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

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

It has the following properties

1. 

2. 
3.  is monotonically decreasing function of Ω

4. As n gets larger,   approaches an ideal low pass filter

5.   is called maximally flat at origin, since all order derivative exist and they are zero at 

The poles of a Butterworth filter lie on circle of radius  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.

where  order Chebyshev polynomial. We have the relationship

with 

Chebyshev filters have the following properties

1. The magnitude squared frequency response oscillates between 1 and  within the passband, the so called equiripple and has a value of , 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

where  is Chebyshev rational function of O determined from specified ripple characteristics.

An n^th 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 function of.

Type From	To	Transformation	Design Formula
Low pass cutoff θp	Low pass cutoff ωp
LPF	HPF
LPF	BPF
LPF	BSF

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 H_p(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 

rounded to 2

For   we get from table 9.1.  From standard tables (or MATLAB) we find standard 2 nd order Butterworth filter with cut off  and then apply the digital transform to get