 So, WAM welcomed the 24th lecture on the subject of digital signal processing and its applications. We have been discussing the Chebyshev-Lopas filter in the previous lecture and we have already talked about the Butterworth filter in a couple of lectures before that where we observed some interesting engineering behavior of the Butterworth filter namely if you wish to place more stringent demands then you need to invest more this is an this is an observation that we made for the Butterworth filter. Now, we have in the previous lecture been able to come up with an expression for the order of the Chebyshev filter. Let us put down that expression. Now, essentially in the Lopas filter design we need to determine epsilon which is what is called the tolerance parameter or ripple and the order which is essentially given by Cauchy inverse, we have seen this last time, have not we? Cauchy inverse d2 by d1 square root divided by Cauchy inverse omega s by omega p. Now, in the tolerance or the ripple epsilon, we saw that epsilon is less than equal to of course, you know you always want to be small if possible less than equal to square root of d1. But we also made an observation on why we should choose epsilon equal to square root of d1 last time. In fact, if we look at this expression, we now make some observations about the behavior of the order with respect to d2 d1 omega s and omega p or rather the ratio omega s and omega p. Let us review a few ideas here. So, you see you recall that this is the specification with which we are working for the Lopas filter. Of course, this is an analog filter. So, it goes all the way and you have omega p here, you have omega s, you have 1 minus delta 1 here and 1 there and delta 2 here and d2 is of course, 1 by delta 2 squared minus 1 and d1 is 1 by 1 minus delta 1 the whole squared minus 1. Naturally, d2 is expected to be greater than d1, otherwise it does not make sense. That is because delta 2 is definitely expected to be less than 1 minus delta 1 and therefore, 1 by delta 2 squared is greater than 1 by 1 minus delta 1 squared. That is to be expected, otherwise it does not make sense. You know if the passband tolerance is such that the passband amplitude goes below the stopband, it does not make sense to design the filter and therefore, it is meaningful to say d2 is greater, strictly greater than d1. That is the least that you can ask. In fact, we expect that it should be reasonably greater. The better the filter, the greater it is. Better in the sense delta 2 goes lower and lower and 1 minus delta 1 goes higher and higher. So, it is make these observations. I mean this is not necessarily only for the cherished filter but any low pass filter. In fact, you know, whenever there is a passband and stopband adjoining, this is true. So, yes, there is a question. Yeah, so the question is in this slide, should we put it from 1 plus delta 1 to 1 minus delta 1 or 1 to 1 minus delta 1? Now, you know in the Chebyshev filter, we are in a position to design it with the upper limit not going beyond 1. So, we are going to put the specification between 1 and 1 minus delta 1 rather than 1 plus delta 1 and 1 minus delta 1 because we can afford to do it here. We know that the Chebyshev magnitude can be constrained upwards by 1 and the same is true of the Butterworth magnitude. So, we do not need to go all the way to 1 plus delta 1 at all. So, the first thing is d 2 is always expected to be greater than d 1 since delta 2 is less than 1 minus delta 1. So, therefore, 1 by delta 2 squared is greater than 1 minus 1 by 1 minus delta 1 squared and better means more d 2 minus d 1, the better the filter the more the d 2 minus d 1. You know now we are stepping into the territory of discrete time processing where we need to appreciate engineering nuances. Before we talked about synthesis, we were essentially looking at analysis where we were looking at system properties and so on. So, there were fewer engineering nuances at that point in time, but now we are looking at synthesis and synthesis as we can see is essentially a process of approximation by different approaches, approximation of the ideal and that is an engineering problem, distinctly an engineering problem and therefore, we begin to see engineering nuances everywhere. Now, also omega s is clearly greater than omega p, I mean that is obvious the stop band edge must be after the pass band edge and therefore, omega s is clearly greater than omega p and therefore, both d 2 by d 1 and omega s by omega p are clearly greater than 1 and therefore, Cauchy inverse of both of these quantities are real, Cauchy inverse for each of these quantities are real. You see because if you take the inverse hyperbolic cosine of an argument which is less than 1, it would turn out to be complex. There is no choice, but for it to be complex, it is only for arguments greater than 1 that you can have a real inverse hyperbolic cosine. In fact, let us recall a few properties of the hyperbolic cosine, Cauchy squared x minus sin squared x is equal to 1 or Cauchy squared x is 1 plus sin squared x is true for all x, for all complex x in fact and in particular, if you want a real value for x here, you see sin squared x would be real for real x and therefore, if you want you know Cauchy inverse and it is very clear that Cauchy x has to be greater than 1 because sin squared x for real x is going to be a quantity which is positive, non-negative at least and therefore, Cauchy inverse must take an argument greater than 1 if you want the output to be real and that is what we verified when we had d 2 by d 1 and omega s by omega p clearly greater than 1. Is that right? Now also Cauchy just like Cauchy is a monotonically increasing, Cauchy is monotonically increasing is strictly monotonically increasing. In fact, it follows as a corollary that Cauchy inverse is also strictly monotonically increasing. This is easy to see because what strictly monotonically increasing means between 0 and infinity is that as you increase the argument from 0 to infinity, the hyperbolic cosine also increases. So it means if you are taking the inverse hyperbolic cosine of an argument that is increasing, the answer would also increase. I mean you know the independent variable increasing strictly increasing means the independent variable and the dependent variable increase simultaneously and so both the inverse and the function itself must be increasing. And therefore, inverse hyperbolic cosine of Cauchy inverse as we have called it is also a strictly increasing function and now we have some interesting insights into how the order behaves. You see the order n which is given by inverse hyperbolic Cauchy inverse d 2 by d 1 square root divided by Cauchy inverse omega s by omega p would clearly increase if the numerator increases and decrease if the denominator increases. Now, let us first look at the denominator fixing the numerator. When would the denominator increase? The denominator would increase if omega s by omega p is greater. Omega s by omega p being greater means that the top band is further away from the pass band which means you allowed a wider transition band. So wider transition band translates to an increasing denominator which means you are asking less and therefore, you expect the order to decrease. Of course, one must remember that decrease is always in steps because the order needs to be an integer and you know the order n instead of saying it is equal to this we should say it is equal to the ceiling of this here. So ceiling function is a stepped function does not suddenly change I am sorry it does suddenly change and it does not you know with a small within between two integers the ceiling does not change that is what I meant yeah. So, you know where there is the possibility of transition from one integer to the other there these factors do play a role. So, if you ask for a larger tolerance or if you make the pass band wider and sorry if you make the transition band wider your in effect asking for less and therefore, the order would tend to go down. Now on the contrary let us look at the numerator. So, the numerator would increase if d2 by d1 is increasing now d2 by d1 would increase as you can see if the filter becomes better quote unquote better. Better means that either delta 2 squared comes down or 1 minus delta 1 squared goes up or both of them happen that means you are asking for a more stringent tolerance in these pass band and or the stop band in either case even if you fix the stop band and ask for more from the pass band or if you fix the pass band and ask for more from the stop band you are increasing d2 by d1 and that means that the numerator is increasing and consequently of course you are likely to be increasing the requirement of order. So, what we saw in the Butterworth filter holds good for the Chebyshev filter too ask for more and you have to invest more right. So, let us write that down again here to asking for more that means more d2 by d1 or less omega s by omega p implies investing more more n, n is a direct measure of investment in fact now we have seen why n is directly a measure of investment n translates also into the order of the discrete time filter how many delays you will need how many multipliers all that is implied by n we have seen that in the Butterworth filter. So, it is very clear that n implies the resources required.