 So, welcome to the 25th session. We will now continue with linear shift invariant discrete systems and we would establish that unit impulse response characterizes the system completely. And that means we have to show that if I give you the unit impulse response for the unit for the discrete system, it tells me about the input output relationship completely. Let us begin the formal proof. So, once again, we have this linear shift invariant system and I am going to abbreviate linear shift invariant with LSI. I am going to use this abbreviation all for the future. By the way, shift invariant is replaced by time invariant if the independent variable is time. The time invariant is a special case of shift invariant. Anyway, let us get back to work. So, we have this x of n being given to the system and you have y of n. And in particular, we know y of n for x of n equal to delta n. Call it h n. Now, let us first invoke one of the properties turn by turn. So, let me invoke shift invariant first. So, if I know that delta n produces h n, shift invariant tells me delta n minus d produces h of n minus d for every possible d. Now, let us make an observation about any input that you can have, any xn. I will first illustrate with an example and then I shall write down a general statement. The example is the following. I have a sequence xn. Now, I am also going to introduce a convenient notation for sequences. You see, when I write something like this, what I mean is that x of 0 is equal to 2. The sample before that is 3. The samples after that are minus 4 and 7 in sequence. That means, x of minus 1 is equal to 3, x of 1 is equal to minus 4, and x of 2 is equal to 7, and xn is 0 else. Else meaning n greater than 2 or n less than minus 1. This is a convenient shorthand notation for finite length sequences, and we shall use it very frequently. I mean, one can adapt the notation if the sequence is located somewhere else, but this notation will be used repeatedly in our discussions. So, for example, let me write down in this notation what a unit impulse will look like. Therefore, the unit impulse delta n should look like this simple. Now, let us take this very xn and express it as a linear combination of unit impulses. Let us go back to this xn that we have here. Now, the sequence xn is very easily seen to be 3 times delta n plus 1. Now, delta n plus 1 means the unit impulse shifted backward by one step plus 2 times delta n plus minus 4 times delta n minus 1 plus 7 times delta n minus 2. This is a simple example which tells you how you could associate a combination of unit impulses or appropriately weighted impulses with any sequence. Now, here I have illustrated this for a finite length sequence, but you can do this forever. So, in fact, if you look at this particular example, what is x, what is 3 really? 3 is x of minus 1. So, you are saying x of minus 1 times delta n minus minus 1 plus what is 2? 2 is really x of 0. So, x of 0 times delta n minus 0 plus x of 1 times delta n minus 1 plus x of 2 into delta n minus 2. So, in general what formula are we coming up with? We are saying that you should essentially have any xn is essentially a sum over all integer k x of k times delta n minus k. And of course, in general k goes from minus to plus infinity. And therefore, now we have a case for linear shift invariant systems. We know what happens when delta n minus k is given to the system. So, let S be a linear shift invariant system. You know that delta n minus k is going to produce hn minus k as a consequence of shift invariance. Let us invoke the properties one by one. Let us now invoke the property of homogeneity. That tells me that if I multiply, remember for a given k, x of k is a constant for a given k. If I multiply delta n minus k by x of k and give it to the same system, I expect the output to be x of k times h of n minus k. And finally, I invoke the property of additivity. And that tells me that when I add over all k, k going from minus to plus infinity x of k delta n minus k, the same thing should happen at the output. So, I should get summation k going from minus to plus infinity x of k h of n minus k. And what is really this particular expression that we have here? This is essentially x of n. So, for any input, what I have said is if I know the impulse, unit impulse response, I know the output. In fact, not only do I know the output, when I look at this expression, I am also able to tell you what that output is. So, in fact, we have just completed a proof of this theorem. What we have shown is that we of course are able to construct the output for any input once we know the unit impulse response. What is more, I also know how to construct that output. So, this proof is constructed. This is what we call a constructive proof. I can tell you that the output is y of n given by summation k going from minus to plus infinity x of k h of n minus k. So, the proof is complete. Now, I am going to leave a couple of things for all of you to do. And I dare say some of my students will probably give you an answer. Here, we have an operation between two sequences. You know, there is a sequence x of n and the sequence h of n and operation between x and h. This operation can be given a name. In fact, we will give it a name convolution as we did for continuous variable systems. So, we will say when we convolve x with h, we get the output y and remember x, h and y are sequences, not numbers. We shall see more on how this convolution works. But I am going to ask you to answer. Is convolution commutative? Is convolution associative? We expect something from the continuous variable, but I would like you to answer this question when you actually write down the expression for convolution and find out by writing down what commutativity and associativity would mean. So, I leave this to you as an exercise. Show that this operation between x and h is both commutative and associative. We will come back to convolution in the next session. Thank you.