 Today we are having the 23rd lecture continuing with passive filters that already discussed the first order and second order low pass filters using passive networks. Let us continue with that it is important that we understand the first order and second order in order to nicely get an intuitive aspect of any higher order filter design. So, first order and second order low pass filters were the ones that were discussed in the 22nd lecture. But whatever maximally flat magnitude characteristic was the feature that was emphasized in filter design and maximally flat delay these are the 2 important characteristics that you would like to have for our filters or the amplifiers or I mean whether they are RF amplifiers or RF amplifiers or the base band amplifiers these are the characteristics we would like to have for these. In some cases we would like to have better noise attenuation property which are characteristic of Tebyshev and inverse Tebyshev filters particularly inverse Tebyshev filters wherein we can locate certain zeros outside the band to get rid of narrow band noise color noise. Elliptic filters again narrow band noise can be very nicely removed and we can permit certain amount of steep characteristic which is steeper than inverse Tebyshev filters for the same order using peaking or ripple in the pass band. So, the ripple in the pass band ripple in the stop band are elliptic maximally flat in the pass band ripple in the stop band are the inverse Tebyshev ripple in the pass band okay is the characteristic of Tebyshev but about is maximally flat in the pass band. So, basis filters are the corresponding filters where maximally flat delay characteristic is got. Now when we discuss passive filters we had considered RC, RL filters and RLC filters now we will further go into an understanding of these passive filters we saw that by having a magnitude characteristic of this type corresponding to maximally flat magnitude characteristic Butterworth filter the numerator polynomial is just a constant it has only denominator polynomial and it has only the highest order which has a coefficient as far as x squared is concerned x squared or x to power 4 x to power 6 like that these are all Butterworth filters. Then we would like to have certain amount of ripple in the pass band then it can have 1 minus k 1 x squared or in the numerator 1 plus k 1 x squared for the second order filter characteristics 1 minus k 1 x squared plus say 0.21 x to power 4 or something like that and numerator can have no polynomial in x or you have numerator polynomial which is 1 plus k dash x squared divided by this kind of denominator polynomial or you can have k 1 and k 1 dash both existing this being positive this being negative that will further enhance that. So, in a sense the kind of characteristic that the Chebyshev filter has is something where peaking can occur. So, if it is only can constrain attenuation it can be 1 minus k 1 x squared kind of thing then when you have 1 minus 0.25 x squared here you can locate a 0 for example at x equal to 2 that is the point where we have let us say a neighborhood noise transmission occurring okay. So, we want to get rid of that or this may be the 50 hertz noise in a biomedical signal right. So, we want to get rid of that fully because it is large in magnitude. So, then we can locate a 0 in the numerator polynomial okay. So, that at a certain point in the frequency domain it goes to 0 of transmission. Now that particular 0 along with a k 1 coefficient which is negative which is greater than okay magnitude than this is going to be if this 0.25 this is 0.5 it is equivalent to having no constant denominator and 0.25 here minus 0.25 here same thing. So, that means actually it is going to peak there considerably okay. So, that means it has a steeper attenuation characteristic at the pass band H that is the elliptic filter. Now coming to first order high pass filters we have this with R and L. So, earlier low pass means R and C now high pass means you just replace the C by L then it becomes high pass right or we can have C in series and R in shunt here. So, that at high frequencies this becomes a short and output is going to be equal to input okay and at low frequencies it is going to decouple the output from the input so it is 0 of transmission. So, the 0 of transmission exists at DC okay and therefore and it is going to go towards 1 as frequency goes to infinity here this particular inductor is going to make the DC appear as 0 here and at very high frequencies it is going to be an open and the output is going to be equal to input. So, both have the same characteristics okay if for example the RC time constant of this network is made the same as L by R of this network that is RC is made equal to L by that means if a capacitance C exists you select an L equal to C R square right then these two will have exactly same characteristics which is 1 by 1 plus S by omega say bandwidth. So, this will be omega bandwidth this is the 1 over omega bandwidth is RC time constant or L by R. So, this is the nature of first order high pass it is easy to design. So, V naught over VI is