 Welcome to the 18th lecture on the subject of digital signal processing and its applications. We have spent quite some time on discussing the Z-transform, the systems whose impulse responses have a Z-transform and in particular a class of those systems called rational systems. We had a good reason to discuss rational systems. Rational systems are realizable, we have also seen how to realize rational systems and in one possible way of realizing rational systems. Later on we will see more ways of realizing rational systems. For all this while we have been talking about analysis of systems. So it is time we now got down to the important theme of synthesis. What we are going to begin with today is synthesis or design and design of course of discrete time filters. It is filters that we wish to synthesize for quite some time in this course. We have already introduced the idea of a filter, a discrete time filter but let us recapitulate. A discrete time filter is a linear shift invariant system which has a frequency response. Now this is interesting, what you mean by it has a frequency response or it is having a frequency response. Essentially you are saying of course it has an impulse response and the impulse response characterizes the linear shift invariant system completely but what you are saying is that the impulse response has a discrete time Fourier transform which converges almost everywhere on the unit circle. Almost everywhere means except for some isolated and a finite such isolated set of points it converges everywhere. So we are saying a linear shift invariant system which in other words an LSI system, a linear shift invariant system whose impulse response has a Fourier transform has a discrete time Fourier transform which exists almost everywhere. Of course the moment you say discrete time Fourier transform you are talking about the angular frequency axis. So it exists almost everywhere on the angular frequency axis. In other words on the angular frequency axis between minus pi and pi or on the unit circle in the z plane there is at most a finite number of isolated points where there could be non-convergence. But otherwise everywhere else it converges, yes there is a question that is a very interesting question. In fact I am glad that question was raised. The question is when we force this are we also forcing the system to be stable? Now we shall answer this question as we go along right it is interesting the answer is interesting. We will see the answer. In fact this is one of the things that we are going to do in this lecture we are going to try and answer this question. Does it immediately imply the stability of the system? Now you must not forget that we are dealing we are talking here about both rational and irrational systems. It is not only rational systems that have a frequency response, irrational systems can also have a frequency response right. So let us make that remark so that is a very good question raised by Ashish. So note the system does not need to be rational. Rational systems also have a frequency response can irrational systems can have a frequency response. You know we will see why we need to make this remark as we go along because our purpose is design our purpose is synthesis and when we want to synthesize of course we must synthesize to meet certain specifications we do not synthesize arbitrarily.