Learn the fundamentals of digital signal processing theory and discover the myriad ways DSP makes everyday life more productive and fun.

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Processamento Digital de Sinais

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Learn the fundamentals of digital signal processing theory and discover the myriad ways DSP makes everyday life more productive and fun.

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Module 4: Part 2 Filter Design

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

Let's begin our survey by looking at some IIR design techniques.

So filter design was an established art long before digital signal processing came

around and

lots of nice designs were developed along the years using discrete electronics.

There are some methods that can translate

analog designs into a rational transfer function.

We will not look at the details of these methods but

most numerical packages such as Mathlab provides you with ready-made routines that

will help you design filters according to this classic template.

The design often involves a trial and error phase.

You specify some parameters from the beginning such as the type of filter and

the desired cut-off frequency.

And then you guess other parameters, usually the filter order.

You run the routine and you verify that the provided filter fulfills the specs.

If that is not the case, you will probably have to change and

increase the filter order until the specs are met.

Let's now look at a few classic filters mutuated from the analog world.

We will concentrate on a low pass prototype.

But be aware that design techniques exist for high pass and band pass as well.

The first filter we'll look at is the Butterworth lowpass.

The Butterworth filter has a magnitude response which is maximally flat and

it is monotonic over the [0,pi] interval.

The Design Parameters are very simple.

The order of the filter and the desire to cut off frequency.

You run the algorithm and you get a prototype filter.

You test the filter with respect to the width of the transition band and

the passband error.

If any of these parameters do not fulfill the specifications

then you increase the order and you run the algorithm again.

The frequency response of the Butterworth filter looks like this.

It is, as we said, a monotonic curve that decays smoothly from 0 to pi.

This is an example of order 4, with a cut off frequency of pi over 4.

The Chebyshevf lowpass has a magnitude response

which is equiripple in the passband and monotonic in the stopband.

The design parameters are the order, the passband maximum error and

the cutoff frequency.

So, we have one extra parameter with respect to the Butterworth.

We run the algorithm and we check the result against the specs with respect to

the width of the transition band and the stopband error.

If any of these parameters do not fulfill the specs, we increase the order and

we run the algorithm again.

The frequency response of a Chebyshev filter looks like this.

Here we have again order of 4 and cut-off frequency equal to pi over 4 and

we specify a maximum pass band error of 12%.

Since the desired volume is 1.

This amplitude here is 0.12.

With respect to the Butterworth ilter, we have a steeper transition band and

this is the reward for accepting an equiripple error in the passband.

Finally let’s look at the Elliptic lowpass filter.

The Elliptic lowpass filters equiripple both in the passband and then the stopband

and the design parameters are the order, the cutoff frequency,

the passband maximum error and the stopband minimum attenuation.

We run the algorithm and

we check the result in terms of the width of the transition band.

So you see, this design lets you control all the parameters of the filter

except the transition band.

So, it's the most complex of the three.

The frequency response looks like this.

And again, our order is 4.

The cut off frequency is pi over 4.

And here we specify a maximum pass band error of 12% and

a minimum attribution of 0.03, so this is 0.03.

And this is 0.12.

The elliptic filter gives you the steepest transition band for

a given order of all the filters we have seen so far.

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