SCR by 1 plus SCR for the RC network and it is SL by R divided by 1 plus SL by R. So, the magnitude function squared is this okay into conjugate of this in a domain where you have put S equal to G omega so when you put S equal to G omega this becomes the magnitude which is omega C R square divided 1 which is nothing but our normalized frequency capital omega X is equal to capital omega hence forth we will use this because this is the way we have started our filter theory right. So and it is X square by 1 plus X square which is what you have as X equal to omega by omega naught or omega by let us say bandwidth. So, the normalizing frequency is the bandwidth of the low pass or high pass which right whichever it is that is the normalizing frequency omega naught in this case is 1 by RC if it is a RC network. That has been plotted here this is the magnitude and this is the delay function right. So, the delay function goes to point 5 of this and magnification because it is square root it goes to 0.7 here because it is square root. So, the delay function is just there is no delay here actually there is a pi by 2 okay minus this delay okay as the phase shift the phase shifting function it is pi by 2 minus tan inverse okay X okay which is differentiated and that is how we get this function from the tan inverse X function design of high pass band pass and band stop filters can be done starting from the corresponding low pass factor type that is an important theory that we would like to discuss today that how frequency transformation can be easily used to convert a low pass prototype into an equivalent high pass okay or given a high pass how to convert it into a low pass prototype and do the design in the low pass domain. So, since we know all the filter types can be easily designed in the low pass domain we can after converting we can do the design in the low pass domain and to convert it back to the required high pass band pass or band stop filter topology. Now one such transformation for low pass to high pass for example is 1 by 1 plus X square okay is the low pass tongue okay where it is equal to 1 at X equal to 0 and goes to 0 at X equal to infinity. Now do the transformation from low pass to high pass that transformation is X is change to 1 by X so X square changes to 1 by X square right. So, we have this now becoming equal to X square by 1 plus X square that is what is plotted here and you can clearly see that at this point okay it is 0.5. So, it finally goes to 1 so this is the transformation how does this transformation work in terms of a passive network that we would like to use ultimately. So, let us say we want to design ahh first order high pass filter with F naught equal to 40 hertz okay omega naught then becomes equal to 2 pi into 40 which is 80 pi let us design the first order low pass filter prototype okay with omega naught equal to 80 pi and then using the transformation will get the characteristic and also in terms of the network we know what to do that is the C let us say omega C equal to omega okay into ahh omega by omega naught which is 80 pi. So, we get this therefore from this for a C equal to 1 micro farad for R of 5.7 K we get this 40 hertz cut out for the low pass prototype and high pass filter okay is also going to have the same bandwidth okay that is this one will be having its upper cut off frequency this one will be having the lower cut off frequency at the same point okay if L is made equal to CR squared how did this come about because we know that our earlier experience that as far as the time constants are concerned they must be made the same that means L by R is equal to C into R. So, the choice of L is based on this CR squared so it gives you 15.7 Henry's which is too huge a value realistically to have as a circuit particularly now right and therefore even in older days about 40 years ago this kind of inductance is too bulky to be used with electronics of ahh even 40 years ago. So, this has been simulated okay using ahh active networks and RC networks okay. So, that is one way of solving this problem of ahh size as far as the inductor is concerned. So, inductor was the first component to be rejected in electronics okay at baseband particularly. So, it is too huge now let us consider building up of a second order RC low pass filter that can be done theoretically using two first order networks in cascade that means actually we have one RC low pass filter cascaded to another RC low pass filter let us say it is R1C1 and this is R2C2 then the transfer function is very easily written as DC transmission is 1 so 1 by 1 plus rest of the things are functions of S okay and this coefficient should be such that the whole thing is dimensionless that means if this is frequency this should be time constants okay. So, first time constant with just one capacitor is R1 into SC1 okay this is one integration you can say 1 by SC1 R1 is one integration with one capacitor this is the other integration okay that current goes the all way to these R1 plus R2 and then into C2. So, C2 coefficient is R1 plus R2 as far as this is concerned as one integration using individual