 A warm welcome to the 31st session of the fourth module in signals and systems. We have now also established a relationship between the Laplace transform and the Z transform. And therefore, we also see that there are several similarities and they are not unexpected from what we discussed in the previous session. There is therefore, one detail which we need to complete in this session and we shall do that. Namely, just as we had a formal inverse for the Laplace transform, we must now query whether we can have a formal inverse for the Z transform. So, let us set down to getting a formal inverse. Now, before we start the discussion, we should be clear why we seek a formal inverse. You see, we have been talking about rational Z transforms, rational Laplace transforms and we have by and large chosen to ignore the irrational case altogether. Now, in case we want to deal with irrational systems and there might be occasions when that is necessary. We must have a general expression to invert the Z transform, because if we are depending on experience all the time, we could run into trouble. So, that is the reason. You see, that is the reason why we are bringing it in at this phase. You know, we of course, brought in the inverse Laplace transform earlier. We could have done it at that point, but we are bringing in it now just for some variety. We are showing that just as you have a formal inverse for the Laplace transform, we can construct a formal inverse for Z transform and we also relate it to the connection between the Laplace transform and the Z transform. So, that is the reason why we are bringing it in at this phase. So, let us get down to it now. Now, what is the Z transform? The Z transform of a sequence X of n is capital X of Z. This is by notation with the region of convergence R X and we know the expression X of Z is summation n going from minus to plus infinity X of n Z raised to power minus n. Now, let us write down Z. Z in polar form as usual, that is what determines our understanding of the variable Z is of the form R e raised to power j omega. And therefore, we can rewrite this as summation n going from minus to plus infinity X n R to the power minus n e raised to power minus j omega n. And now, we can recognize this as a discrete time Fourier transform. So, this is nothing but the discrete time Fourier transform of X n into R raised to power minus n and we know how to invert the discrete time Fourier transform. So, let us do a quick recall. If we had for variety a sequence Y n now and we took its discrete time Fourier transform, which would be capital Y e raised to power j omega given by summation on all integer n Y n e raised to power minus j omega n, then we can obtain any particular Y n. You see, these are all orthogonal functions with respect to the variable omega. So, you know let us make a remark there just to recapitulate. So, we recall that for different integers n e raised to power j omega n are orthogonal functions over any contiguous interval of 2 pi and in particular the interval from minus pi to plus pi. So, with that we can simply multiply. So, we multiply Y e raised to power j omega n rather you know you do not have to write n. You multiply it by the corresponding vector integrate. So, you know what are you doing here? The component that is the interpretation component. Now, you know it is a set of so what we are doing here is it is essentially now how do we interpret this here? This was an inner product. An inner product of the sequence Y n with e raised to power j omega n. What are we doing here? We are multiplying the component along e raised to power j omega n, but now you are thinking of it as the dimension being indexed by omega not n. You know there is a little tricky issue here and you are multiplying by a vector for that omega and combining over all omega and we need only go from omega equal to minus pi to omega equal to plus pi and we bring in a factor of 1 by 2 pi essentially for the purpose of normalization. So, this was the inverse. Now, of course you can also think of the same expression in a slightly different way. So, you know interpretation is that for different omegas omega going from minus pi to plus pi I have different components that is a continuum of components and I multiply each component by a unit vector to make it a unit vector I need to divide by 2 pi and integrate over all omega. The other interpretation is that we can multiply by e raised to power j omega m for a specific m and because of the orthogonality all the other terms drop out. So, I shall show you that argument too. I am saying when we multiply y e raised to power j omega by e raised to power j omega m you know let us use a different index variable now and integrate not from minus to plus infinity, but from minus to plus pi and divide by 2 what are we doing of course you need a d omega here and you need a d omega here. Now, let us interchange the order. So, let us take each n separately and the integral is on omega. So, it can bring y n outside and put this in brackets. Now, we can easily evaluate this integral this integral equal to 2 pi for m equal to m and 0 else essentially this integral is an inner product that is how you can interpret it you can of course, evaluate