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All right. Okay. Today we're gonna continue with our study of convolution. And let me remind you

of the star of the show and how we got there. So last time we introduced convolution in a,

what I hope you thought, was a natural way to answer a reasonable, natural question in

signal processing. So we talked about how do you combine two signals in such a way that

their Fourier transforms multiply. We are led to convolution by asking for the Fourier

transform of a combination of f and g is the product of the Fourier transform. So the

Fourier transform of f times the Fourier transform of g. And what we found was that the

combination was certainly not obvious, but actually quite least compactly written. So the

answer was given by, the convolution of two functions. I can either look at the

convolution of g with f or f with g; it doesn't matter. The convolution is the integral from

-8 to 8 of, I think just to be consistent with how I wrote it last time I believe, g of x minus

y, f of y, dy. That is to say, if this combination is defined, if this is the way you combine f

and g according to this integral, then the Fourier transform of the convolution is the

product of the Fourier transforms, which is quite a remarkable statement. I mean, all

these operations are not to be taken lightly.

Certainly, the Fourier transform is a complicated enough operation, involving an integral

from -8 to 8, a complex exponential, the rest of that stuff. This integral, although, it

doesn't involve any complex quantities, is certainly, again, nothing to be taken for

granted. And the fact that you combine these two complicated operations, and they

combine in such a simple way is pretty impressive. And not only pretty impressive, it's

pretty useful. In fact, before talking anymore about any general properties of the Fourier

transform, let me give you an example of just this sort of thing in action. So let me give

you an example of this in filtering. And I'm gonna take a particular – I'm not gonna spend

a lot of time on this, but I just want to show you that I'm not making this up. Because

we're gonna return to a lot of these ideas repeatedly throughout the course, and also

similar examples, and sometimes study them in greater depth as we go through the

course. The example that I have in mind, though, to start with was one that I borrowed

from a book by Briggs and Henson on the discrete Fourier transform. I think I put this in

a list of references that's on the website, and it's called something like The DFT: An

Owners' Manual. It's very well written, and has all sorts of good examples and good

problems in it. And the one problem they study is the problem – they use as an example,

actually of filtering, is the study of terbidity. Now, I think we actually had some people in

earth sciences in the class at one point, I think I remember that. Anybody know what

terbidity is? Anybody from earth science in here today?

Terbidity is sort of a study of, I don't know whether it's a measure of the clearness of

water or the murkiness of water, but it has to do with measuring the clarity of water. And

the idea is, that particles are suspended in water, and light scatters off of particles, and

you sort of measure how light scatters; that's a measure of the murkiness. It's a measure

of the more particles, the more scattering and the more murky it is. And terbidity varies

over time. And one of the problems is to study how it varies over time. So they presented

of the star of the show and how we got there. So last time we introduced convolution in a,

what I hope you thought, was a natural way to answer a reasonable, natural question in

signal processing. So we talked about how do you combine two signals in such a way that

their Fourier transforms multiply. We are led to convolution by asking for the Fourier

transform of a combination of f and g is the product of the Fourier transform. So the

Fourier transform of f times the Fourier transform of g. And what we found was that the

combination was certainly not obvious, but actually quite least compactly written. So the

answer was given by, the convolution of two functions. I can either look at the

convolution of g with f or f with g; it doesn't matter. The convolution is the integral from

-8 to 8 of, I think just to be consistent with how I wrote it last time I believe, g of x minus

y, f of y, dy. That is to say, if this combination is defined, if this is the way you combine f

and g according to this integral, then the Fourier transform of the convolution is the

product of the Fourier transforms, which is quite a remarkable statement. I mean, all

these operations are not to be taken lightly.

Certainly, the Fourier transform is a complicated enough operation, involving an integral

from -8 to 8, a complex exponential, the rest of that stuff. This integral, although, it

doesn't involve any complex quantities, is certainly, again, nothing to be taken for

granted. And the fact that you combine these two complicated operations, and they

combine in such a simple way is pretty impressive. And not only pretty impressive, it's

pretty useful. In fact, before talking anymore about any general properties of the Fourier

transform, let me give you an example of just this sort of thing in action. So let me give

you an example of this in filtering. And I'm gonna take a particular – I'm not gonna spend

a lot of time on this, but I just want to show you that I'm not making this up. Because

we're gonna return to a lot of these ideas repeatedly throughout the course, and also

similar examples, and sometimes study them in greater depth as we go through the

course. The example that I have in mind, though, to start with was one that I borrowed

from a book by Briggs and Henson on the discrete Fourier transform. I think I put this in

a list of references that's on the website, and it's called something like The DFT: An

Owners' Manual. It's very well written, and has all sorts of good examples and good

problems in it. And the one problem they study is the problem – they use as an example,

actually of filtering, is the study of terbidity. Now, I think we actually had some people in

earth sciences in the class at one point, I think I remember that. Anybody know what

terbidity is? Anybody from earth science in here today?

Terbidity is sort of a study of, I don't know whether it's a measure of the clearness of

water or the murkiness of water, but it has to do with measuring the clarity of water. And

the idea is, that particles are suspended in water, and light scatters off of particles, and

you sort of measure how light scatters; that's a measure of the murkiness. It's a measure

of the more particles, the more scattering and the more murky it is. And terbidity varies

over time. And one of the problems is to study how it varies over time. So they presented

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