 So, WAM welcomed the 20th lecture on the subject of digital signal processing and its applications. We have been discussing the synthesis of discrete time filters and we should recapitulate for a couple of minutes what we have done so far on that theme. We have been looking at the ideal filter first, the ideal discrete time filters and we realized that they are unrealizable they have they are disqualified so to speak on many counts on three counts to be more specific. We then moved on to putting down a realistic set of specifications and in the realistic set of specifications the passband and the stopband have a tolerance and there is a transition band. These are the two things that need to be taken care of now given that you have allowed for a non-zero tolerance in the passband and the stopband or passbands and stopbands as the case may be and that you have allowed a transition band. The resultant specifications are always realizable we noted that so with these compromises the discrete filter is realizable. Now in fact the good thing is it is realizable both as IIR and FIR that is you could realize it by using a finite impulse response system or an infinite impulse response system. You have the choice between the two. Now obviously the finite impulse response system has several advantages to its credit we shall see them but even right now we can see that the finite impulse response system is stable unconditionally even in the presence of numerical errors it would be stable that is not the case for infinite impulse response system they could become unstable. Moreover we shall see later that it is only the finite impulse response system that can give us the phase response that we desire the phase is unavoidable we noticed that last time we said the specifications are on the magnitude not on the phase. The phase is a necessary evil it normally follows as a consequence of the magnitude response that we are trying to meet right but there again the FIR filter scores over the IIR filter we will see that later. Now in all these circumstances why it all do we choose the IIR filter for design then and we noted that we noted why that was the case we said that you know in spite of all this the IIR filter can offer advantages in the amount of computation the complexity of the system or the computational or the hardware software resource demand and therefore we still consider the possibility of IIR systems and there is another reason IIR systems can be designed by taking inspiration from analog system. So there is a history of analog filter design of which we can take advantage we can take advantage of the methods available for analog filter design and use a universal process to convert from analog to discrete time and that is a huge advantage in the design of IIR filters. So for all these reasons we agreed that we would begin with IIR filter design we decided to begin with IIR filter design and the basic philosophy that we would follow in IIR filter design is to imply known methods from analog IIR filter design. In fact we also made a remark we noted last time that when the when we consider analog filters we have no choice there is no finite there is no non-trivial finite impulse response analog filter in fact that is one of the reasons why we go to discrete time so there is no finite impulse response analog filter and therefore it is only by IIR filter design that we can take advantage of the knowledge of analog filter design right. Now after you design the analog filter what do you do next the analog filter is subjected to what we call an S to Z transformation essentially we said that the analog filter is described in terms of the Laplace variable. So we looked at the system function of the analog filter which would essentially be a function of the Laplace variable S we are also given an interpretation to the Laplace variable the last time we noted that the Laplace variable is essentially representative of the derivative operation and you know we could spend just a minute on justifying that again actually the Laplace transform is an effort to express any function in terms of e raised to power of S t complex exponentials and it is quite clear the derivative of e raised to power of S t is S times e raised to power of S t so multiplication by S is like a derivative operator on e raised to power of S t and if you come if you think of any function as comprised of several e raised to power of S t's then multiplication by S individually on each of these e raised to power of S t's amounts to taking a derivative. So in that sense multiplication by S or the operator S or the variable S is representative of a derivative now we note this because this is what will inspire us to come up with an S to Z transformation in contrast the variable Z or the complex variable Z in the discrete domain relates to shift to movement when we multiply a Z transform by Z inverse we are shifting the sequence by one step forward and we will be multiplied by Z we are shifting it by one step backwards. So Z corresponds to shift is that right let us make a note of that Z in the discrete domain corresponds to shift. So what we need is an approximation of derivatives in terms of shifts in terms of in fact the linear combination of shifts the better we can make the linear the better we can approximate a derivative in terms of shifts the better we will do in going from analog to discrete time by replacing S by of course a rational function of Z. Now we are also put down certain criteria on that rational function of Z.