 Today will be on to lecture 21 on filters we had earlier discussed integrators as building blocks of filters please remember how we had chosen the integrator in preference to the differentiator for filters and also seen the effect of finite gain bandwidth product on integrators and differentiators. Frequency compensation in negative feedback system is equivalent to filter design only op amp and LDO frequency compensation again aspect of filter comes into picture most of these are supposed to be good low pass filters after considering amplifier design with negative feedback we have brought about the effect of gain oriented product how it can cause ahh sort of filtering action in the amplifier that is designed okay and how we can optimise that Q of that combination such that it is suitable for maximally flat frequency response characteristics for a second order we have shown that Q should be 1 over root 2. So filter design has already started commenced when we discuss the frequency response of all these systems log anti-log amplifier frequency compensation again the aspect of filter design comes into picture for small signals analog filters and digital filters now when you come to filtering filtering is a major operation we will see how these filter blocks are absolutely necessary in present day signal processing activities why they are necessary and how these filtering is done obviously the choice now lies between analog filters and digital filters when digital signal processing has become pretty common in signal processing activity overall in most of the ahh systems today electronic systems today can we therefore do the filtering also in digital domain in fact some of the first hardware filters okay in ahh online signal processing try to implement straight away digital filters for filtering digital filter has been also used for offline filtering of data by geologist so lot of digital filter work has been already performed and what are digital filters as against analog filters so digital filters make use of as they are building blocks the digital delay that to the power minus 1 where as analog filter ahh uses for is basic building block 1 by S that is integration integrators as the building blocks. So analog filters try to solve a linear differential equation and digital filters try to solve the same thing in digital domain as ahh linear difference equation instead of differential equation it becomes difference equation and the ahh other block that is normally used for setting the coefficient of filter in analog is a summing amplifier here we have multipliers to fix up the coefficient of digital filters so we can summarize analog filter design as a combination of integrators and summing amplifiers digital filters on the other hand use delay 1 by Z Z to the power minus 1 clock delay and it is easily generated analog filter basic building block integrator is pretty complicated in its design and performance right delay is easily generated in analog filter summing amplifier is pretty easy in digital filter fixing the coefficient requires multipliers and multiplier design is more complicated than delay design. So present day precision filtering is done by digital filters whereas filtering required for band limiting which is a function which must be performed if you are going to sample the data for conversion of the analog information to digital to avoid aliasing error. So these filters pre filters and post filters after doing ahh signal processing activity in digital domain converting it back to the analog request post filters so these filters have to be analog so improvement in initial signal to noise ratio occurring in pre amplifiers or prior to pre amplifiers and in receivers and transmitters even to this day mostly passive filters are used these might be covering RF frequency range or micro frequency range but passive filters are the most reliable ones today okay even in ahh present day electronics or electronic product. So now let us consider the basic theory behind filters now this theory is very important both for analog filter design and digital filter design this is what is really important to understand that the filter theory itself is something that has not changed over the years as far as filter design is concerned. So the only topic that is very important in analog which has perhaps no changes that have taken place is the filter theory over years and that is again another topic that has been systematically tackled from the beginning of electronics okay and this is something that still retains the analog flavor even if you were to do digital filter design for most of the filtering you have to start with analog filter and its theory and analog filter prototype okay transfer function in yes domain and change it over to Z domain and then realize the digital filters. So ahh let us now consider what are the ahh mathematical ahh foundations which we must lay before beginning filter design. So now for example we can consider magnitude or delay of the signal okay on ahh one axis amplitude or delay of the signal on the one axis and frequency on the other axis and then the box like approximation to filter says that this is the passband this is the passband corresponding to the signal that we want to handle it may be audio video biomedical okay RF IF anything. So there is a certain band which is of interest to us in terms of signal processing there is another band okay outside this which is called the stop band where mostly noise exist as far as we are concerned our signal is concerned these signals beyond these ranges correspond to noise for us okay. So in order to improve and actually that noise okay might extend over a large bandwidth and therefore right we want to remove that noise fully and only process the signal within the band okay select only the signal. So there should be full transmission transmission which is not varying with frequency okay within the band and there should be no transmission in the stop band okay full attenuation. So whereas in certain type of filters wherein we want to reject a certain band of