 warm welcome to the sixth lecture on the subject of digital signal processing and its applications. We take up today a further discussion on the properties of systems that we are embarked upon in the previous lecture, all right. So, let us once again put down a few thoughts that we had come up with in the previous lecture. So in the previous lecture, we had essentially looked at linearity and shift invariance and we had seen that linearity comprises of two properties, additivity and homogeneity. In fact, we had defined additivity and homogeneity by using special cases of linearity. So we had defined them as properties which were subsets of the properties of linearity. However, we had remarked and I had left it to you as an exercise to prove that the relationship is two-way. In other words, additivity and homogeneity together come to make linearity once again. That means together they are sufficient for linearity to hold. So it is both ways. We had also looked at shift invariance and I might remark that when we are very clear that the independent variable is time, then shift invariance is sometimes called time invariance. So in many texts or in many references, we encounter the term time invariance and the meaning is shift invariance with the independent variable being time. Now we also explained the context from which these properties arose. The context was that we were trying ultimately to insist or realize the property of behavior with respect to rotating complex numbers of phases and ultimately we were trying to insist on certain properties pertaining to the response to sinusoidal excitation and we explained why that was the case. That was because sinusoids are not only found in nature at least in an electrical context but they are also very smooth functions. Now what we intend to do today? First is to continue what we had just embarked upon in the previous lecture namely to look at a linear shift invariance system and to characterize it completely. We had remarked in the previous lecture that a linear shift invariant system and now we will use an abbreviation for this. So a linear shift invariant system is completely characterized by its response to a unit impulse delta n. You see we had agreed that we would use essentially three steps in this proof and we also agreed on what we mean by characterization. We had also cautioned ourselves that we should not bring in more than we have put in the definition right now. Right now in the definition we have not brought in any sine waves or we have not brought in any rotating complex numbers. We have brought in no system functions or transfer functions. We only have the knowledge that there is a relation between the input and the output and that relationship obeys the properties of linearity and shift invariance. With this we must now prove that if I know the response to one particular input namely the unit impulse I know everything about the system. In other words I know the output that would appear for any input and that is what we are now going to prove. Well the first step in the proof is to observe that one can express any input in terms of the unit impulse and that is very easy to see. So let us take an arbitrary example of an input. So at 0 let the input take the value 5, at 1 let it take the value minus 2, at 2 let it take the value 3 and for the moment let us not confuse matters we would not worry about what is present at 2, 3, 4 and so on beyond 2. At minus 1 let it take the value 4, at minus 2 let it take the value let us say minus 5 and then let it do whatever it wants to after that. And we will show that if we focus our attention on this part between minus 2 and plus 2 we can express this as a combination of appropriately shifted unit impulses shifted and scaled. So you see it is very easy to see that this sample at 0 can be obtained by putting a unit impulse here and multiplying it by 5. So this can be contributed by what is called 5 times delta n so let me write that down. In fact let me take any other specific point so let us take the point n equal to 2. The point at any 2 to 2 can be contributed by shifting the impulse to 2 in other words writing delta n minus 2 and multiplying it by 3. So this is contributed by 3 delta n minus 2 and similarly this is contributed by minus 2 delta n minus 1. Essentially the unit impulse shifted forward by 1 if the unit impulse is shifted backward by 1 you would put an impulse here and if you multiply it by 4 then you would get this sample and similarly for this and similarly for any other. So we make the remark that this xn can be written as well many other things together with minus 5 delta n plus 2 plus 4 delta n plus 1 plus 5 delta n minus 2 delta n minus 1 plus 3 delta n minus 2 and of course many other things. In fact we can now come up with a more general expression here. If we notice a typical term in this summation is really the sample value at a point k multiplied by an impulse the unit impulse shifted by that k. So for example this is the term for k equal to 0 this is the term for k equal to 1 this is the term for k equal to minus 1 and so on and so forth. So we have in general xn being a combination of delta n minus k for all integer k and the coefficient in the linear combination is x of k. Now although it is very easy to understand this particularly because we have just taken an example one must also appreciate some finer points in this discussion. The finer point is that x of k in this discussion is now a constant it is not a sequence for any particular k. Secondly delta of n minus k is a sequence not a constant although it has only one sample it is a sequence right. So what we are saying is that a given input any given input can be thought of as a combination of several sequences each of which have only one sample. These finer points must be understood because they are important in the proof. I must repeat that if there are any questions on the way you must ask them then and there do not wait until the discussion is done to ask questions. So if there are any questions at any point in time you must raise your hand right there and ask the questions. So what we are saying in effect is that x of n the whole sequence xn is a summation over all integer k xk delta n minus k. Note this is a constant this is a sequence and we assume that when we put h when we put delta n into this LSI system we get a sequence h n emerging as the output. So here there is a sequence that goes in and a sequence that comes out please note. Naturally this is the unit impulse response the response of the seek of the system to a unit impulse. Now we first invoke the property of shift invariance. So shifting variance tells us that if I were to give delta n minus k for a fixed k in place of delta n the only result on the output would be a shift of the same k and therefore if delta n produces let us denote the system by s now if delta n produces hn then delta n minus k is going to produce h of n minus k that does half our work. So here we invoke shift invariance. Now invoke homogeneity xk times delta n minus k remember xk is a constant xk times delta n minus k when apply to the same system is therefore expected to produce xk times hn minus k. Note again here h of n minus k is a sequence xk is a constant the interpretation must be very clear at every step. So here we have invoked homogeneity or scaling and finally we invoke additivity. Now additivity says summation over all k what we are saying is invoke additivity pair by pair notionally keep taking pairs and go on constructing the whole sum here right. So summation k over all integers xk delta n minus k would then be expected to produce summation xk hn minus k and in fact this is really the output yn because this is the input xn and what is significant is that here we have put no restriction on xn at all for any arbitrary xn that we gave we have an expression for the output in terms of the impulse response hn and the input xn. So in fact not only have we proved the theorem but we have also given what is called a constructive proof a constructive as against an existential proof. I must distinguish in some situations one can only give an existential proof that means one can prove that a certain solution exists but it is not easy to construct the solution or the proof itself provides not much of a queue on how one might construct the solution. But here we are fortunate to have a proof that is not just existential but constructive in that it actually gives you a process for construction of the output given the input and the impulse response. In fact what we have even now is the construction in a rather raw form in the sense that we have what seems like a fairly complicated expression. Let us spend a few minutes in interpreting what we have got. So let us first draw a couple of conclusions. So conclusion is that the theorem is proved and the proof is constructive in that you can actually explicitly write the output in terms of the input and the impulse response and when we write it like this the interpretation is very easy. What we are saying here is essentially the output is a linear combination of several shifted versions of the impulse response. The impulse response shifted by k samples weighted by the input sample at the point k and then added over all such k that is what the output is. So you can visualize this. You can visualize that if the system were to respond to unit impulse with a sequence h n and what you have is the output essentially this sequence h n shifted by all possible shifts and the particular shift by k is multiplied by the input sample at the point k and all such shifts are added together. This is one interpretation.