capacitors then double integration it means it gets integrated to this and then through this right is coefficient of SC2 so you have SC1 R1 into SC2 R2 which gives you SC2 R1 C2 R2 this is the easy way of straight away writing down the transfer function of most of these networks okay. So, Omega naught the normalizing frequency here SS squared by Omega naught squared okay so Omega naught by comparison becomes 1 by square root of R1 C1 R2 C2 and Q this is S by Omega naught Q normalizing. So, please remember our lecture here this C1 R1 plus C2 into R1 plus R2 C1 R1 plus C2 into R1 plus R2 is equal to 1 by Omega naught Q so Omega naught being equal to this then from that you can get 1 over Q as square root of C1 R1 by C2 R2 square root of C2 R2 by C1 R1 if this is to be made that is Q is to be made very high let us see then we have to maximize this one way of maximizing this Q is minimizing the denominator polynomial because all these things are occurring in the denominator of Q okay this is what 1 over Q. So, this is required to be minimized okay. So, that means R1 by R2 can be selected such that this is very small these are all positive so R1 can be selected to be sort of much smaller than R2. So, that means R2 is large compared to R1 then we can ignore this then you have this as X plus 1 over X kind of thing so which is going to be minimum at X equal to 1 that means when C1 R1 is equal to C2 R2 so the test minimum value for this is equal to 2 it cannot go lower than that in a passive network of this type if it is made second order filter then you cannot have Q okay greater than half it cannot be even made equal to half it is always less than half. So, you do not use okay higher order filters okay particularly second order filter with two Rc networks or two RL networks you will can prove that the same is true with two RL networks LR LR okay. So, it is impossible to get Q greater than half or equal to half. So, what do you do you that must use RLC networks for getting any Q you desire. So, the best circuit for getting higher Q is a resonant circuit this is nothing but a series resonant circuit this has been studied by you in networks right and we have also analyzed this circuit earlier in the previous lecture. So, what happens we can therefore write the magnitude of this from our previous experience as 1 by 1 plus magnitude squared is 1 by 1 plus 1 over Q squared minus 2 into X squared plus X to power 4 where X is omega by omega naught and omega naught is the resonant frequency okay omega naught is nothing but 1 over root L into C okay this is equal to 2 pi into F naught. So, we have omega naught as this and Q equal to 1 over omega naught CR if you write down the transfer function the transfer function is something that you can readily write it is going to be 1 over SC by 1 plus okay SC into R S S squared LC. So, you will notice that Q is equal to 1 by omega naught CR if this is S squared by omega naught square. So, this can be written as okay. So, this is S squared by omega naught squared this omega naught squared. So, this is nothing but 1 okay. Now by making Q equal to 1 over root 2 the coefficient of this can be made equal to 0 okay. So, then it becomes 1 by 1 plus X to power 4 that is the square of magnitude or it is called Butterworth. Now if you do the transformation of X squared okay to 1 over X squared with the bandwidth remaining the same the Q remaining the same it simply becomes 1 by 1 plus okay 1 over Q squared minus 2 into X to power minus 2 plus X to power minus 4 which is the high pass function with the same Q equal to 1 over root 2 maximally flat that means Butterworth. So bandwidth of both are the same again. So, second order low pass filter example so 40 hertz is the bandwidth. So, you can say L is equal to 15.800 again you see that it is too huge okay for maximally flat response Q equal to 1 over root 2 so R becomes 5.62 while the C and R values are reasonable the induct value is too big to be accommodated normally and we have to convert it into high pass of the same order with Q equal to 1 over root 2 so we just put X squared as 1 over X squared so we get the high pass filter characteristics with L dash equal to L C dash equal to C omega naught equal to 1 over root L C Q is omega naught into C into R. So, we get this characteristic which is that of the high pass and the enter key is this right. So, it is 1 minus the low pass is the high pass 1 minus low pass. Now going to band pass filter band pass filter is a combination of low pass and high pass filter characteristic obviously it should be at least second order because the low pass or the high pass characteristic comes into picture first. So, we have a band pass characteristic this is the high pass characteristic comes first and then the low pass characteristics so cascading high pass with low pass results in this kind of behavior so it can be just done this way but we have already mentioned that if it is the same kind of network for example C and R like this R R and C like this does not matter both are band pass this comes first and that right. So, there is another interesting network which also this