noise okay and permit outside the signal to appear in this band the signal exist and this in this band the noise exist. So this is called a band stop filter and this is called a band pass filter. So once again we have a sort of band starting from zero frequency up to a certain high frequency this type of filter okay which attenuates all the high frequency components beyond this okay and retains only the low frequency component is called appropriately ideal low pass filter. And ideal high pass filter okay is going to attenuate all the low frequencies and only permit high frequency to pass through. Please remember that if zero is the uh frequency around which this is being explained right in realistic situation we must also consider the other region okay that is the negative frequency region okay which will be therefore having uh characteristic going like this. So actually speaking this also is a band okay centered around zero this is again a band centered around zero this is the stop band and this is the pass band. So actually speaking the band pass filter and the band stop filter that we have discussed is uh situation these are the uh cases wherein we have shifted the frequency center frequency of these bands from zero to a certain frequency let us say F naught. Similarly this is also a band stop filter which is shifted from zero at that point of time it was a high pass filter with the certain bandwidth on this side and now it becomes a band stop filter with this kind of band okay. So even filter design is primarily initially carried out for low pass prototypes then using suitable frequency transformation of shifting the origin okay to stable frequency we can depict it as band pass okay starting from low pass or band stop okay starting from high pass from low pass to high pass it is equivalent to shifting the zero to infinity that is F to 1 by F okay. So that is the uh what is that domain change okay that we have to do in order to uh perform these designs also. So it is easier to consider that design has to be carried out only for low pass prototypes and then by appropriate frequency transformation we can actually get the prototype for band pass band stop high pass or any other filter types. Now let us consider what the filter does okay what it should do to the signal normally the signal is a periodic only for testing purposes we use these periodic signals and understand the characteristic of the systems. So actual uh case the signal may be some a periodic signal and amplifier simply has to amplify this okay in terms of voltage or current or both so that power amplification is achieved. In the process of amplification what happens is that the parasitic resistors and capacitors make the amplifier itself appear or the system itself appear as a filter and therefore the parasitics cause some filtering effect. In addition we would like to avoid certain noise components which are existing in the pickup signal. So these components of noise may be okay actually superimposed over the signal. So this in reality is the high frequency noise which is there in the signal okay and therefore when we perform an amplification without any filtering these signals the noise signals also get amplified. If for example this noise is too much okay that it is taking considerably then it is obvious that this high frequency noise okay you can see the rate of rise is very high here that is it changes rapidly in magnitude as this time uh this is the time axis so this rate of rise is high that means this high frequency signal okay. So this has to be removed by suitable low pass filters right the low pass filter can only take these low rates of rises right. So it will only permit such low rate of rise and it will attenuate all these high rate of rise voltages which form the noise. And when it is doing that it may not be totally uh permitting all the low frequency component to come it will also attenuate we will see all these problems okay in this process within the band itself the signal frequency components okay which you want to amplify properly they themselves get attenuated okay compared to the low frequency signals okay. So that kind of distortion should not happen in a filter so that means amplification should be uniform for all the wanted frequency components throughout. Similarly if they suffer phase variation okay the phase variation should be proportional to frequency that means in such a situation the delay is constant that means the signal if it is like this okay it will get amplified and retain the shape exactly but there will be certain delay that is all. So linear phase indicates constant delay okay and therefore the signal just simply gets delayed by a constant value and it is not distorted at all it is a faithful reproduction of the original signal without noise. So this is the requirement of filter design. So what should be the characteristic of a filter normally the box like approximation can only be approximated okay the box like ideal characteristic can only be approximated by something close to the box like approximation okay in terms of anyway some variation may occur here that is of no consequence as long as it is within a certain limit okay and here also it may vary but then again it should be within certain limit okay and this variation at the pass band edge okay that variation should be again a rapid variation with respect to frequency it is a close approximation to the abrupt change the abrupt change indicates that this is a multiple value infinite values for a single input okay which is impossible to physically realize. So no system okay which is having multiple values with for a single value is a stable system okay and therefore we have to approximate it to a variation of this type in order to physically realize that. So this is the kind of approximation that we are kind to achieve okay in filter design where are the filters used that we should know I have already explained to you that almost all receivers today radio receivers cell phone receivers okay all of them are using tune circuit at the input that is the antenna okay