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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From the course by École Polytechnique Fédérale de Lausanne

数字信号处理

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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.

From the lesson

Module 4: Part 1 Introduction to Filtering

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

What happens if you take a linear time invariant filter H,

and we use as the input, a complex exponential of known frequency omega 0.

Well, let's write out the convolution.

Then we exploit the fact that the convolution is commutative and

we exchange the order of the terms in the convolution product.

At which point we can start writing out the convolution sum explicitly.

And we have the sum for k that goes from minus infinity to plus

infinity of h of k that multiplies e to the j omega 0 and minus k.

We can take out the constant term that doesn't depend on k out of the sum and

we have a leading term of e to the j omega 0n that multiplies the sum for k that

goes from minus infinity to plus infinity of h[k] times e to the minus j omega 0 k.

But now this guy here, we know very well what it is.

It's the DTFT of the impulse response computed in omega 0.

So, it's big H of e to the j omega 0,

which multiplies our original input e to the j omega 0 n.

Hence the name Eigen sequences.

Just like an Eigen vector is a vector that when multiplied by a matrix gives a scaled

version of itself.

An Eigen sequence is a sequence that when input to a linear time invariant filter,

returns the sequence itself times a scaling factor.

Which happens to be the value of the DTFT of

the impulse response at the frequency of the input.

So the first fundamental property that we can glean from this derivation is that

linear time invariant system cannot change the frequency of a sinusoidal input.

And so, since the sinusoidal input is a pure frequency component.

It's clear that the DTFT of the impulse response fully determines

the frequency characteristic of a filter at a given frequency.

Let's examine a little bit more of what happens when

we process a complex exponential with a linear time invariant filter.

If we write the value of the DTFT of the impulse

response at omega 0 as A times e to the j theta,

where A is a real number.

And theta, of course, is an angle between minus pi and pi, if you will.

Then we have that the processed complex exponential is the original

complex exponential, which has been scaled by the amplitude A,

and has been delayed or advanced by a phase term theta.

So, if A is larger than 1, then we have an amplification.

If A is less than 1, then we have an attenuation of the of the sinusoid.

And the phase shift again if it’s bigger than 0,

then we have an advancement of the sinusoid.

And if it’s less than 0, then we have a delay.

The convolution theorem tries to generalize this result by asking

the question, what is the DTFT of the convolution of two sequences?

Or in other words, what is the DTFT of the output of a filter?

The intuition here is that the DTFT reconstruction formula tells us that any

signal is made up of infinitely many sinusoidal components of the form

x to the e to the j omega times e to the j omega n.

And, if I were to process a single sinusoidal component independently via

a filter h, I would get a value like so.

H, j omega, e to the j omega, n.

So I wouldn't be surprised if the DTFT of the convolution of two sequences

was just a product of the Fourier transforms.

So let's go see how we can derive this more formally.

If we write out the DTFT of the convolution of two sequences,

well, we can write out initially the formula for the DTFT.

Which is the sum from n from minus infinity to infinity of

the value of the convolution in n times e to the minus j omega n.

Then we expand the formula for the convolution inside of the DTFT summation.

And we get a double summation over the indices n and

k of the convolution product here times this complex exponential here.

In the next step, we do a little trick whereby we add and

subtract k from the argument n.

So instead of putting n, we write (n- k) + k.

And in so doing, we manage to split the complex exponential in two components here

that we will distribute across the summations.

So now here we collect the terms that do not depend on n, and here we put the rest.

This first term here is the DTFT of x of n, that's really easy to see.

The second term looks like a DTFT.

We might be confused by the fact that the indices here and

here are (n- k) instead of n.

But the fact that the index n ranges from minus infinity to plus infinity makes this

term completely irrelevant.

And so, indeed, we had the product of two DTFTs,

the product of the DTFT of the impulse response times the DTFT of the input.

The Fourier transform of the impulse response is called the frequency response.

And just like in the case of the single complex exponential,

we can split the significance of the product in the frequency

domain by separating the effects of the magnitude and those of the phase.

So the magnitude of the frequency response will determine whether certain frequencies

of the input signal are going to be amplified or

attenuated according to whether the frequency response is larger than 1 or

smaller than 1 in magnitude in certain frequency bands.

Defects of the phase are a little bit more difficult to qualify right now.

But it will be clearer later on that the phase will determine whether the signal

will conserve its shape in the time domain, or its shape will be altered.

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