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The frequency response characteristic of |Ha(Ω)|-Power(2) is as shown. As the order of the filter N increases, the butterworth filter characteristic is more close to the ideal characteristic.

**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.**

N= 5.88

1.
Order of the filter should be integer.

2.
Always go to nearest highest integer vale of N.

Hence N=6

**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

Cutoff
0-60Hz

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 10^{4} sps. Use BZT and Butterworth
approximation.

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