5.4.d Sampling strategies

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En provenance du cours de École Polytechnique Fédérale de Lausanne

Digital Signal Processing

305 notes

École Polytechnique Fédérale de Lausanne

Digital Signal Processing

305 notes

Digital Signal Processing is the branch of engineering that, in the space of just a few decades, has enabled unprecedented levels of interpersonal communication and of on-demand entertainment. By reworking the principles of electronics, telecommunication and computer science into a unifying paradigm, DSP is a the heart of the digital revolution that brought us CDs, DVDs, MP3 players, mobile phones and countless other devices. The goal, for students of this course, will be to learn the fundamentals of Digital Signal Processing from the ground up. Starting from the basic definition of a discrete-time signal, we will work our way through Fourier analysis, filter design, sampling, interpolation and quantization to build a DSP toolset complete enough to analyze a practical communication system in detail. Hands-on examples and demonstration will be routinely used to close the gap between theory and practice. To make the best of this class, it is recommended that you are proficient in basic calculus and linear algebra; several programming examples will be provided in the form of Python notebooks but you can use your favorite programming language to test the algorithms described in the course.

À partir de la leçon

Module 5: Sampling and Quantization

5.4.a Raw sampling5:47

5.4.b Sinusoidal aliasing4:03

5.4.c Aliasing for arbitrary spectra8:25

5.4.d Sampling strategies4:07

Rencontrer les enseignants

Paolo Prandoni
Lecturer
School of Computer and Communication Science
Martin Vetterli
Professor
School of Computer and Communication Sciences

This leads us to sampling strategies for different signals.

So if we are given a sampling period T s, and

the nyquist frequency omega n is equal to pi over T s,

if the signal is bandlimited to omega n, then raw sampling is fine.

It is equivalent to sinc sampling up to a scaling factor.

If the signal is not bandlimited, we have two choices.

Either we bandlimit it first,

with a lowpass filter that will cut off all frequencies beyond omega n,

in the continuous time domain before we do sampling.

Then, everything will just fine as we have seen.

Or, we raw sample the signal, and we have aliasing.

Now let's face it, aliasing usually sounds horrible so

usually it's not the first choice.

So let's investigate a little bit how we sample signals that are not

strictly bandlimited to begin with.

So how do we do sinc sampling and interpolation?

Well, the sample x hat n, is simply the inner product of the sinc function

shifted to the location n times T s with x of t, the input signal.

This can be written simply as the convolution of the sinc with x at

location n T s.

Then the extrapolation x hat of t is the sum of

the samples multiplied by the same function shifted to the location n T s.

We see this in this block diagram.

From left to right we have a continuous time signal x(t).

It goes through an ideal low pass filter with a cut off frequency capital omega n.

It is sampled every T s seconds.

So samples are interpolated by sinc interpolation and

the result is x hat of t.

It is interesting to look at this scheme in purely geometrical terms.

So we have an input signal x, it's a vector in a Hilbert space.

We have the subspace of band-limited signals BL.

So first thing we do is we have an orthonormal basis for BL, it's given by

the sinc function and it shifts by integer multiples of capital T s.

Based on this orthonormal basis, we write an orthonormal expansion,

which is the orthonormal projection of x onto the space of band limited signal.

So, it's written as a sum of the sinc inner product with x

times the sinc functions.

And so we see that the scheme we had on the previous slide is simply

the projector of x onto the subspace of band limited signal.

That projection, of course, can be written in terms of the sampling theorem.

But we cannot write x, we can only write x hat, the projection.

Now, let's look at the concrete example.

One that we have encountered before, which is a non band limited signal,

here is this Gaussian shaped signal.

And we're going to do least squares approximation

on the space of band limited signals, using sinc sampling and interpolation.

So we start with capital X C,

is a continuous time spectrum, non band limited.

We are going to band-limit it to minus omega N to omega N so

this is given by this green ideal filter.

After filtering, the spectrum is strictly band-limited,

it's 0 outside this interval of size 2 omega N.

Then we do the sampling and this reads to a periodic spectrum,

X til C, given in this pink shape here.

And it repeats at multiples of 2 omega N as we have seen before.

Then, we look at the DTFT of this sequence.

And the DTFT is 2 pi periodic.

It's given by this blue function.

And finally, we re-interpolate continuous time spectrum,

using sinc interpolation that gives us x hat c, which now is exactly

the reconstruction of the band limited version of the initial x c.

This band limited version of course is the orthogonal

projection of the initial spectrum onto the space of band limited signals.

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