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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来自 École Polytechnique Fédérale de Lausanne 的课程

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.

从本节课中

Module 5: Sampling and Quantization

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

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.