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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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.

從本節課中

Module 3: Part 2 - Advanced Fourier Analysis

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

And indeed some things would lead us to believe so.

For instance, if you take the DFT of the delta function,

you remember you have the constant 1.

And similarly, the DTFT of the delta function expressed as the inner product

between this pseudo basis function and delta function, is again 1.

However, some things are not okay at all.

The DFT of the constant 1 is very well defined, and

it's equal to N times the delta function in frequency.

But the DTFT of 1 is by definition,

the sum from n goes from minus infinity to plus infinity of e to the minus j omega n.

Now to see that there is a problem with this sum, just put omega equal to 0,

Aad you see that the sum diverges.

The problem is that there are too many interesting sequences that are not

square summable.

And of course, the constant 1 is 1 of them.

So in order to be able to keep using the change of bases paradigm, even for

sequences that are not square summable.

We have to introduce a little mathematical trick called Dirac delta function.

This little animal here which is usually indicated by the symbol delta.

But now delta of a real variable t not the delta sequence,

is defined by the sifting property that looks like this.

If we take a delta function, we center it in s where s is a variable in r.

And then we multiply this delta functional by any function of a real variable t.

And then we take the integral from minus infinity to plus infinity,

then what we get is the value of f in the point s.

Graphically, we usually represent delta function as an upwards point and

arrow, we center this in s.

We multiply this by any function of real variable t.

And then we integrate this from minus infinity to plus infinity and

we get the value of the function in s.

In order to develop some intuitions for the properties of the Dirac delta

function, let's consider a family of so-called localizing functions rk(t).

Where k is an integer index and t is a real valued variable.

The properties of this family of functions are two.

The support of each function is inversely proportional to the index k.

But regardless of an index k, the area of each function,

the integral from minus infinity plus infinity of each function is constant.

As an example take the rect function, rect function is a classic

indicator function that is equal to 1 from minus 1/2 to 1/2 and 0 everywhere else.

So this function has the support of 1 and an area of 1.

We can use this function to build a family of localizing functions like so.

We multiply the rect by a factor k, and

we shrink the support of the rect by factor k.

So rk(t) in this case will have support that goes from

-1/2k to 1/2k so the support is 1/k and the area is 1.

If we plot some functions in this family this is what we get for

k=1, the value here is 1.

For k=5 the support has shrunk to 1 over 5 and

the area is still 1, because the value here is 5.

We got 15 it will look like this, and

to 40 will go like this and we go on to infinity.

Now consider the integral between minus infinity and plus infinity

of the product between rk(t) and any function f of a real variable t.

Rk(t) is known 0, only between minus 1 over 2k and 1 over 2k.

So these are the new integration limits of this product.

The value of rk(t) over the integration interval is k.

So we can bring this outside of the interval and

inside we have simply, the integral of the function over this integral.

Now we invoke the mean value theorem.

You can go back to your calculus textbook to revise its proof.

And the mean value theorem says that the value of this integral here

will be equal to f of gamma for some point gamma within the integration interval.

Now we don't know where gamma is inside this interval, but

we do know that it exists.

Now as k goes to infinity,

the support of the indicator function becomes smaller and smaller.

And so gamma which is somewhere in the integration interval will be

sandwiched between interval limits that grow closer and closer.

And in the limit, f of gamma will be f(0), because the width

of the interval has shrunk down to an infinitesimal width.

So the delta functional is really a shorthand for this limiting operation.

Instead of writing the limit of the integral for

a family of localizing function we just use the delta notation.

And what's interesting is that the shape of the base function that we used to build

the family of localizing function is not really critical.

We can use pretty much any shape and as long as the two properties of shrink and

support and constant area are satisfied,

the limit will converge to the point wise value of the function.

So now a last technicality before we understand why we are doing all this.

We will be using the Dirac delta functional in the frequency domain.

Now we know that all DTFD spectre are 2pi periodic.

So if we want to use this tool in the frequency domain, we have to periodize it.

The periodic version of the Dirac delta functional is called a pulse train.

And it is built by placing copies of the delta function every 2pi and

by scaling the whole signal by 2pi.

So if we want to represent that in the frequency domain with usual

upward arrow notation, we see that there will be a pulse every 2pi.

Now we can let the show begin.

Let's consider the inverse DTFT of the pulse train.

Well we apply the definition and we have 1 over 2pi times the integral between

minus pi and pi of the pulse train times e to the j omega n in the omega.

So the first thing to remark is that since the paradise delta is scaled by a factor

of 2pi, this cancels the normalization factor in front of the integral.

Then we are integrating only between minus pi and pi.

In this interval we only have one pulse, so

we can remove the implicit periodization.

And then because of the sifting property of the delta functional.

This integral will be just the value of this function of the real

value variable omega in 0, because this delta is centered in 0.

And so the value of e to the j omega n for omega equal to 0 is equal to 1.

This formula is similar to the fact that the inverse DFT of

n delta of k is actually equal to 1.

So by using the delta functional,

we have established another formal parallel between the DFT and DTFT.

So if the inverse DTFT of the delta functional is 1,

then it means that they direct DTFT, the forward for

Fourier transform of the constant 1 is formally equal to the pulse train.

Does this make sense?

Well we could try to compute numerically the partial sums that are involved

in the computation of the DTFT of the constant 1.

So define Sk of omega as the sum that goes from minus k to

k of e to the minus j omega n.

Sk goes to infinity, sk of omega should converge to the DTFT of the constant 1.

So if we plot these partial sums in magnitude for increasing values of k,

we have something like this.

For k = 5, we have this shape.

And as we increase the index, we see that this family of

partial sums looks like a family of localizing functions.

So the support gets narrow and the area stays constant.

So it really makes sense to say that in the limit

these partial sums will converge to the DIrac delta function.

With this fundamental result in our pocket we can now proceed to derive some

other interesting DTFT pairs for non square summable sequences.

With the same technique we used before,

we can show that the inverse DTFT of a shifted pulse train.

A pulse train shifted by a frequency omega 0 gives a complex

exponential of frequency omega 0 in the time domain.

So if the DTFT of 1 is the pulse train centered in 0.

The DTFT of an arbitrary complex exponential of frequency

omega 0 is the pulse train shifted by omega 0.

By using Euler's relation and the linearity of

the DTFT we can derive the DTFT of the cos of omega 0n,

this is just 1/2 times the sum of 2 pulse train.

1 centered in omega 0 and the other 1 centered in minus omega 0.

And the DTFT sin omega 0n which is minus j over 2 times a pulse

train centered in omega 0, and another pulse train centered in minus omega 0.