is used in what is called wind bridge this is also a band pass. So, if for example this is for all these network the transfer function is the same what is it SCR divided by 1 plus 3 SCR R 1 is equal to R 2 so we get and C 1 equal to C 2 that means Q of this is going to be equal to 1 by 3 I told you that the highest Q it can have by design is less than 0.5. So, these are all pretty useless circuit for most of the general filter characteristic that we would like to generate that means we should be able to get a Q which is anywhere from low value to very high values low pass to band pass transformation. So, the X in this case is going to be replaced by X is going to be replaced by X minus K squared by R this is going to be the same as X squared minus K squared by X as depicted here. So, X is replaced by X minus K squared by X then the entire characteristic gets shifted to where X was 0 that gets back to X equal to K here. So, this is the shift from X equal to 0 this maximally flat magnitude square characteristic or the Butterworth characteristic of first order gets shifted to X equal to K with the same bandwidth right. So, you can see that so, we have substituted here is 1 by 1 plus X squared becomes X squared minus K squared by X whole square. So, that is the transfer that is plotted here. You can see this this is 1 by 1 plus X squared and this is 1 by 1 plus X minus 25 by X okay that means K is equal to 5 K squared is 25. So, you can see that it is peaking the speak at X equal to 0 has been transferred to X equal to 5 and the bandwidth of this is unity let us say it is normalized with respect to the bandwidth and the same bandwidth is replicated here after the shift. So, that is 4.5 to 5.5 bandwidth. So, this normally has the negative side getting replicated at X equal to minus X equal to minus 5 okay X squared equal to 25 there. So, X equal to plus or minus 5 this gets replicated. So, the bandwidth remains the same as that of the low pass prototype. Design a second order band pass filter with center frequency equal to 5 kilo hertz let us say and the bandwidth of 1 kilo hertz. Design first order low pass prototype for a bandwidth of equal to bandwidth of the band pass filter which is 1 kilo hertz. So, the low pass prototype should have a bandwidth of 1 kilo hertz. So, we have selected okay in this case omega equal to 1 by RC this is the bandwidth as 2 pi into 10 to power 3 selecting 1 micro farad we have appropriately chosen R as 159 so that the bandwidth of this comes out as about 1 kilo hertz here that is 0.7 because this is square root of the magnitude okay the 0.5 becomes 0.7 or 7. So, then we can note that if you sketch this this way this bandwidth corresponds to exactly okay the 1 kilo hertz that we have designed it for right as the low pass prototypes. So, this is the simulated result okay and it is exactly located at 5 kilo hertz this is what is called the low pass to band pass transformer. Please remember this fundamental thing it is the RC that determines the bandwidth okay if this is remaining the same it is 1 over RC which is also the bandwidth okay of the so called band pass filter okay this just gets replicated this way. Now low pass to band stop transformation okay again if you start with RC right you have to let us just go back to this and see how this particular thing is designed basically we have a low pass that we start with this is C and R. So, this reactance is substance is SC so S E is replaced by okay SC plus 1 over SL what is it this is an admittance okay originally SC okay it is replaced by SC plus 1 over SL which means actually its frequency dependence is J omega C. So, this now becomes equal to J omega C okay into minus 1 okay over plus 1 over J omega L or this is J omega C into 1 minus right what happens here this becomes omega C has been taken out so it is omega square L into C okay this 1 over LC is really is what is called 1 minus okay omega naught square normalizing frequency right. So, this is the this CR is the one that determines the bandwidth okay of the low pass prototype and this is the shift this is what is called as X okay that is 1 over X square for example you can call this as 1 over X square that is K square over X square okay. So, this is the centre frequency shift that means it gets shifted to the resonant frequency of the system. So, when we just put this way this is C this inductor is okay nothing but 1 over omega naught square into C that is the value of inductor. So, we can easily compute this equivalent okay in the frequency transformation and that is how we have designed the whole thing and you can see that that is what this resonant frequency is at 5 kilo hertz 1 micro farad and 1 milli henrys roughly corresponds to very nearly equal to 5 kilo hertz. So, replace the capacitor by a parallel resonant circuit replace the inductor by a series resonant circuit okay. So, that way actually a parallel resonant circuit if it is replaced by a series resonant circuit then you have a transformation of things from band pass to band stop. So, the same thing only thing is this frequency scaling effect now is the resonant frequency is 1 