec . Use BZT method to design digital filter of 20 Hz bandwidth at sampling frequency 60 sps.

Q) Second order low pass butterworth filter whose bandwidth is known to be 1 rad/sec . Use BZT method to obtain transfer function H(z) of digital filter of 3 DB cutoff frequency of 150 Hz and sampling frequency 1.28 kHz.

Q) The transfer function is given as s²+1 / s ²+s+1 The function is for Notch filter with frequency 1 rad/sec. Design digital Notch filter with the following specification

(1) Notch Frequency= 60 Hz

(2) Sampling frequency = 960 sps.