 Welcome to the 40th session in the first module of the course signals and systems. In the last two sessions, we have now convincingly proved the commutativity and associativity of convolution. Let us once again recapitulate the consequences of this. The consequences of commutativity and associativity of convolution in linear shift invariant systems. Essentially, we ask the following question. What happens if I have a number of LSI systems, as I call them, LSI systems in casket. Now, these LSI systems could all be continuous time or continuous independent variable or all discrete and we put them in casket, which means we put them one after the other with the previous output being given to the next input. So, let us consider three systems in casket. Let us call them S1, S2 and S3. And all of these are linear shift invariants. The respective impulse responses are H1, H2 and H3. I am going to write them in green here. I am first considering continuous time, but the same arguments will go for discrete. Now, suppose you look at the overall input-output relationship here. So, I take the overall input to be X of t and the overall output to be Y of t. What is the relation between them? It is very simple. Y of t is X convolved with H1 first and then convolved with H2 and then convolved with H2 and all of this evaluated Now, here, we can use commutativity and associability one by one and get some interesting results. In fact, what we will do is, first let us consider only the first two systems. So, let us focus on only the first two systems and look at the intermediate output. Let us call this Y1 t and let us write down an expression only for Y1 for the moment. Let us not bring in too much. We will take three systems later. So, Y1 t is clearly X convolved with H1 first and then convolved with H2 evaluated at t and we now invoke associativity. That tells us that X convolved with H1 evaluated at H2 is the same H1 convolved with H2 first done. Now, this is of course true point by point for everything and essentially what we are saying here is equivalence. So, we are saying that instead of keeping two separate systems, you could put Xt into one system, one LSI system with impulse response H1 convolved with H2 and you would get the same output Y1. So, essentially associatively tells us that you could replace a cascade of the first two systems by one system with impulse response given by the convolution of the two. And now the moment the impulse response becomes the convolution of the two, you can invoke commutativity. So, let us do that. Now, we invoke commutativity. So, H1 convolved with H2 is the same as H2 convolved with H1. So, the order of H1 and H2 can be interchanged. So, we notice that the order of H1 and H2 can be interchanged and there is an equivalent system comprising of the two systems in cascade. It is the same as one LSI system with an impulse response equal to the convolution of the two impulse responses and it does not matter in which order you convolve these impulse responses because convolution is commutative. So, we have two things that have happened for the first two systems. First, that there is an equivalence system that is a consequence of associativity. Secondly, it does not matter in which order we put them. Now, let us bring in the third system also. So, let us now consider all three systems together. So, we have Xt given to LSI system 1 given then to LSI system 2 and then to t with impulse response with H1t. I am not saying impulse response explicitly that is understood. And we have just called this output yt and this output y this output is y1t and this output is yt. Now, we have just seen that you could combine these two and let us call it S12. So, we have Xt given to S12 with impulse response H12t which is H1 convolved with H2 evaluated at t or vice versa. And then you have S3 still. Now, we shall show here that we can permute. See, so far of course, we have combined two of the systems into one and now again we have two systems in cascade. So, we can have the same argument. The overall, you can combine all the three systems into one system. You can do that. Let us do that now. We can combine this and this two by the same argument. So, we could have essentially S12 followed by S3 and here I have H12t as the impulse response that is the convolution of H1, H2 and H3t as the impulse response. And we have seen before that we can commute. You see, we have two systems here. We have argued that we can commute. So, we could have S3 first and then S12 or S12 first and then S3. Both are equivalent. What is more is we can also make internal changes. So, let us see what each of these gives us. You know, here we are talking about H1t first, H1 and H2 first being convolved and then being convolved with H. You see, what we are trying to show is that we can make any reordering of the systems here. So, I am just, I have given you two of the possibilities. One where S3 comes first and one where either S1 or S2 comes first. So, which comes first? Based on that we have two possibilities either S1 or S2 or S3. If S3 comes first, then there are only two possibilities. S2 follows and then finally S1 or S1 follows and finally S2. Let us look at this again. So, here two possibilities are covered. S3 followed by S2 followed by S1 and S3 followed by S1 followed by S, both of these are covered. Now, here what are the possibilities that are covered? You could have S1 followed by S2 and then followed by S3 or S2 followed by S1 followed by S3. Now, there are two more permutations that we need to worry about where essentially where S3 comes in the middle. Now, how can we get S3 coming into the middle? Here we now have to invoke associativity on the convolution of impulse responses. So, let us do that. So, now let us look at this expression. We have H1 convolved with H2 and then convolved with H. Let us invoke associativity of that. So, H1 convolved with H2 and then convolved with H3 is the same as H1 convolved with the result of convolution of H2 and H3. So, essentially this followed by this S1 to followed by S3 is equivalent to first S1 appearing and then S2 and S3 appearing. So, what possibility does this give you? This gives you the possibilities. Let us write them down. This gives you the possibilities S1, S2, S3 which of course was the original one or S1, S3, S2. So, we have already covered one more possibility where we have S3 coming into the middle and now there is one possibility left. That is S3 remaining in the middle but S1 and S2 being interchanged around it. That is also possible. How do we do that? Well, you could first bring S3 into the middle as you have done here and invoke associativity and commutativity together. So, in fact, I am going to leave this to you as an exercise. So, exercise, there are six permutations of S1, S2, S3. We have covered most but left out some. Prove that those are also valid and with this exercise I leave you to understand further the consequences of commutativity and associativity. Thank you.