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 Escola Politécnica Federal de Lausana

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 3: Part 1 - Basics of Fourier Analysis

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

In the previous lesson, we introduced the Fourier basis for CN.

We showed that it has a dual nature.

You can interpret it as a family of signals,

finite-length signals of length n.

This family's indexed by an index k.

And each element of each signal is given by e to the j 2pi over N times nk.

And both n and k will move from 0 to big N minus 1.

Another way to look at this family of signals is to write them up

in vector notation in which case you will have a family of vectors indexed by k.

And each element of the vector will be again e to the j 2pi over N times nk.

So once again, it's a complete isomorphism between the signal notation and

the vector notation.

We showed that the vectors, these N vectors are orthogonal, and

therefore the constitute bases for CN.

They're not orthonormal and

therefore we will have to normalize the inner products explicitly.

Okay, so we're in CN.

And given an arbitrary element of CN, call it x,

we want to express x in this new four-year basis.

Well, so we know that the analysis formula will give us N new coefficients for

the vector in the new basis.

And each coefficient, which we'll call big X of k,

will be simply the inner product of x, which each vector of the new basis.

If we want to go back to original representation for

X we simply have to sum a scaled version of all the basis functions.

And the scaling factor is none else then the coefficient

that we found in analysis formula.

Please note here that since the basis is not orthonormal,

we will have to normalize somewhere.

And we choose to normalize in the synthesis formula by dividing

the sum by big N.

Also this is the Fourier transform really.

Think about it, we usually assume that a single x lives in the time domain.

In vector notation, that means that we can express x as the sum of

each component of the vector x of k, times the kth canonical vector ek.

And ek is nothing but a delta in n minus k.

So each canonical vector does nothing but extract one precise time instant.

Here we're saying that we can express the same vector x

as a linear combination not of delta functions but of synthesis functions.

What will change are the coefficients of this expansion and

these coefficients are given to us by the analysis formula.

So we're moving from the time domain to the frequency domain.

To stress that this is just a linear algebra operation,

we can express the change of basis in matrix notation,

as we always can in the case of euclidean spaces.

So if we define big W of n as e to the minus j 2pi over n,

or simply W when big N is evident from the context.

Then we can this big matrix where we put

basically the congregant of each basis vector in each row.

And with this notation the analysis formula will give us a vector of

Fourier coefficients as the product of the matrix w times the signal vector x.

Vice versa the synthesis formula will allow us to retrieve the original vector

in the canonical basis simply by applying the Hermitian operator to the matrix W.

And I remind you that the Hermitian operator is a combination of transposition

and conjugation of each element of the matrix.

So we multiplied the Fourier coefficient vector by this matrix, and

normalized by 1/N.

And we get back the original vector.

A third way of looking at discrete Fourier transform

is to consider explicitly the operations involved in the transformation.

In this case, we will consider signals explicitly and this is a notation that is

particularly useful if we want to consider the algorithmic nature of the transform.

So here we will consider that the frequency domain signal

that we obtain after the transformation is obtained as the sum for small and

it goes from 0 to begin minus 1 of each element of the signal vector,

times e to the minus j 2pi over big N times nk.

So what we are performing here is the inner product in an explicit fashion.

We move from an end point signal in the time domain

to an end point signal in the frequency domain.

The synthesis formula is the dual of that with the only difference

that here we have to remember to put the normalization coefficient in front.

And so from an endpoint signal in the frequency domain,

we go back to an endpoint signal in the time domain.

Maybe this is a good point to remind you that, although we talked about

time domain, in the discrete time world, time is really adimensional.

So it won't be time in the sense of seconds or any other physical unit of

measure it will be just an index that moves over the range of integers.

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