it directly. In fact, let us do that let us evaluate and do that in green. Now, obviously for m not equal to n e raised to power j pi into m minus n is the same as e raised to power minus j pi into m because e raised to power j pi is equal to e raised to power minus j pi which is equal to minus 1. So, it does not make a difference and therefore, this is identically 0 for m not equal to n and when m is equal to n this integral becomes integral from minus pi to pi d omega which is simply 2 pi. So, that justifies this answer that we have here and therefore, all other terms drop out from the summation and division by 2 pi then takes care of this factor it makes it 1 and therefore, what is left finally is that this is equal to y of m. So, that is another way to understand this inversion of the discrete time Fourier. Now, we will employ this in so we said we took the z transform which is the discrete time Fourier transform and it can be inverted. How do we write down the inverse x n into r raised to the minus n is essentially integral from minus pi to pi divided by 2 pi x z e raised to power j omega n d omega. Now, remember you should treat z now as a function of omega. So, z is r e raised to power j omega and now you should multiply. So, you know this integral is on omega but the variable is z here that is fun. So, how can I bring the same variable into play? So, I would like a d z here and what is z indeed? z is r e raised to the power j omega. So, what would d z be? d z remember it is omega that is changing. So, d z is r j e raised to the power j omega d omega and therefore, we can write down d omega. d omega is minus j times r inverse times e raised to the power minus j omega d z and we can substitute this here and therefore, we would get x n times r raised to the power minus n is 1 by 2 pi. Now, you know you have an e raised to the power j omega n here. Let us bring this r to the power minus n also on the other side and then we will be able to take care of that. So, for the moment we will think of e raised to the power j omega n as being combined with an r to the power minus n and r to the power of n. So, we multiply and divide r to the power n r to the power minus n times e raised to the power j omega n and we know what d omega is. d omega is minus j times r inverse e raised to the power minus j omega and then d z and now it is all very clear. We can combine this and this and this becomes z raised to the n and we have an r to the power minus n on this side and r to the power minus n on this side. The r to the power minus n cancels from both sides and of course, this is nothing but z inverse and we will keep minus j as it is minus j can be written as 1 by j if you like. So, altogether what we get is 1 by 2 pi j. Now, when you say omega goes from minus pi to pi, you are talking about integrating x z over a circle. You see, you are essentially saying, take a circle with radius r centered at the origin and we are making a counter clockwise integration from minus pi to pi. You see, you can visualize it. You have this circle of radius r and you are essentially traversing the circle like this. You can think of minus pi here and plus pi. They are really the same point, but you are traversing the circle like this. So, it is a counter clockwise integration along the circle. In fact, now we can invoke our knowledge of complex analysis. If you want to integrate on the circle, you can also integrate over any other contour which can be joined to that circle in such a way that between these contours, there is no singularity. And where does this circle lie? This circle lies in the region of convergence. So, in fact, we chose the particular closed contour in the region of convergence given by a circle of radius r, but we could choose any closed contour in the region of convergence. That is the more general choice of contour, clear? Because after all, what we are saying is integrate on a circle and we could employ the knowledge of complex analysis which tells you that as long as you are integrating over a closed contour in the region which is inside which the function is entirely analytic, you have a zero integral. Now, of course, some of you may not have a background of complex analysis and do not worry if you do not fully understand this, but for those of you who do, you could appreciate that you could choose a more general contour. But anyway, otherwise you have one formal inverse ready for you. Choose a circle centered at the origin inside the region of convergence and you are doing well. For those of you who, of course, have a background in complex analysis, you could make it a more general contour. I think that is good enough for the moment. Anyway, so let us write down a formal inverse for the Z-transform now with this knowledge. So, this is the formal inverse and all that I would like to do before we conclude this session is to point you again to a similarity. In the Laplace transform, we integrated over a vertical axis in the formal inverse, any vertical axis in the region of convergence. And here too, we are integrating over any closed contour in the region of convergence which could in particular be a circle centered at the origin. We shall see more in the next session. Thank you.