 Welcome to 28th session on signals and systems. Let us take a brief recapitulation of what we have done in the previous lectures. We have looked at the impulse response. We have looked at the idea of a unit impulse in the context of discrete systems and we have also established that the unit impulse response is an adequate descriptor of a linear shift invariant system. We know everything about a linear shift invariant system once we know its impulse response. Now, an important question which we had kept in the back burner both for continuous and for discrete systems is the following. Suppose I am told that a system is linear and shift invariant and I am asked whether it has the other properties that we could investigate for a system, namely causality and stability. If the unit impulse response is a complete characterization of the system, it should be possible to infer causality and stability or otherwise by looking at the impulse response. How can one do this? This is the question that we set out to answer now. So, in fact, let me frame two questions to make the matter simpler to handle. The first question is given a system to be linear and shift invariant. Establish its causality or otherwise from its impulse response. Now, please note that I am asking this question both for continuous variable systems and for discrete variable systems. And the reason why I am doing that, the reason why I am taking these two issues together for continuous and discrete variable systems is that the answers are very similar and we must look at the parallels in the answer. So, let me proceed to answer this question. You see, one way to understand causality is that the output depends only on the pass and the current input. Now, recall the input-output relationship in a linear shift invariant system. So, for a continuous variable system, the output at a particular time can be expressed in terms of the input at all times by using the expression that we have here, the convolution integral at the end. So, what we are saying here is that you involve inputs, you know, the interpretation is that for a given point t, you need to involve inputs at all t minus tau, tau both positive and negative. Now, tau positive corresponds to the pass, tau negative corresponds to the future and tau equal to 0 corresponds to the current input. And therefore, what we want is that the pass and the current should be allowed and the future disallowed. That means, tau negative should be disallowed. And how can we disallow it? By making h tau 0 for tau less than 0, as simple as that. So, the condition is very simple. For causality, we need only pass and current input. Hence, h tau is equal to 0 for all tau less than 0. In other words, we are saying y of t should be simply integral 0 to infinity h tau x t minus tau d tau. So, in fact, this is clearly a necessary and sufficient condition, but we need to establish that. You see, we have clearly established sufficient for if h tau is equal to 0 for all tau less than 0, the system is causal. We have established that. However, we have not quite established necessity. In other words, we have not established that if for some reason h tau is nonzero somewhere at negative tau, then the system cannot possibly be causal. Now, how do we need to establish? How do we need to establish necessity? I am going to leave this to you as an exercise and I am going to leave you with a hint to carry out this exercise. So, let me write down the exercise and let me give you a hint and I encourage you to complete this. In fact, it will be easier for us to handle this exercise if we also look at the corresponding discrete time derivation. So, let us do that and then let us take the exercise together. So, for discrete time or discrete variables, again we have y n is summation k going from minus to plus infinity h k x n minus k. Note, we have used the commutativity of convolution here. In other words, we know the following relationship and we have used the following from commutativity. Now, once again h k for k less than 0 corresponds to the future. For k equal to 0, it corresponds to the current input and for k greater than 0, it corresponds to the past. Therefore, if causality is desired, h k is equal to 0 for all k less than 0 should be a sufficient condition. In fact, let us write down the expression that will make it easier. Let us note that y of n given that h k is 0 for all k less than 0 will simply become summation k equal to 0 to plus infinity h k x n minus k. And here we are involving the past and the current inputs, but not the future ones. So, we will prove that this is sufficient. In other words, let us formally prove that this is sufficient. So, given that this is true, let us show that if x 1 n is equal to x 2 n for all n less than equal to n 0, then y 1 n is also equal to y 2 n for all n less than equal to n 0. So, indeed consider y 1 n for n less than equal to n 0. It is clearly equal to summation k going from 0 to infinity h k x 1 n minus k. But you see, you will realize that x 1 n minus k is the same as x 2 n minus k for k positive or non-negative more appropriately and n less than equal to n 0. This is a given from the assumption. Thus, y 2 n which is the same as summation k going from 0 to infinity h k x 2 n minus k is equal to k 0 to infinity summed h k x 1 n minus k. And that is the same as y 1 n provided n less than equal to n 0. Very interesting. So, we have proved sufficiency here. Now, I have proved sufficiency formally for the discrete case because it is easier to do. I shall now leave you with both the exercises that I wish you to pursue and answer. Let me write down the exercises and give you the corresponding thing. It will help you work out the exercises. So, the first exercise is to prove the same, prove sufficiency for the continuous time case or continuous variable case. That is the easy exercise. In fact, let me give you a hint. Let x 1 t be equal to x 2 t for all t less than equal to t 0. Now, let us consider y 1 t given the system given that h t is equal to 0 for all t less than 0. Please remember this is the symbol for all and we will use this frequently in future. So, y 1 t given that h t is equal to 0 for t less than 0 becomes summation integral sorry 0 to infinity h tau x 1 t minus tau d tau. Now, x 1 t minus tau is equal to x 2 t minus tau given tau is greater than equal to 0 and t less than equal to t 0. Now, you can use this and go further to complete the proof for sufficiency in the continuous time case or continuous variable case. The proof is very similar. This is the easy exercise. Now, I am going to give you the slightly difficult exercise and the slightly difficult exercise is as follows. Exercise 2, prove that this condition is necessary. By necessary, we mean that unless this condition is true, the system cannot be causally. In other words, we need to prove if this condition is violated, the system cannot be causally. And what do we mean by saying the system cannot be causally? You must be able to show account example. You must produce two inputs x 1 and x 2 which are identical up to a point n 0, but which could possibly differ afterwards and which produce different outputs for up to n 0 or t 0. Now, I am going to give you a hint and leave you to complete the proof. So, the hint is consider two inputs. Let us take this script first, x 1n equal to x 2n for all n less than equal to say 10. Now, x 1n and x 2n differ say for n equal to 10 plus say l and let h of minus l be non-zero. Show a contradiction. What I mean by this is you have to show that you need every h l for l less than 0 to be 0 if causality needs to hold it. Now, suppose you are adverse here, you need to adopt that, you know, adversary and proponent-oponent kind of argument. So, if your opponent says here, I have h l less than 0 and h l not equal to 0. You will then produce a pair of inputs which are identical up to n 0, but which differ l steps afterwards and then look at the output at that point n 0 and establish that the output is different and therefore causality is violated. I have given you a hint. Now, complete this proof and also do the same for continuous variable systems. We shall look at the next property in the next discussion stability. Thank you.