 In fact the S to Z transform should do the following, preserve rationality that means the transform should be rational on its own you see because an analog filter system function is expected to be rational, rationality corresponds to realizability even in the analog domain. So the analog system function itself is expected to be rational. Now you want to retain that rationality when you go into the discrete domain by making an S to Z transformation. So the only way you can do it is if you replace S by a rational function of Z. So that transformation must be rational. Secondly, the transformation must preserve stability and now we want to interpret what preservation of stability means. Preservation of stability means a proper mapping from different parts of the S plane to different parts of the Z plane. In fact, let us now put down specifics there. So you see what does it mean? It means in the S plane let us mark the regions of importance of the S plane and the Z plane. So you have the Z plane here and you have the S plane there. The contours of importance are the unit circle in the Z plane or mod Z equal to 1 and the imaginary axis in the S plane S equal to j omega. We had seen that S equal to j omega corresponds to sinusoids. So the imaginary axis and the unit circle must be a mapping one on the other. That is a requirement. Of course that is actually not a requirement. By itself it is nothing to do with stability. It is a requirement that when you have a frequency response already constructed in the analog domain, it moves to a corresponding frequency response in the discrete domain. But what is important for stability is that the left half of the S plane go into the interior of the unit circle and the right half of the S plane go to the exterior of the unit circle. This is what is important for stability. If this happens, if the right half of the S plane maps into the exterior of the unit circle and if the left half of the S plane maps to the interior of the unit circle and of course the imaginary axis maps to the unit circle, then we will very easily be able to show that one cannot result in, it cannot happen that we design a stable analog filter and it becomes an unstable discrete time filter. We can show that without too much of difficulty. In fact let us prove that. So what we are saying is S equal to j omega must map into mod Z equal to 1. Real part of S greater than 0 must map to mod Z greater than 1 and real part of S less than 0 that is the left half plane must map to mod Z less than 1. This is the right half. Now remember what we are saying is we are making the transformation on S. S is the independent variable in the system function in the analog domain. Suppose it could happen that you have, now you see remember we are retaining now again we assume the filter to be causal. So let us also I think we should make a note of that. We should emphasize that. What we are saying is that we take a causal analog filter and use this S to Z whatever this transformation be. We will have to now come to that transformation but we must know what we want out of that transformation. So a causal analog filter must move into a causal discrete time filter or discrete filter. Now if the discrete filter is causal we know what to expect from its poles. We are assuming the discrete filter is causal. We want it to be rational because it should be realizable and we have a causal rational discrete time system. We know what to expect of its poles. If it is to be stable the poles must lie inside the unit circle. They have no choice. Suppose a pole happen to go outside the unit circle. Now what is a pole? A pole is the point where that system function diverges goes to infinity and that must have come from some point in S. Under the transformation some point in S must have brought you to that point in Z. Now a point on the if the point in Z happens to be outside the unit circle it could have only come from a point on the right half plane in S. It could not have come from a point in the left half plane of S. If we have ensured that there is a closed relationship of left half plane to interior of the unit circle, imaginary axis to unit circle and right half plane to the exterior of the unit circle. If we have ensured that there is this disjoint relationship then you cannot have a pole on the exterior of the unit circle because that should have come then from a point in the right half plane. It could not have come from a point in the left half plane. But you cannot have a pole in the original system in the right half plane because it is stable. Of course that filter has to be stable. So if we ensure that this is satisfied that left half goes to interior, imaginary axis goes to the unit circle and right half goes to the exterior we are guaranteed that a stable and lock filter will go into a stable discrete time filter. Is that clear to everybody? Yes. So that is of course but you see we of course now as a consequence of that you know because left half goes to interior, right half goes to exterior. Naturally the unit circle and the imaginary axis will have a one to one relationship. However, we do not want any arbitrary relationship there too. We had made a remark on that too. We said that relationship must also be monotonically increasing. The relationship, so this is the condition number we had already given two conditions so far. The third condition is that there is a monotonically increasing sinusoidal frequency relationship, increasing one to one sinusoidal frequency relationship between omega and small omega. This is the unlogged sinusoidal frequency and this is the discrete sinusoidal frequency normalized of course. Small omega is a normalized discrete frequency. Now why does it need to be one to one? We have also put down the explanation. It needs to be one to one because we want to keep the nature of the filter intact. One to one and increasing to retain the nature of the filter. You see for example in the unlogged domain if you happen to have a low pass filter with its pass band and stop band edges as we talked about the last time and tolerances and transition bands. Then under this transformation, 0 must come to 0. As we move from 0 towards infinity we must go from 0 to pi. Therefore infinity must be in one to one with pi. As we go from 0 to minus infinity we should be moving from 0 to minus pi and therefore minus infinity should be one to one with minus pi. And because of the increasing nature we are going to have the same pattern translated albeit with a non-linear distortion. It has to be non-linear. There is no option but the nature of the filter is retained. So we noted that this transformation has no choice but to be non-linear. The transformation between capital omega and small omega or the analog sinusoidal frequency and the normalized discrete angular frequency. It has no choice but to be non-linear. So all these are the conditions that this transformation must satisfy. And now we need to use our intuition and our insights to construct such a transformation. We also have a hint from where to begin. The transformation must essentially be an approximation of the derivative operator in terms of shifts. And what simpler example of a derivative approximation can we have than to subtract the current or the pass sample from the current sample. You see what is the derivative? A derivative is essentially a rate of change. And the smallest unit of time on which we can measure change in this discrete system is one sample time. So if we look at the change in one sample time we have an approximation to the derivative. So we are tempted to consider the following candidate for this transformation.