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Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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Some frequently used analog filters

In the previous two examples we have used Butterworth filter. The Butterworth filter of order n is described by the magnitude square frequency response of

Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

It has the following properties

1. Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

2. Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
3. Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is monotonically decreasing function of Ω

4. As gets larger, Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  approaches an ideal low pass filter

5. Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  is called maximally flat at origin, since all order derivative exist and they are zero at  Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The poles of a Butterworth filter lie on circle of radius Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) in s-plane.

There are two types of Chebyshev filters, one containing ripples in the passband (type I) and the other containing a ripple in the stopband (type II). A Type I low pass normalizer Chebyshev filter has the magnitude squared frequency response.

  Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

where Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) order Chebyshev polynomial. We have the relationship

Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

with 

Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Chebyshev filters have the following properties

  1. The magnitude squared frequency response oscillates between 1 and Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) within the passband, the so called equiripple and has a value of Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE), the normalized cut off frequency.
  2. The magnitude response is monotonic outside the passband including transitionand stopband.
  3. The poles of the Chebysher filter lie on an ellipse in s-plane.

An elliptic filter has ripples both in passband and in stopband. The square magnitude frequency response is given by

Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

where Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is Chebyshev rational function of O determined from specified ripple characteristics.

An nth order Chebyshev filter has sharper cutoff than a Butterworth filter, that is, has a narrower transition bandwidth. Elliptic filter provides the smallest transition width.

Design of Digital filter using Digital to Digital transformation

There exists a set of transformation that takes a low pass digital filter and turn into highpass, bandpass, bandstop or another lowpass digital filter. These transformations are given in table 9.1. 

The transformations all take the form of replacing the  Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  some function of.

Type FromToTransformationDesign Formula
Low pass cutoff θpLow pass cutoff ωpSome Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
LPFHPFSome Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
LPFBPFSome Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
LPFBSFSome Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

 

Starting with a set of digital specifications and using the inverse of the design equation given in table 9.1, a set of lowpass digital requirements can be established. A LPF digital prototype filter Hp(z) is then selected to satisfy these requirements and the proper digital to digital transformation is applied to give the desired.

Example

Using the digital to digital transformation, find the system function H(z) for a low-pass digital filter that satisfies the following set the requirements (a) monotone stop and passband (b)-3dB cutoff frequency of 0.5π(c) attenuation at and past 0.75π is at least 15dB.

Because of monotone requirement, a Butterworth filter is selected. The required n is given by 

Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

rounded to 2

Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

For  Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) we get from table 9.1. Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) From standard tables (or MATLAB) we find standard 2 nd order Butterworth filter with cut off Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) and then apply the digital transform to get

Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The document Some Frequently used Analog Filters & Digital to Digital transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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