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 4: Part 1 Introduction to Filtering

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

Now that we have the ideal low pass in place,

we can obtain the series of derived filters such as the ideal highpass filter.

You can here from the plot that we are now reserving

the frequencies from Omega C to pi, and of course symmetrically,

those from minus omega C to minus pi, and we're killing everything in between.

Now, formally, we can write the frequency response of the high pass as 1 for

omega between pi and omega c, and

symmetrically from minus pi to minus omega c, and 0 otherwise.

The 2 pie periodicity is always implicit.

We can also write that as 1 minus the complementary low pass.

So if we have a low pass with cutoff frequency omega c and

we take 1 minus that low pass, we obtained the high pass.

So that the impulse response in time domain is actually

a very simple modification of the impulse response of the complementary low pass.

It's just a delta function minus the sinc function scaled by omega c/pi.

Another variation is the bandpass filter.

The bandpass filter preserves the frequencies

in a band center around omega 0.

And 2 omega C wide, and the response is, of course,

symmetric in the negative part of the spectrum.

And we can obtain the band pass filters, starting from prototype low pass filter,

with band width again, 2 omega C.

And by modulating this low pass filter, using a cosine modulation.

So, if we take a cosine at frequency omega 0 and we multiply that by the inpulse

response of the prototype low pass, we get one copy centered in omega 0,

2 omega c wide, and we get another copy in minus omega 0.

We sum the two, we scale them back to unit amplitude, and

we have our ideal band pass filter.

Formally, we can write the frequency response as 1 for omega plus or

minus omega 0, less than omega c in magnitude, and 0 otherwise.

Again, the whole thing is 2 pi periodic.

And if we look at the time domain, we see that we have the standard impulse

response of the prototype low-pass filter with cutoff frequency omega c.

Modulated by the cosine of omega 0 n.

And the factor 2 here,

scales the filter back to unit amplitude in the frequency domain.

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