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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Do curso por École Polytechnique Fédérale de Lausanne

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 1 Introduction to Filtering

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

In general when we talk about signal processing, we imagine a situation where

we have an input signal, x of n, an output signal, y of n, produceD

by some sort of black box here that manipulates the input into the output.

We can write the relationship mathematically Like so

where y then is equal to some operator H that is applied to the input x of n.

Already when we draw this block diagram

we are making assumptions on the structure of the process in device

in the sense that we consider a system with a single input and a single output.

We could imagine system with multiple inputs or multiple outputs.

But even with these limitations, the possibilities for

what goes inside this box here are pretty much limitless.

And unless we impose some structure on the kind of processing that happens in this

block, we will not be able to say anything particularly meaningful

about the filtering operation.

So the first requirement that we imposed on a filter is linearity.

Linearity means that if we have two inputs and we take a linear combination

of said inputs while the output is a linear combination of outputs.

That could have been obtained by filtering each sequence independently.

This is actually a very reasonable requirement, for

instance take a situation where you processing device is an amplifier and

you connect a guitar to your amplifier.

Now if you play one note and then you play the same note louder,

you expect the amplifier to produce just a louder note.

[SOUND] Similarly, if you play one note and

then another note, and then you play two notes together,

you expect the amplifier to amplify the sum of two

notes as the sum of two independent amplifications.

[MUSIC]

Now note that this is not necessarily the case in all situations.

For instance, in some kinds of rock music you want to introduce some distortion and

so you add a fuzz box.

That will distort the signal non-linearly to create very interesting effects but

that belong to a completely different category of processing.

[MUSIC]

The second requirement that we impose on the processing device is time invariance.

Time invariance, in layman terms, simply means that the system will behave

exactly in the same way independently of when it's switched on.

Mathematically, we can say that if y[n] is the output of the system when the input is

x[n], well if we put a delayed version of the input inside the system,

x of n minus n-0, what we get is the same output delayed by n-0.

And, again, we can use a guitar amplifier as an example.

If I turn it on today, well I expect it to

amplify the notes exactly in the same way that it amplified them yesterday.

But again, some types of guitar effects exploit

time variance to introduce a different flavor to the music that's being played.

For instance, the wah pedal is a time varying effect that

will change the envelope of a sound In ways that are not time invariant.

[MUSIC]

So what are the ingredients that go into a linear time-invariant system?

Well, linear ingredients.

Addition, which is a linear operation, scalar multiplication,

another linear operation, and delays.

Another requirement that is not mandatory, but makes a lot of sense if you want to

use linear time variance system in your real time, is that the system be cause

by that we mean that the system can only have access to input and

output values from the past.

In that case, you can write the input output relationship as follows.

The output is a linear functional of past values of the input and

past values of the output.

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