 warm welcome to the 25th lecture on the subject of digital signal processing and its applications. We begin with another aspect of infinite impulse response filter design today. So far, we have designed the Butterworth and the Chebyshev low pass approximations for the analog domain. Now we would like to design other kinds of piecewise constant filters of importance namely high pass, band pass and band stop. We have of course assumed that we have brought the specifications into the analog domain already. So using the bilinear transform, we have put down the specifications of the analog filter that we wish to design, whether it is high pass or band pass or band stop. But now we wish to get down to actually designing the analog filter of given nature with appropriate nature for the pass band and stop band. By nature I mean equi-ripple or monotonic. So we want the nature and the type of the filter that is the kind of bands in the filter to be designed explicitly. So therefore, the process that we are going to follow is to again as we have done before take advantage of the designs that we have until date and then use those designs to go further to build other kinds of filters. So we have already designed Butterworth and Chebyshev low pass filters. Now we would like to use a transformation on these low pass filters to get filters of other kinds. And therefore, what we are going to do today is to discuss frequency transformations in the analog domain. Now as the name suggests, frequency transformations essentially refers to a transformation on sinusoidal frequencies and sinusoidal frequencies come from the imaginary axis in the S plane. So what we expect is a transformation that goes from the imaginary axis to the imaginary axis. And of course the transformation must obey certain requirements. Let us first put down what we mean by a frequency transformation and what we insist that it should obey. So what do we want of an analog frequency transformation? We want 3 things once again as we did when we went from analog to digital or digital to analog. First thing that we want is preservation of rationality. So essentially what you are saying is that if you denote the complex frequency variable for the low pass filter by SL, then you need a transformation on SL. Now please note, so far we have always been just writing S for the complex frequency variable of the analog filter. But now what we are saying is we want indeed to have the complex frequency variable S, the Laplace variable S. But then you want to go through an intermediate step. You want to design a low pass filter and you want to replace the complex frequency variable S in the low pass filter by another function of S. So we need to distinguish between these two variables. The complex frequency variable in the low pass filter shall therefore henceforth be referred to as SL, S sub L and S corresponds the complex frequency variable in the filter of desired nature. This is what we mean by an analog frequency transformation. Replace the complex frequency variable in the low pass filter by an appropriate function of S. And obviously if you want rationality to be preserved, so if you want the low pass filter which is of course an analog rational function to continue to be a rational function after the transformation, the only way is that F S itself must be rational. So F must be rational, there is no choice. Of course we do not need it to be bilinear or we do not need it to be first it could be any degree that does not matter but it has to be rational. Secondly we want it to preserve stability and that implies a correspondence between left half planes and right half plane. You see it is the left half plane and the right half plane in S in the complex in the low pass variable S that determines stability or otherwise. So we need left half plane to left half plane and right half plane to right half plane mapping. That would ensure that if the low pass filter is stable as it is, then this transformation would not create an unstable analog system due to the rational change. Finally we need a correspondence between the imaginary axis. After all we are talking about a sinusoidal frequency transformation. Point the imaginary axis to go to the imaginary axis otherwise there is no meaning in this transformation. After all what we want is that a certain set of specifications on the kind of filter that you want to design should translate into corresponding frequency domain specifications for the low pass filter. So the absolutely essential thing is that sinusoidal frequencies from the filter of desired nature must go into sinusoidal frequencies of the low pass filter and that implies that the imaginary axis in the S plane must go on to the imaginary axis in the SL plane. Now we might ask what is a transformation or what is the set of transformations that meets all these requirements and how do we search for such transformations. Here again our experience of analog functions should come to our aid. You see after all if we just reflect a little bit, what are those analog functions of S? Let us begin with the third requirement. What are those analog functions of S which when you substitute S equal to j omega give you imaginary quantities in SL? Now it immediately rings a bell if we think of impedances. We might always visualize the function f of S to be an impedance or an admittance and if we do that, then it immediately rings a bell if we ask that S equal to j omega should put us on to the imaginary axis. In other words that the impedance must be imaginary or the admittance must be imaginary. We know what kind of networks give us imaginary impedances. Essentially networks that comprise only of capacitors or inductors. So the first clue that we get from the third requirement it is clear that we are asking for f S to be the impedance or admittance of an LC network. A network comprising only of inductances and capacitances and of course we might ask whether if we do indeed choose such LC impedances, they satisfy the other two properties as well and we are fortunate actually that they do. Now the property of moving from the left half to the left half and the right half to the right half is actually a very important property of all impedance and admittance functions. It is called the property of positive realness. Many of us might have encountered this property when we dealt with network theory, network synthesis in particular, but we probably have never looked at it from that perspective. What we are saying in effect for property 2, we want real part of f S to have the same sign as real part of S and that is essentially a so called positive real property of network function. So even without explicitly verifying this property, if we do indeed use LC networks, we are very likely to be satisfying this implicitly by design, but we shall when we look at the frequency transformations explicitly verify this property for each of them. However, this tells us this positive real property of impedance or admittance functions assures that we are likely to succeed in that endeavor. Finally, we go to the first of the requirements rationality. Now rationality is not a problem at all. The impedance or the admittance of LC networks is of course rational in S. So we do not need to worry about that at all. So with that then we have all the properties satisfied and we can look at the simplest of the three possibilities. We wish to go from low pass to high pass, low pass to band stop, low pass to band pass and therefore the simplest of them as we expect is low pass to high pass.