over 2 pi root L dash C dash okay and the original C that determines the bandwidth that gets scaled to okay C into bandwidth square divided by center frequency this normally called okay 1 over Q square okay. So, the case scaled to okay this kind of thing and the inductor is to be chained to okay L dash by 1 by bandwidth squared into C okay. So, that L dash C dash becomes equal to 1 over center frequency squared as so the resonant frequency remains the same. So, if I now chain these capacitors and inductors in this manner then the Q remains the same okay as before okay and this becomes the band stop. It is equivalent to changing X to X square minus K square by X okay to the power minus 1 now. So, first transformation is okay X to 1 over X and then that X can be chained to X square minus K square by X okay. So, the parallel resonance becomes a series resonant circuit and L dash and C dash get scaled accordingly so that the Q remains the same. So, that is the band stop characteristic exactly produced. Let us do the same thing 5 kilo hertz and 1 kilo hertz okay 5 kilo hertz is the in place where we want the notch and 1 kilo hertz is the bandwidth of the notch. So, actually design first order low pass filter prototype for a bandwidth of and is equal to bandwidth of the band pass filter. So, this is the low pass filter prototype with 1 kilo hertz bandwidth somewhere here right 0.707. So, if you extend the same thing here this is going to be again 1 kilo hertz and this notch frequency is going to be at 5 kilo hertz. The network is again same 159 ohm resistance this one micro farad gets divided by 5 square because that is the Q 5 by 1 kilo hertz okay 0.04 micro Henry's and for it to be resonant at 5 kilo hertz the inductor value is 25 milli Henry's at micro farad. So, we can easily design this scheme starting from the low pass filter prototype and this is the notch filter next. Let us consider using inverse TBCU filter as given in the figure right this is an important network now inverse TBCU filter okay means what we already understood this I want the characteristic like this let us say it as maximally flat characteristic up to the bandwidth and at this point right where I want to get rid of a specific frequency that is going to be like this let us say okay. So, this particular point is going to be the frequency at which it is rejected totally this component of frequency is rejected. So, this can be obtained by a magnitude function which is 1 minus n1 times omega squared the omega squared is now currently being replaced by us as omega is equal to X. So, that means this is our X squared so one square root of 1 plus X squared plus X to the power 4 this is the kind of characteristic like to have right. So, you can note that at X equal to 0 is going to be just 1 okay and transmission and as X goes to infinity right this is going to result in let us say if you have something like this right this particular thing is going to result in this is square root of omega to the power 4 is going to dominate as X increases to a large value. So, this also becomes X squared and this becomes very nearly equal to okay at X equal to 0 it is 1 at X equal to tending to infinity it is going to be n1 squared that is the transmission because it is going to be dependent upon n1 squared because this omega squared gets cancelled with this omega squared this is asymptotic to n1 squared. So, this is a very simple design therefore it uses just a tapped inductor along with the capacitor that is all with the resistance. So if you write down the transfer function of this again right you can note that this inductor in series with this is the resonance circuit which means it is series resonance that means actually it becomes 0 okay at a certain frequency when omega L1 is equal to 1 by omega C. So, that means omega L1 equal to 1 by omega C these two reactances get cancelled together and it causes the thing to become a short circuit. So, the transmission is 0 that means the 0 is equal to 0 of the numerator polynomial is at 1 by L1C. So, that means actually we have this the n1 okay which is easily controlled by us okay with respect to omega is really equal to say small omega by omega naught omega naught is the bandwidth it is that of the pole okay omega naught. So, what is omega naught omega naught is the series resonance frequency of this what is it? It is equal to 1 by root of total inductance L1 plus L2 into C. So, this frequency is lower than the 0 frequency the bandwidth is lower than it. So, this let us say is roughly the point at which we have this the bandwidth okay. So, the bandwidth is located when this series resonance occurs there okay the denominator polynomial goes to the minimum value that means it has a peak it has a tendency to peak. So, it can have a peak also depending upon the queue right. So, we will select the queue such that that the it is maximally flat let us say. So, that value of queue at which is maximally flat for a given problem let us work out this problem it is interesting I want for biomedical applications let us say a 40 hertz bandwidth I have a predominant component of 50 hertz