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 5: Sampling and Quantization

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

So, let's study continuous time for a while.

So, this is a new object, because we have looked only at sequences so far.

So we have a real-time variable t, we have a signal x(t),

which is typically a complex function of that real variable t.

As announced, we would like them to be of finite energy.

So the square integral of x(t) is finite, and on L2(R),

we can very naturally define an inner product between two functions,

x(t), y(t), as the integral of x*(t)y(t)dt.

And here x*, of course, is the conjugate of x(t), it's a complex conjugation.

The energy is defined as the inner product of x with itself.

We have spent a module studying discreet time filtering.

So naturally, there are also continuous time filters.

These are called analog linear time in varying filters.

So are given by a box H, the notation is x*H, is a convolution.

So this can be written as an inner product, or conveniently,

as integral from -infinity to infinity, of x(tau)h(t- tau)d tau.

Please note, very importantly, it the (t minus tau), so

that corresponds to a time reversal when you compute this integral.

We promised a Fourier transform for continuous time signals.

Remember, in this discrete time, there was this maximum angular frequency of +- pi.

In continuous time, there will be no such maximum frequency,

it can growth to infinity.

So it's a concept of frequency is the same,

we take an inner product between the function x(t) and e to the -j omega t dt.

Now this is a capital omega, to be sure we understand this is a continuous time,

Fourier transform.

So capital X(g), capital omega, is such a Fourier transform.

We can invert this Fourier transform by re-normalizing with 1 / 2pi,

taking the integral of the Fourier transform,

with respect to e to the j capital omega t.

We will not study in great details, the conditions when it exists,

under what conditions it will converge, etc.

But let us assume for the sake of practicality,

that this transform for the object of interest for us, does actually exist.

So this real world frequency's very important,

because it does interact with what we have an intuition for.

Capital omega is expressed in radians per second, but we can also talk about Hertz.

So, that would be capital omega / 2pi.

Hertz is 1 / seconds.

And the period is 1 / the frequency in hertz, or, 2pi / capital omega.

Let's look at an example.

This is a good old Gaussian bell shaped curve.

So x(t) is this exponential of t squared,

normalized by sigma squared, and it has the usual look.

It's continuous time, Fourier transformed,

happens to be also a bell shaped curve, simply rescaled appropriately.

This is rather an oddity, usually the Fourier transform of a function doesn't

look like itself, as we will see in the sequel.

We have seen the convolution theorem for discrete time convolution,

is the very same theorem holds for continuous time as well.

So if we have x(t) convolved with H leading to y(t), its Fourier transformed

is simply the product of the two Fourier transformed, capital X and capital H.

We introduce a new concept, namely the concept of bandlimited functions.

Omega N bandlimitedness means that the Fourier transform of a function,

that this band limited is strictly 0,

outside an interval from -capital omega N to capital omega N.

This is a very simple function, It's a rect function between -omega N and

omega N, centered around 0, and of height G.

So, this was the Fourier transform of this bandlimited functions,

a prototype of a bandlimited function, what is a time domain function?

Capital phi is a rect function, so phi(t) is going to be the inverse

continuous time Fourier transform written here.

We had calculated this, so let's not do it again.

Please look it up if you don't remember it, and

the time domain function is up to a scaling, one of these sinc functions.

We will normalize this sinc function, so

that area is equal to 2pi in the Fourier transformed domain.

Then the inverse continuous time, Fourier transformed, will have a maximum of 1,

at the origin.

So let's fix this, G is equal to pi over capital omega N.

Total bandwidth is 2 capital omega N.

And we define Ts, as 2 pi / the bandwidth, or

that's equivalent pi / capital omega N, and

that will be a very important sampling interval for this sinc function.

So now we have this prototypical band limited function.

On the one hand, capital phi is the rect function, and

phi(t) is the sinc function, t / capital Ts.

We see it now, in normalized fashion,

so it's a box function from -capital omega to capital omega.

The height is pi / capital omega.

Its time domain function, phi(t), has a maximum, at the origin, equal to 1.

And it has 0 crossings at multiples of Ts.

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