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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來自 École Polytechnique Fédérale de Lausanne 的課程

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.

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Module 4: Part 2 Filter Design

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

Let's move on now to a new method to approximate

an ideal filter called frequency sampling.

The idea here is to say,

what if we draw the desired frequency response in the frequency domain.

Then what we do is sample this frequency response at regularly

spaced intervals and then we compute the IDFT of these values.

So the inverse, DFT, which we can always do for a finance set of frequency points,

and use the result as an M-tap impulse response, hat h of n.

Look at an example,

we start with our classic idea of low pass with omega c equal to pi over two.

And say we pick M = 11 and we take 11 samples of

this frequency characteristic between minus pi and pi.

We rearranged them according to the DFT notation and

we compute the inverse DFT with a numeric algorithm.

So if we do that we get the following impulse response.

This is an 11 point impulse response obtained by computer

inverse DFT of the samples of the ideal frequency characteristic.

So if we take the fully transform now, so

the DTFT of the finance support filter, well what do we get?

We need to convert the DFT representation to the DTFT representation

we know how to do that from the lecture about relationships between transforms.

And we know that the frequency response will be interpolation in frequency

of the frequency samples that we took in the first step.

The interpolator turns out to be once again

the transform of an N-tap rectangular window.

So we're not really escaping from the indicator function that

we used in the in-post-truncation method.

And because of that, we will have no control over mainlobe and

sidelobes of the interpolator.

Graphically, it looks like this.

Here is the original DTFT in green.

In grey we have the frequency sound poles and

the resulting DTFT will be the interpolation

of these frequency samples using this interpolator here in blue.

When we do the interpolation, that is what we obtain in the end, and

again we have a filter that will exhibit

an error around the pass man that we are not going to be able to control.

In short, these methods to approximate ideal filters are certainly

very useful when we want to derive a quick and dirty prototype, and we don't have

time to use more sophisticated filter design methods that we will see later on.

So these are good methods to know in order to quickly try something out

when we're faced with filtering problem.

But definitely they are not optimal and they leave a lot to be desired

in terms of the fine control that we can have over the maximum error