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All right. Today, I want to continue down the path that we started last time, of talking

about general properties of the Fourier transform. Remember, as I've said now a couple

of times, there are two tracks that we're following in developing our understanding

techniques of using the Fourier transform. One is to develop specific transforms,

transforms of specific functions that you need to have at your fingertips, so to speak, the

kind of things that come up often enough that you want to know what that answers are,

what the formulas are. And the second path is to understand how to take the Fourier

transform and different combinations of functions, different combinations of signals that,

again, come up often.

So that's what we're gonna do today, we're going down more the second path, including

an extremely important operation. So we're gonna have three big items today, each of

which are important in themselves and come up all the time. One is delays, what to do

with a Fourier transform when the signal is delayed. One, a formula for what happens to

the Fourier transform under a stretch, and finally, a very general operation, which we

have now seen a couple times in different forms, but today we're gonna see them today in

the context of the Fourier transform in its full glory, so to speak, and that is convolution.

So I want to take each one of these in turn, and I want to derive what the formulas are for

you. The book has – the notes have examples of how they're used in practice, and you'll

also have plenty of chance to practice some of these in problems. What I want to do is

show you what the general formula looks like, and how it comes about.

So let's first look at the question of delay. And there, the question is: If a signal is delayed

by – I'm not sure which term you prefer, I'm not even sure what term I prefer, delayed or

shifted – say shifted by an amount b, what happens to the Fourier transform? In other

words, if the signal is F of t, and that corresponds to a Fourier transform F of s, so I'm

gonna use the capital notation here. Then if the signal is delayed by an amount b, and for

the purposes of this discussion, that means I consider F of t minus b, not F of t plus b.

And, again, it's always a question of what you mean by delay, whether you take a plus or

minus, but I think this is probably a fairly standard way of looking at it. Then the

question is what happens over here? What happens to the Fourier transform?

Now, this is the sort of question that you have to be able to answer for yourself routinely.

These are very simple cases, but there are other cases where similar sorts of things come

up, not too many, fortunately. And, again, we have no recourse here, other than to deal

with the definition of the Fourier transform. So the Fourier transform of F of s is the

integral from -8 to 8, e to the minus 2pi, st, F of t, dt. And then the Fourier transform of

the signal F of t minus v is the integral from -8 to 8, e to the minus 2pi, st, F of t minus dt.

Now, when you get enough experience, and by now, you probably have that experience,

when you look at something like this, the thing that cries out to be done here is a change

of variable. You want this to look as much as possible like the ordinary formula for the

Fourier transform, that means F of a single variable here rather than F composed with

anything.

So what you do is, the simplest thing is to make a change of variable. So u = t - b, b is a

constant.

about general properties of the Fourier transform. Remember, as I've said now a couple

of times, there are two tracks that we're following in developing our understanding

techniques of using the Fourier transform. One is to develop specific transforms,

transforms of specific functions that you need to have at your fingertips, so to speak, the

kind of things that come up often enough that you want to know what that answers are,

what the formulas are. And the second path is to understand how to take the Fourier

transform and different combinations of functions, different combinations of signals that,

again, come up often.

So that's what we're gonna do today, we're going down more the second path, including

an extremely important operation. So we're gonna have three big items today, each of

which are important in themselves and come up all the time. One is delays, what to do

with a Fourier transform when the signal is delayed. One, a formula for what happens to

the Fourier transform under a stretch, and finally, a very general operation, which we

have now seen a couple times in different forms, but today we're gonna see them today in

the context of the Fourier transform in its full glory, so to speak, and that is convolution.

So I want to take each one of these in turn, and I want to derive what the formulas are for

you. The book has – the notes have examples of how they're used in practice, and you'll

also have plenty of chance to practice some of these in problems. What I want to do is

show you what the general formula looks like, and how it comes about.

So let's first look at the question of delay. And there, the question is: If a signal is delayed

by – I'm not sure which term you prefer, I'm not even sure what term I prefer, delayed or

shifted – say shifted by an amount b, what happens to the Fourier transform? In other

words, if the signal is F of t, and that corresponds to a Fourier transform F of s, so I'm

gonna use the capital notation here. Then if the signal is delayed by an amount b, and for

the purposes of this discussion, that means I consider F of t minus b, not F of t plus b.

And, again, it's always a question of what you mean by delay, whether you take a plus or

minus, but I think this is probably a fairly standard way of looking at it. Then the

question is what happens over here? What happens to the Fourier transform?

Now, this is the sort of question that you have to be able to answer for yourself routinely.

These are very simple cases, but there are other cases where similar sorts of things come

up, not too many, fortunately. And, again, we have no recourse here, other than to deal

with the definition of the Fourier transform. So the Fourier transform of F of s is the

integral from -8 to 8, e to the minus 2pi, st, F of t, dt. And then the Fourier transform of

the signal F of t minus v is the integral from -8 to 8, e to the minus 2pi, st, F of t minus dt.

Now, when you get enough experience, and by now, you probably have that experience,

when you look at something like this, the thing that cries out to be done here is a change

of variable. You want this to look as much as possible like the ordinary formula for the

Fourier transform, that means F of a single variable here rather than F composed with

anything.

So what you do is, the simplest thing is to make a change of variable. So u = t - b, b is a

constant.

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