 a warm welcome to the 30th session of the fourth module in signals and systems. We are now rather well equipped as far as determining properties goals. So, once I have a rational system, I know whether it is causal or stable, whether I am talking about continuous independent variable or I am talking about discrete independent variable that means I am talking about the Laplace transform or the Z transform. In the case of the continuous independent variable, I look at the imaginary axis and the right most extreme axis so to speak, real part of Z tending to infinity and real part of Z equal to 0 in the Laplace transform. These are the two critical contours. So, let us make a note of that, critical contours in the Laplace transform, real part of S equal to 0 or the imaginary axis and this is critical in determining stability. Real part of S tending to plus infinity or the right extreme vertical axis is critical in determining causality. Similarly, we can identify the critical contours of the Z transform, mod Z equal to 1 or the unit circle. This is critical in determining stability and mod Z tending to infinity, it has to be plus infinity, it cannot be minus infinity anywhere. Outermost circle so to speak, critical in determining causality. So, you see we have identified these critical contours. Now, a natural question is when I put these on the table before you in comparison, can we see an underlying relationship and this question becomes even more pertinent if we invoke some of our knowledge of module 3. Towards the later part of module 3, we had brought a union between continuous and discrete time systems. So, we said in some situation you could build a discrete time system which would do more or less what the continuous time system did. And that means there must be a way to relate the Laplace transform of the corresponding impulse response of the continuous time system or the system function of the continuous independent variable system and the corresponding system function of the discrete independent variable system. So, in a broader sense there should be some relation between these two transforms. The Laplace transform and the Z transform must share certain mutual characteristics. So, there must be a relation which tells you why it is these two contours which are critical when the other two contours are critical in the other transform and that is the relationship that we are now going to establish. Now, obviously it has to be something that brings together the continuous and discrete independent variable system and transforms work on the signal first. So, we have to see what is it that brought together continuous and discrete independent variable signals. It was the process of sampling. So, let us now consider that. So, let us take an xt with a Laplace transform assuming it exists given by capital X of S with the region of convergence Rx. Let us sample xt at a sampling interval of TS. In other words if we are doing ideal sampling, we are multiplying xt by a train of ideal impulses. This is the same as you know by using the fact that if you multiply a continuous function with an impulse it essentially picks the value of the function at the point of the impulse. So, you could now look at the Laplace transform of the sample signal and here we can invoke the property of the impulse term by term. So, what we will do is we will interchange their order and we will operate the impulse on e raised to the power minus xt. So, that gives us very interesting. So, you see we could think of a discrete sequence X square bracket n given by X at the point n TS. So, the sampling creates a discrete sequence and we could think of the Z transform of this discrete sequence. And we can now compare these two expressions. This expression and this expression. In fact, the expressions are very similar. If we accept this correspondence X of n is X at n TS and if we also make the mapping Z is essentially e raised to the power s TS that is the mapping that the expressions are identical. So, this is a central idea here. The central idea is the mapping Z is e raised to the power s TS. In fact, this mapping almost says it all it brings a correspondence between the contours because after all it is mode Z which is relevant in determining regions of convergence. And if you write down S in terms of its real part and imaginary part. So, sigma plus j omega real part imaginary part then mode Z is essentially mode e raised to the power s TS which is mode e raised to the power sigma plus j omega times TS which is equal to mode e raised to the power sigma TS because mode e raised to the power j omega TS is just one. So, what does it tell you? It tells you that the real part only the real part determines mode Z. So, this is one interesting correspondence. You see you notice that in the Laplace transform it is these vertical axis real part of S sigma which made all the difference as far as regions of convergence went. And here too you see that situation mode Z has only to do with the real part as a correspondence. So, it is not surprising that if you are talking only about the vertical axis of the S plane that implies talking only about uned or circles. Circle center at the origin mode Z in the Z plane let us make a note of that. It is the reason why vertical axis of the S plane corresponded to circle center at the origin in the Z plane. Now, let us take specific instances. Let us consider the unit circle. The unit circle comes from the unit circle comes from sigma equal to 0 and you can see very clearly that sigma equal to 0 is essentially the imaginary axis. Similarly, let us consider the other important contour of critical contour real part of S tending to plus infinity or mode Z tending to infinity. You see mode Z tending to infinity comes from of course, it has to be plus infinity comes from mode e raise the power sigma T s tending to plus infinity or T s being finite sigma tends to plus infinity and that is essentially the real part of S tending to plus infinity. So, that means this is this correspondence is also established. In fact, we can go a step further. We can establish a more general correspondence. Mode Z is constant which means a circle centered at the origin in the Z plane comes from mode e raise the power sigma T s is a constant and that means sigma must be a constant because T s is a constant and that means a vertical axis in the S plane. So, in fact, we have a much more general correspondence here. Vertical axis real part of S equal to constant in the S plane go to mode Z equal to constant in the Z plane and there is a one to one map and it is therefore not surprising that the critical contours also map. This is a beautiful correspondence that we have established between the continuous general transform variable S and the discrete general transform variable Z. We will say a little more in the next session. Thank you.