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 Escola Politécnica Federal de Lausana

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 3: Part 1 - Basics of Fourier Analysis

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

>> So what is the problem with tide prediction, well

the tides are a phenomenon that is linked to the movement of the sun, of the moon,

of the rotation of the earth and so on so all these things are periodic.

But are not all of the same period so

we have a phenomena that is only to first order simple periodic phenomenon.

Now tides are of course important because they can lead to some trouble and

here in Venice is a case in point.

Every now and

then the tides plus other natural phenomena lead to floods in Venice.

And this is either for the fun or the tourist or for

the annoyance of the people that try to make a living there.

So can we try to predict when there will be flooding in Venice?

And the first step, is as we mentioned, to approximate the tides.

So data set where we look at is hourly measurements taken in the Grande Canal

during the year 2011.

So that's very recent data and we can look for

example here at three days of measurements, so that's 72 hours.

You can see it's a smooth curve and

it has a periodic feel to it but it's not completely clear what a period is,

what the components would be in a Fourier transform.

So we shall take a Fourier transform of such a signal and

then see how to approximate this periodic signal.

So if we take one year of measurement and we take a Fourier transform,

that was the frequencies in one over hours we can see that there are a few peaks and

then there is a lot of noise.

The DFT magnitude is plotted on a log scale here, it's very important, so

you take the DFT, you take the magnitude of it's coefficients and

then take the leveling dB or in logarithm in base ten,

therefore you can see all the components on a very short scale.

And you can see there is a peak at 0 then a couple of peaks moving

towards 400, a couple of peaks around 700, and

a lot of other information that we don't quite understand.

So let's do now an approximation of this periodic signal.

And we consider either 1 Fourier coefficient, 3, 5, or 11.

So that's a very small number with respect to the potential

number of Fourier coefficients needed to exactly represent a signal.

And we're going to start with light gray approximation,

which is one coefficient, so the one coefficient is the constant.

So that's when you simply average out all the single values and

you get this sea level in centimeter approximation in grey,

which is a constant, not a very good approximation, just the average.

The second one is the constant plus cosine wave,

so that's three Fourier coefficients plus or

minus that frequency, the first fundamental frequency.

Then we have five Fourier series coefficients,

that's a couple sine waves and then we have a darker gray value and

we see it starts to look like the original signal.

Finally we see 11 only 11 coefficients, we are almost there so

the black curve is very close to the original red curve.

So we get the precise approximation with only a very limited number of

coefficients.

And that is good because now if we needed to just look a little bit beyond 72 hours,

we probably could have a pretty good prediction based on this model.

But we're not going to go into the prediction,

this was just a goal of showing approximation,

something we have seen previously when we studied Hilbert spaces.

We had look at polynomial approximations, for example.

Here we have Fourier coefficient based approximation of a periodic signal.

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