 Welcome to the 9th lecture on the subject of digital signal processing and its applications. We will spend a minute or two in recalling what we did in the previous lecture and then invite questions if there are any before we proceed. In the previous lecture, we had put down the properties of systems and we had also just begun our discussion of going into an alternate domain. We said that if we look at the expression which relates the input to the output with an input equal to a complex exponential in an LSI system, the relation involves a constant which depends only on frequency. That constant we saw could be viewed as an operation between the impulse response and the sequence e raised to the power j omega n. And we had concluded the previous lecture by noting that we could perhaps give it an interpretation which relates to projection, projection of the impulse response or taking a dot product of the impulse response in some way alright. So we now need to take these ideas more concrete that is the objective of the lecture today and also to take us to the first transform domain that we are going to discuss. We shall also in this lecture identify what we mean by a transform domain in general. And in particular we shall be talking about the discrete time Fourier transform. So with that then we begin the lecture today but before we continue our discussion since this is going to be a very important new aspect of this great time signal processing we could spend a couple of minutes in asking whether there are any questions which immediately need to be answered. Yes. So the question is in the previous lecture we had looked at the proof of the condition for stability being necessary. That is you know we had seen that we if a system if an exercise system needed to be stable then it is impulse response needed to be absolutely summable. We have proved sufficiency with great ease but to prove necessity we needed to use what we called a troublesome input. So the question is that choice of troublesome input and the reason why we proceeded with that input was not very clear. So let us quickly identify the main steps or the main reasoning that went into that so-called troublesome input. You see this is essentially a review and clarification of points from previous lectures. So the first point is necessary condition for stability. Now you see what we did was to take an LSI system with impulse response h n. We gave an arbitrary input x n and noted that the output y n in general is summation k going from minus to plus infinity x k h n minus k or h k x n minus k whatever you please. Now in particular we said consider n equal to 0. We said for that n y of 0 is summation k going from minus to plus infinity x k h minus k. Now our reasoning was that we can choose a bounded a troublesome bounded input x n and we chose that troublesome bounded input as x k for x n is h conjugate minus n divided by mod h conjugate minus n. Well I mean conjugate or it does not matter here whenever h minus n is not 0 and 0 otherwise. Now one point which perhaps may not have been so clear in the previous lecture is that this input is actually bounded. Maybe that point was not emphasized enough. In fact if you happen to look at this x n the modulus of x n for any n here.