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Instructor (Brad Osgood): Okay. Let me remind you where we finished up last time. We took an important first step in understanding the analysis and undertaking the analysis of periodic phenomena and trying to represent a general periodic phenomena by the sum of much simpler periodic phenomena under this is complex exponential? So think in terms of sines and cosines, all right?

So last time, I say we took the first step in analyzing general periodic phenomena via the sum, so several combination, a linear combination of simple building blocks, simple periodic phenomena. So let me remind you what we did because it’s very important that you realize what we did and what we didn’t do. We said suppose that you can write a periodic signal in a certain form what has to happen. So we start off by saying F of T is a given periodic function, periodic signal. Function, signal same thing. And just to be definite we took it to have period one, all right? And the question is can it be represented in terms of others and suppose it can be represented in terms of other simple signals of period one, namely the complex exponential.

So suppose we can write F of T as a sum, say something like this. Okay? K from minus N to N, cease of K, E to the two pi I, KT. There was a question, by the way, somebody sent me an e-mail about why the sum is symmetric and so why does it go from minus N to N and we talked a little bit about this last time. This is also discussed a little bit more in the notes. You can think in terms of sines and cosines, all right? And the idea is that if you have a real signal the coefficient [inaudible], but it bears repeating. The coefficient satisfying an important symmetry relationship. The complex numbers they cease of minus K is equal to cease of K bar and because of that the positive frequencies combines with the negative frequencies. The positive terms combine with the negative terms to give you a real part. So will give you essentially a sum of cosines. Okay?

Or sum of sines and cosines because the values are complex. And it’s a symmetric sum. Instead of going from one to N or zero to N, if you use complex exponentials it goes from minus N to N. The proof of the helpfulness of this representation will just become apparent as we use it, all right? As I said before, the algebraic work in the analysis is just made incomparably easier by using complex exponentials than real sines and cosine’. Just the calculations become that much easier. All right.

So, once again, suppose we can do this. Then what we found is the coefficients had to be given by a certain formula. So then the coefficients are given by C sub-K is the interval from zero to one, E to the minus two pi I, KT times F of T, DT, all right? It’s an explicit formula for the coefficients. And in principle that’s known, all right? If you know the function then you can carry out the integration in principle. And I want to remind you that this formula – there were two parts of deriving that formula. There was a sort of algebraic part, where we just try to isolate the K coefficient and that certain way, but then the analytic part invoked a little calculus where to solve for the coefficient I had to integrate. This depended on a very important relationship of the complex exponentials. This I’ll write over here – oh, right here.

The interval from – I’ll write it over here. The interval from zero to one, E to the, say, two pi I, NT, E to the two pi minus two pi I, MT, DT.

So last time, I say we took the first step in analyzing general periodic phenomena via the sum, so several combination, a linear combination of simple building blocks, simple periodic phenomena. So let me remind you what we did because it’s very important that you realize what we did and what we didn’t do. We said suppose that you can write a periodic signal in a certain form what has to happen. So we start off by saying F of T is a given periodic function, periodic signal. Function, signal same thing. And just to be definite we took it to have period one, all right? And the question is can it be represented in terms of others and suppose it can be represented in terms of other simple signals of period one, namely the complex exponential.

So suppose we can write F of T as a sum, say something like this. Okay? K from minus N to N, cease of K, E to the two pi I, KT. There was a question, by the way, somebody sent me an e-mail about why the sum is symmetric and so why does it go from minus N to N and we talked a little bit about this last time. This is also discussed a little bit more in the notes. You can think in terms of sines and cosines, all right? And the idea is that if you have a real signal the coefficient [inaudible], but it bears repeating. The coefficient satisfying an important symmetry relationship. The complex numbers they cease of minus K is equal to cease of K bar and because of that the positive frequencies combines with the negative frequencies. The positive terms combine with the negative terms to give you a real part. So will give you essentially a sum of cosines. Okay?

Or sum of sines and cosines because the values are complex. And it’s a symmetric sum. Instead of going from one to N or zero to N, if you use complex exponentials it goes from minus N to N. The proof of the helpfulness of this representation will just become apparent as we use it, all right? As I said before, the algebraic work in the analysis is just made incomparably easier by using complex exponentials than real sines and cosine’. Just the calculations become that much easier. All right.

So, once again, suppose we can do this. Then what we found is the coefficients had to be given by a certain formula. So then the coefficients are given by C sub-K is the interval from zero to one, E to the minus two pi I, KT times F of T, DT, all right? It’s an explicit formula for the coefficients. And in principle that’s known, all right? If you know the function then you can carry out the integration in principle. And I want to remind you that this formula – there were two parts of deriving that formula. There was a sort of algebraic part, where we just try to isolate the K coefficient and that certain way, but then the analytic part invoked a little calculus where to solve for the coefficient I had to integrate. This depended on a very important relationship of the complex exponentials. This I’ll write over here – oh, right here.

The interval from – I’ll write it over here. The interval from zero to one, E to the, say, two pi I, NT, E to the two pi minus two pi I, MT, DT.

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