FREQUENCY RESPONSE CHARACTERISTIC
The frequency response characteristic of |Ha(Ω)|2 is as shown. As the order of the filter N increases, the butterworth filter characteristic is more close to the ideal characteristic. Thus at higher orders like N=16 the butterworth filter characteristic closely approximate ideal filter characteristic. Thus an infinite order filter (N ∞) is required to get ideal characteristic.
Ap= attenuation in passband.
As= attenuation in stopband.
Ωp = passband edge frequency
Ωs = stopband edge frequency
Specification for the filter is
To determine the poles and order of analog filter consider equalities.
Q) Design a digital filter using a butterworth approximation by using impulse invariance.
Filter Type - Low Pass Filter
Ap - 0.89125
As - 0.17783
Ωp - 0.2∏
Ωs - 0.3∏
Step 1) To convert specification to equivalent analog filter.
(In impulse invariance method frequency relationship is given as ω= Ω T while in Bilinear transformation method frequency relationship is given as Ω= (2/T) tan (ω/2) If Ts is not specified consider as 1)
|Ha(Ω)| ≥ 0.89125 for Ω ≤ 0.2∏/T and |Ha(Ω)| ≤ 0.17783 for Ω ≥ 0.3∏/T.
Step 2) To determine the order of the filter.
1. Order of the filter should be integer.
2. Always go to nearest highest integer vale of N.
Step 3) To find out the cutoff frequency (-3DB frequency)
cutoff frequency Ωc = 0.7032
Step 4) To find out the poles of analog filter system function.
For stable filter all poles lying on the left side of s plane is selected. Hence
S1 = -0.182 + j 0.679 S1* = -0.182 - j 0.679
S2 = -0.497 + j 0.497 S2* = -0.497 - j 0.497
S3 = -0.679 + j 0.182 S3* = -0.679 - j 0.182
Step 5) To determine the system function (Analog Filter)
Step 6) To determine the system function (Digital Filter)
(In Bilinear transformation replace s by the term ((z-1)/(z+1)) and find out the transfer function of digital function)
Step 7) Represent system function in cascade form or parallel form if asked.
Q) Given for low pass butterworth filter
Ap= -1 db at 0.2∏
As= -15 db at 0.3∏
a) Calculate N and Pole location
b) Design digital filter using BZT method.
Q) Obtain transfer function of a lowpass digital filter meeting specifications
Stopband > 85Hz
Stopband attenuation > 15 db
Sampling frequency= 256 Hz . use butterworth characteristic.
Q) Design second order low pass butterworth filter whose cutoff frequency is 1 kHz at sampling frequency of 104 sps. Use BZT and Butterworth approximation.