present. So, I will locate this what is the 0 at 50 hertz. So, my omega 1 is going to be omega z is going to be 2 pi into 50 hertz. So, that is equal to 1 by square root of L1 into C. So, this omega z is fixed based on my desire to get rid of this as much as possible. Therefore, I put a 0 of transmission at this frequency. Then I select the queue of this network in order to make okay. So, N1 has already been fixed because of the choice for this right. So, that is the thing N1 is equal to okay this yeah because it is 2 minus 1 by Q square 2 N1 is equal to 2 minus 1 by Q square this is the way we are going to select the 0. So, that it becomes maximally flat N1 is equal to omega P pole if you consider this as the pole frequency what is it omega P is equal to 1 by square root of L1 plus L2 into C as the pole. So, omega P by omega z okay is D N1 which is less than 1 okay because omega z is at a higher frequency than omega P. So, this is 50 and that is 40. So, F that P divided by F z okay P by F z therefore is 0.8. So, N1 is 0.8 square okay and you can therefore note that this when we square it this becomes square. So, this becomes 1 minus 2 N1 omega x square okay plus N1 square omega to the power 4 that is why that 2 N1 coefficient has to be same as minus 2 plus 1 by Q square. So, that makes the coefficient of x square and numerator and denominator same that means it maximally flat that occurs at a Q of 1.18 okay and R to make that Q equal to 1.18 is 3.34. So, I would like you to simulate this and see the characteristic of this which exactly resembles what I am showing here. So, this is the characteristic that is normalized of course of the filter that is the Chebyshev filter that I have designed now you can see if 1 corresponds to the bandwidth. So, if this is equal to 40 hertz this is equal to 50 hertz. So, it is 50 by 40 which is 1.25 scaled to the bandwidth. So, this is the filter that we have designed okay and it is going to be this kind of characteristic that we should have in order to make it work as good filter for let us say biomedical applications. It can be improved by cascading one more first order filter. So, that this gets attenuated and the characteristic looks now like this right. So, most of the white noise gets removed by one more first order that is cascaded to it and here it does not touch. So, this is the characteristic that finally you can get by having a second order filter cascaded to a first order filter. You can improve upon it. So, this is the kind of design that normally gets carried out okay buyers very easily in fact we do not need any help of higher order equations and higher order mathematics for understanding filter design. Let us now just for conclusion sake start with some random specifications right. Let us say I have a filter design that I have to do for getting rid of two major neighborhood frequency components okay in let us say baseband signal process that means I want a low pass filter okay which outside the band has two major narrow band noise components okay that must be got rid of. So, if I have to get rid of two major narrow band noise components and white noise of course. So, what I have to have minimum to start with and what is the topology how can I intuitively get this topology okay just by knowing the requirement right. So, let us start with this problem it is bandwidth is known the signal bandwidth is known let us see. So, I want to approximate it to a box like characteristic with this is the bandwidth okay this is the low pass filter I want to design. So, I have the bandwidth as let us say I will say omega naught earlier I have been normalizing always the whole thing with respect to the bandwidth of the low pass filter prototype. Now let us say this is going to be omega one noise component this is the other noise component so these are major then apart one that I have to attenuate the noise considerably uniformly. What is the minimum order that I should select and how what should be the nature of this obviously I want to locate my two zeros here that means first thing that you do is the numerator polynomial is okay containing one plus s square by omega z one square. So, write down the numerator polynomial first this may be some constant H naught one plus s square by omega z two square then the denominator polynomial right let us see denominator polynomial greater than the numerator polynomial that means this is already fourth order. So, the minimum thing that I should have is let us say I am cascading to second order okay omega let us say p one then one plus s by omega p two q two plus s square by omega p two square. So, we have seen in the earlier this thing the numerator order if it is young or young the denominator should be at least one order higher okay so that means I have to have this as at least okay I can do better by having higher orders but this is the minimum that I require. So, straight away write down the transfer function of the required polynomial that you have to that means it can be just that this one thing can be realized by using one second order network another can be realized by another second order network