 here we conceive of the output as a sum of many shifted impulse responses that is one way to look at it, but often that is not what we want to do, we do not want to sit and put so many sequences down and add them, we would much rather find out the output point by point and for that we need to carry out a little more work on the expression that you have just got. Therefore, we will introduce some terminology now, you see this output Yn has been obtained by an operation between the input sequence and the impulse response sequence here and we will give that operation a name, it is an operation not between two numbers, but between two sequences. So, we are saying that the output sequence some operation is obtained by some operation acting on the input sequence and the impulse response sequence, it is an operation between two sequences and therefore let me underline sequence everywhere, it is an operation between two sequences, we will give this operation a name, we call this operation convolution and there is a reason why we give it that name, you see we typically use the term convoluted in common discussions or in colloquial terms where something is complicated or where something twists and mixes, you know one sometimes refers to a convoluted argument to denote a set of you know statements or a set of utterances or a set of justification which are very complicated to understand in which intertwined together in some complicated way and perhaps lead to a conclusion, but it is not at all straight forward to see how they start from the beginning and come to a conclusion, we say it is a very convoluted argument, right. Convoluted in general tends to refer to twisting and mixing together in a highly intertwined in a highly interactive way and in a minute we will see that is exactly what we are doing in a way between the input and the impulse response here. If you think of the input as a string of samples and if you think of the impulse response as a string of samples, what we are doing is to intertwine those strings together in a very complicated and a very fine manner to obtain the output, yes there is a question. Yes, so the question is that should we be writing the operation between the input sequence and unit impulse response sequence? Well the answer is yes, you know we have sort of ignored the word unit, but we will write that to be clear here. So it is the response to a unit impulse that is correct, one should in a way one should emphasize that it is the response to a unit impulse because otherwise you know it could be a scaled impulse too. Now you know we will try and understand both of these interpretations with a specific example. We will take a very simple input sequence and a very simple unit impulse response sequence. So we will take the input sequence and here I am going to introduce some notation once again for convenience. When we have a very short input an input which is non-zero only for a few samples we tend to use this notation where we denote the location of one of the points and then we denote all the other points around it. So we indicate it for example by showing that at point 0 I have the sample let us say 6 here and then I have 1-1 and 2 obviously 1 occurs at the point n equal to 1-1 at the point n equal to 2 and 2 at the point n equal to minus 1. Needless to say I mean just for the sake of notation I am trying to explain this notation. This is the same thing is writing all of them are the same. So we can write them in any manner that we please, but what is important is to mark one point and all other points follow from there. So here the question is you see the sequence is 261-1 respectively at the points minus 1, 0, 1 and 2. So therefore you could say that this sequence takes the value 1 at the point n equal to 1 and the other marks of course follow and similarly it could be marked at minus 1 and the other points obviously follow. So similarly let us take a very simple impulse response I mean of course a unit impulse response if you please, but henceforth when we say impulse response we will mean the unit impulse response we would not want to keep repeating unit every time. The unit impulse response is very simple it is just 1 and 1 respectively at 0 and 1. Now I shall obtain the output in 2 different ways and we will explain therefore the operation of convolution why it is called convolution. One way as I said is to use this expression with the interpretation that was shifting and adding the impulse response several times. So what would this mean? This would mean I have taken the impulse response I have shifted it by each k I have scaled that shifted impulse response by the value of the input sample at the point k and I have summed up all the shifted versions of the impulse response. So let us sketch that output let me remind you of the input once again what it would mean is I must shift the impulse response backward by 2 keep it as it is shifted forward by 1 forward by 2 and respectively multiply by 2 6 1 and minus 1. So here I am so you know x k h n minus k so k equal to I will write k and I will write this sample here. So k equal to minus 1 and I have 2 2 and this occurs at 0 k equal to 0 and I have 1 1 here for convenience I will put the 0 in at k equal to 1 I have I am sorry so this should be multiplied by 6 that is 6 and 6 all right let me you know let me complete this. So 1 at k equal to 1 of course I have 1 and 1 occurring at 1 and k equal to 2 I have minus 1 minus 1 at k equal to 2 these are the 4 x k h n minus k and now of course I add them. So I will have to you know let me draw it again let me put them down since the first time we are solving such an example we have 2 2 we have 6 6 1 1 and we have minus 1 minus 1. So of course and this is the point I will I will mark it from above this is the point 0 here. So I have of course 2 8 7 0 and minus 1 this is the output sequence y of n is that clear yes there will be follow this very doubts what is going on here is very clear from this example is that you can obtain the output point by point unlike what we are doing here where we are putting sequences and adding them up by looking at what is expected at each point you see what I mean by that is we reinterpret and let us do that.