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

To see sampling as an orthonormal expansion, we take our sample of

orthonormal vectors phi n, taking a product with x and we look what comes out.

So write this as sinc t minus n in a product with x t.

We use symmetry again, and then this is a convolutional thing with x at location n.

Now, use the convolution theorem.

So these convolution of sinc with x is a product of rec with the fully

transform of x.

We take the inverse for a transform of this product.

That's or 1 or 2 pi integral x e to the j, omega and d omega.

And this, of course, is exactly equal to x(n).

So the inner product between phi n and x t is simply the sample of x at location n.

We have now the coefficients of the ordinal expansion.

That's x[n], the samples, which is in a product between sinc and

x(t), and this gives us the synthesis formula in an ordinal expansion,

namely x(t) simply as a sum, x[n] sinc(t-n).

Not all band limited functions are precisely band limited to minus pi to pi.

So what happens to omega N unlimited functions.

Well, we will need to rescale the sinc function by Ts,

Ts is equal to pi over capital Omega N.

And with this rescaled sinc we have alterable basis,

for, as a set of omega n band limited functions.

So, formulas are given here, both for analysis,

how to sample, in a product between sinc and the function.

And, the synthesis formula,

is a linear combination of x n times the rescaled sinc function.

We can now conclude on the sampling theorem.

We have seen the space of omega N bandlimited function is a Hilbert space,

we can set Ts is equal to pi over omega n.

So, functions phi n properly riskade form a basis for that space and

any x n, belonging to the space of unlimited functions

has coefficients in the sinc basis that are simply given by the samples.

So in conclusion for any x(t), which is an omega in band limited space,

a sufficient representation is a sequence x[n] = x(nTs).

An unlimited is always a subspace of another band limited space

set as a wider bandwidth.

This also means that if we have x(t) that belongs to omega n band limited

a sufficient representation is the sequence taken at multiples of

Ts for any Ts that is smaller or equal to Pi over capital omega N.

So when we take more samples than necessary, and

this is something that is called oversampling, that we are going to

see again when we talk about practical analog to digital conversion and sampling.

Maybe this is a good point to describe the sampling theorem in terms of hertz

frequencies, so any signal x(t) band limited to Fn in Hertz can

be sampled with no loss of information by using a sampling frequency Fs greater or

equal to two times FN.

FN is the so called Nyquist frequency, that is the maximum frequency.

Present in the signal.

Or the sampling period t(s) has to be smaller or equal to 1 over 2 times f(n).

To make a concrete example,

speech over telephone lines is typically band limited to less than 4000 Hertz.

Therefore, digital telephone users 8000 hertz sampling rate,

which is a sufficient representation of speech, and allows to reconstruct

the speech at the other end of a telephone conversation adequately.

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