and third is a first order network. So, I know that I can cascade this together and definitely get what I want after that how do I optimize the filter design obviously the location of all these poles and zeros are already fixed the poles will make it such that I should get for example maximally flat here and obviously there will be ripple here okay. So, what is the kind of ripple you get if it is maximally flat it is going to be just this and then it is going to be and because of the this order dominating ultimately it will all go to zero ultimately. So, this is the characteristic I am going to get and it can be maximally flat here that is I want to have better attenuation characteristic what should I do with it I will permit okay this cannot peak because this is going to be just going on decreasing this is a going to have 1 by 1 plus x squared under the root this one will have obviously a tendency that can cause a peak this one will also have a tendency to cause a peak so that can be two peaks okay apart from the peak at zero caused by this first order that can be two more peaks that means basically I can have it going like this okay and going like this actually it may therefore increase above this right so it may be the kind of characteristic you may be able to get. So ultimately I know that it will be some kind of if it is equiple like this and rippling at this point this is the best I can do with selection of cues and selection of these different pole frequencies omega p1, omega p2 and omega p3 so I know what I should get so please know that all this peaking frequency will be close to the bandwidth so that means omega p1 is going to be let us say the first peaking frequency and omega p2 is the second peaking frequency and the q of this q1, q2 should be greater than q1 because every such thing can be concluded very intuitively this way that this is always attenuated like this and then that attenuation of this first order can be compensated for by next peaking circuit okay. So let us therefore see how it is going to be looking like so first order is going to go like this because it is attenuating continuously peaking only at omega equal to 0 right we would like to have the next one which is going to peak like this so this decrease is compensated for by this increase so it becomes maximally flat okay up to this point so there after both of them are decreasing right so at most it can therefore become maximally flat only up to this point and there after okay both of them are decreasing so it will be decreasing at double the rate right and then I have to have another peak which is higher than this okay. So that this decrease can be compensated for by this increase so it will have to be higher so the q of the second order should be higher than that of the and it is going to occur close to the bandwidth there after of course all these three things are going to cause attenuation and therefore they are going to decrease only. So this is the way we are going to have the characteristic of all these blocks individually consider and that is how they become maximally flat or they can have a ripple okay that means if this is located further close to the bandwidth this may be permitted decrease and then increase similarly this one okay can be permitted to decrease and then increase and that is how you get the ripple okay. So the q's of elliptic filters will be higher than the q's of let us say inverse tables filters okay and all these inverse tables and elliptic q's will be higher than that of the Butterworth filters blocks. So these are the intuitive concepts that you can derive out of just knowing okay about the second order and first order. So try out this as trial and error just have the second orders and first order cascaded okay and see what happens to the multiplied higher order network this is the way right higher order filters are designed and that particular thing does not require the help of any sophisticated tool or anything else okay for understanding okay. So this is the conclusion that we are going to have that passive filters are very efficient and power efficient I am saying they do not require any biasing arrangement etc and precision passive components can be easily got therefore you can actually build these filters okay very easily if you understand the rudiments of the filter theory okay. In the next class we will just see how and by active filters came into existence and one natural way of thinking filter design because most of the passive filter designs have already been standardized and elaborate tables exist for variety of filter design okay these are all in normalized frequency domain what should be the q what should be normalized frequency ratios in order of ripple of so many decibels okay etc. And therefore one ready way of converting all this into active is just to get rid of the inductor and simulate the inductor using active device resistor and capacitor. So in the next class we will see how inductor simulation can be carried out with the help of the active device primarily the op amp and resistors and capacitors inductor simulation using capacitors.