 A warm welcome to the 34th session of the second module in the course signals and systems. In the previous module, of course we had looked at several properties of linear shift invariant system then in this module we have been looking at a transform domain and at this point we have come to a very special property where we are actually bringing the two domains together and we are seeing something very interesting about the two domains that we can interchange their roles. You know in the previous session we had talked about the property of duality and duality is something that takes from the previous module and the current module and says that you have a beautiful equivalence when you go from time to frequency the same thing can be done by interchanging roles of time and frequency with a little bit of adjustment. So if I have done one movement from time to frequency I get an other possible movement free. If you ask me duality is a very beautiful property in electrical engineering. Let us first review the duality that we had learnt the last time and then make a few remarks. We had talked about duality in the Fourier domain and specifically what duality said was that if you had x t with a Fourier transform capital X of omega then capital X treated as a function of t has the Fourier transform 2 pi small x evaluated at minus omega. Let us take an example to illustrate this point. So in fact let us take the very same rectangular pulse that we had. Recall it was of height a from minus capital T to plus capital T and we had worked out its Fourier transform capital X of omega was 2 a t sin omega t by omega t. So here you have this rectangular pulse which I will call x of t and x of t this rectangular pulse had the Fourier transform capital X of omega given as above. Now let us invoke duality and in doing so let us write down capital X of t would look like 2 a capital T of course as usual sin t times capital T divided by t times capital T. Now the problem is capital T is essentially a time quantity. Let us replace it by capital omega or let us replace it by some other symbol which essentially suggests frequency. So let us replace it by W suggestive of frequency. What we are saying is effectively capital X of t is of the form 2 a W sin t W by t W and let us in fact now find its Fourier transform. So from duality its Fourier transform is 2 pi small x evaluated at minus omega. Now you see x t is even is an even function of t therefore x of minus omega in this case is equal to x of omega and therefore the Fourier transform of capital X t described by 2 a W sin t W by t W is 2 pi small x of omega itself. So essentially the same kind of a Fourier transform that means essentially a rectangular pulse but in the frequency domain but with a scaling a multiplication by 2 pi that is not equal to 2 pi. So let us draw that. So on the capital omega axis I have from minus W to plus W and the height is essentially it would have been just a if I did not have the scaling but here it is 2 pi times a that is the Fourier transform of capital X. Now you see the point to be noted here is that we have already derived a convolution multiplication association. So we said if I convolve that rectangular pulse with itself I get a triangular function and I know the Fourier transform of the triangular function immediately it is essentially the square of that sin x by x kind of function where x is of the form omega times capital t. Now you know in this case the convolution multiplication association was useful in calculating a Fourier transform. So if you were to try and find the Fourier transform of that triangular pulse directly by integration it would be a little more cumbersome it would require to you know it would essentially require evaluation of something like say t times some you know t times e raised to power minus j omega t and so on so forth. Not a very complicated integral but I am a little more difficult than just a constant integrated multiplied by e raised to power minus j omega t. So in this case the use of that association, convolution multiplication association was in finding out a Fourier transform. Now we can see how this can be used to evaluate a convolution. So you see if I convolved we may ask the question you have that sin t by t kind of pattern here. If I want to convolve capital X t with itself and capital X t is of the form some kind of sin t by t kind of thing. Good gracious if you try and evaluate the convolution by any kind of integration it is going to be a nightmare but here we have a very simple answer. What is the Fourier transform? You can easily find out the convolution of X t with X t let us do that right here. So we now use this to evaluate the convolution X t convolved with X t. Now of course you know if I take the Fourier transform X t convolved with X t has the Fourier transform 2 pi small x minus omega the whole squared and we can easily write this down we can sketch this. So it is simply 2 pi a whole squared that is all it is the same it is a rectangular pulse but with a different height. Now it is very easy to evaluate the inverse Fourier transform of this. In fact there is you know nothing much to be done all in all that you have done is to multiply the original function by 2 pi a that is all. So in fact I can straight away write down the convolution of X t with X t. Very then X t convolved with X t must be 2 pi a times X t which is 2 pi a times capital X t you know it was 2 a w sin T w by T w answer without any calculation with almost no calculation is not that beautiful if you were to try and integrate this complicated expression I am sure it would take you a long time at least the first time. Now here in one shot you have written the answer. So now duality has been used to show you that you can use this convolution multiplication association to evaluate a convolution in fact to evaluate a fairly complicated convolution. Now in fact you know you could use the same principle to evaluate slightly easier convolutions but do them efficiently. For example you can show you know you can find out the Fourier transform of say a rectangular pulse convolved with a sin X by X kind of pattern in fact let me give you an exercise in that direction show that a rectangular pulse convolved with a sin X by X or sin Y by Y whatever you want to call it sin Y by Y pattern. In fact we have given this a name and we will give this name again this is often called a sinc pattern can easily be Fourier transform find the Fourier transform so I mean you could choose your width of the rectangle and the size of the main load that is yours in fact play around with that this exercise is meant for you to understand play around with it as I said so this exercise is meant to teach you what happens the convenience of using the Fourier transform in convolution. You could choose your width of the rectangular pulse and the size of the main lobe and side lobes and so on that you could choose you could play around with them in fact and then decide on an appropriate in fact come up with some general guidelines on what happens to the Fourier transform anyway what I am trying to illustrate is that this is one instance where duality shows you two different aspects of the same principle. It shows you the aspect of how a convolution can be easily evaluated using the Fourier transform and a Fourier transform can be easily computed for a convolution where the convolution is not difficult to do you know if you convolve the rectangular pulse with itself it is not a difficult convolution but its Fourier transform is easily evaluated both have been shown by duality we shall see much more about duality in the next session duality is a very powerful property thank you.