 Now let us assume that the impulse response and the input both do have a discrete time Fourier transform. So let us assume this situation. You have an unicyle system, impulse response h n, input x n, output y n and we assume that x omega exists, h omega exists and y omega also exists, all of them exist. We ask what is the relation between y omega, x omega and h omega? We would again first like to obtain this relation without the mathematics and then we shall do the mathematics. Without the mathematics, what is the interpretation of x omega? x omega is nothing but the component of the sequence along ea is the power j omega n. So focus your attention on that particular component of the input along the angle of frequency omega. When you apply x n to this LSI system, you see what are you applying? You will know that when we apply x n, you are actually applying minus pi to pi, x omega e raised to pi j omega n d omega to the system. Now invoke the property of homogeneity and additivity, only additivity if you like. You can think of x omega as a constant and the integral is required to or rather additivity is required to act on the integral. So you are saying take a combination of many different e raised to the power j omega n here for different values of omega and that combination is taken by the integration. Now x omega requires homogeneity, it is a constant. So multiply each of these vectors by a constant x omega, integrate or add over all such components, make the addition to its limit and make an integration. Now you know the property of LSI systems. What will be the output b to e raised to the power j omega n? It is going to be h omega times e raised to the power j omega n. Because of homogeneity when you multiply it by x omega, the output is going to be x omega times h omega e raised to the power j omega n. Because of additivity when you integrate this over all omega, the output also gets integrated over all omega. So it is very clear that why omega is going to be integral in fact let us write on the next page. Why omega is going to be integral of course you know I have forgotten a factor of 2 pi, I did not pay too much of attention to that. So if you put that factor of 2 pi back again here, why omega is going to be 1 by 2 pi x omega times h omega e raised to the power j omega n d omega from minus pi to pi. But you see y omega is also equal to 1 by 2 pi into I am sorry y n is also equal to 1 by 2 pi integral minus pi to pi y omega e raised to the power j omega n d omega and y n is also equal to this. So y n must have this expression from our reasoning of Eigen sequences and y n must have this expression from the inverse discrete time Fourier transform. And therefore it is very clear that why omega must be equal to x omega times h omega very important result. So here you have an interpretation of the result before we even derive it mathematically. The interpretation is that we have thought of the input as a linear combination of several rotating complex numbers rotating with different angular velocities, normalized angular velocities omega. The output is going to be the linear combination of the same complex exponentials. But the complex exponentials are going to get multiplied by their Eigen value h omega for h omega and then you are going to integrate this over the omegas. And here we are using four things in the context of linear shift invariant systems additivity, homogeneity, shift invariance and finally the Eigen sequence property that is e raised to the power j omega n when it goes into the error sign system comes out as h omega times e raised to the power j omega n. Now here we have seen effectively that when you convolve x n and h n and then you take the discrete time Fourier transform it is equivalent to multiplying the discrete time Fourier transform of the individual sequence. Now this will be in the context of an LSI system and we now have a beautiful interpretation for it. But we want to prove this in general. So what we are saying is convolution in time, convolution in the natural domain it could be time corresponds to multiplication in the frequency domain. So you see here we are now introducing this whole idea of domain. You know the same signal or the same sequence can be viewed in different domains. All this while we have been viewing it in what is called the natural domain, the domain in which that signal occurs that could be time or it could be space or whatever. But we can take the sequence of the signal to a different domain where you can equivalently view it. Now you see this is a reversible process as we have seen. You can go from the natural domain to the frequency domain by using the discrete time Fourier transformation and you can come back from the discrete time Fourier transform to the natural domain or the frequency domain to the natural domain by using the inverse discrete time Fourier transform. So it is invertible. Yes? Now we need to prove this. We have proved this or we have explained or we have justified this in the context of an LSI system but let us prove it in general algebraically. So let us prove this, prove in general that if I take X1n and X2n and if they respectively have the dt of t's X1 and 2 of omega and if X1 convolved with X2 also has a dt of t let us call it Y then Y omega is equal to X1 omega times X2 omega. You want to prove this in general. You want to prove it algebraically. Now that is easy. In fact, consider the dt of t of X1 convolved with X2. Of course it is summation k running from minus to plus infinity, summation l running from minus to plus infinity. Now you need to first evaluate the convolution at the point k and then multiply this by e raised to power minus j omega k and I have assured myself that this double summation converges. That is why I said that the dt of t of the convolution exists. Now if it does let us collect all the summations into 1. So really this becomes summation k over all the integers, summation l over all the integers X1l X2k minus l e raised to power minus j omega k and we use a standard trick. You see the troublesome term is this. So let us put k minus l equal to m. So now what we are going to do is consider instead of l and k we are going to move to l and m. Remember that l and k independently run over all the integers and of course from this we also see that m is equal to I am sorry we want to eliminate k. So k is equal to l plus m. You see l when l is of course as it is. So l runs independently over all the integers. But what about m? For a fixed l when k runs over all the integers so does m and therefore a double summation where l and k independently run over all the integers is equivalent to a double summation on l and m both of them independently running over all the integers. And therefore this becomes summation on all l summation on all m X1l X2l e raised to power minus j omega l plus m and of course it is very easy to decompose this. This is e raised to power minus j omega l into e raised to power minus j omega m. Now you observe that it is only these two terms that depend on l and it is only these two terms that depend on m. I can bring the summation on m inside to act only on the terms that depend on m and I have summation l X1l e raised to power minus j omega l into summation m X2m e raised to power minus j omega m but this is very familiar this is X2 omega. So this is X2 omega it is independent of l and I can bring it outside and once I bring it outside I see that this is nothing but X1 omega here. So there we are. So this becomes X2 omega times summation l over minus to plus infinity X1l e raised to power minus j omega m and that is X2 omega into X1 omega. So we have proved this. So we have a very significant result here but when you convolve two sequences if both of them have discrete time Fourier transforms and of course then assuming that the convolution does too, the discrete time Fourier transform of the convolution is the product of the discrete time Fourier transforms of the individual sequences.