itself forms a tune circuit so as to select the station we want in and reject the neighbouring stations which we do not want to receive that is an important the neighbouring station is a real problem as far as our reception is concerned okay if you want interference free reception the first noise component that interferes with us is the neighbouring station which is pretty close to us receiving frequency ECG the main noise component in this ECG is nothing but the power line frequency which is a major component which is occurring within the pass within the signal frequency range itself and therefore we have to very carefully remove this with the very sharp what is called as notch or band stop filter if it is not sharp then we are also attenuating some useful information of our signal okay. Graphic equalizers are well known right as filters in most of the music systems graphic and parametric equalizers and these days with the usage of cell phone proliferating everybody has come to know about graphic equalizers FM receiver as I already pointed out say typical most of the transmission is FM today okay 87.9 megahertz to 107.9 megahertz FM transmission several radio stations operate within the spectrum okay and 200 kilohertz apart these stations may be located and FM 102.9 for example can mean that the center carrier frequency used by this station is 102.9 megahertz the radio station has to filter signals outside this 102.9 megahertz plus or minus let us say 75 kilohertz using band pass filter before transmitting okay FM receiver should have a tune circuit which is a band pass filter. So like this therefore whether it is RF or IF all of them may use of band pass filters okay the center frequency by bandwidth is crudely called for a second order filter as Q of the filter okay intermediate frequency of such system is typically 10.7 megahertz local oscillator is adjusted to 113.6 so the mixer produces this these are all the technicalities involved this just signifies how important the filters are okay. So the typical magnitude of ECG signal is anywhere from 0.1 millivolts to 300 millivolts okay interference signal is 50 hertz from power supplies major interference motion at fax due to patient movement is a very low frequency radio frequency interference from electro surgery equipments okay these deflation pulses pacemaker pulses all these are interfering signal components. So ECG monitor mode okay high pass filter is set at either 0.5 hertz or 1 hertz and the low pass filter is set at 40 hertz. So this one eliminates all the low frequency noise and this one eliminates all the high frequency noise okay including the 50 hertz or 60 hertz. Again music systems equalization okay so that requires filters you have heard of what is that bass level controller and treble level controller that is for high frequency filtering and this low frequency filtering. So hearing aids for example one has to have the filters designed to adopt themselves to the condition of the ear so that okay certain range of frequencies which are not very clearly heard can be boosted up and certain other range where the sensitivity of the ear is good can be attenuated. Now filters okay can be passive I told you mostly the earlier filters used to be purely passive that R L and C particularly L and C okay filters are the picture because of finite source resistance and finite load resistance coming into picture. So the in between thing is purely L and C and non idealities of L and C bring about some resistance okay into picture in the actual filtering. So however historically most of the telephone filters were purely passive filters are very bulky because the time constant 1 over 2 pi RC corresponds to the frequency or 1 over 2 pi root LC corresponds to the frequency. So just remember this basically if it is a LC filter frequency corresponds to 1 over 2 pi R into C okay and omega corresponds to 1 by RC and if it is LC omega corresponds to 1 by root LC this is radiance per second this is in terms of hertz. So this is what we should remember that the time constant RC used to determine the frequency of omega. So these were very stable filters and precision was very easily achievable because R's and C's could be made precision components and precise design was possible. However if you evaluate for F equal to let us say some few under sub hertz or kilo hertz evaluate RC then you will realize that the LC component or RC components used become very large in size and it is now that finitely not compatible at all with the kind of component sizes that we can we would like to have okay in the electronic systems. So the size was the main cause for the downfall of passive filters okay at low frequencies or base bands that means audio signal filtering video signal filtering etc had to deal with large size components and definitely biomedical filters okay required to huge a component to be used. So all these base band signal processing started adopting filters which are digital filters or digital signal processing came into prominence okay and resulted in the downfall of all passive filters. And anyway most of the base band signal processing is now done using DSPs and therefore there is no need to worry about filtering in analog domain as far as these are concerned except for pre filters and post filters which are still going to be definitely not passive okay they better be active what does it mean with the advent of transistors and integrated circuits there has been a requirement for size reduction of filters at that time itself in 1960s itself okay active RC filter became very popular and size reduction took place. So active RC simply means the active device transistor or op amp in association with R and C L was totally eliminated L was the major component which was too huge in size for base band. LC filters are the most reliable filters in micro range even to this day and even in RF okay because passive components in this frequency range become somewhat compatible okay even though they are still large in size compared to other signal processing components in integrated circuit. So ideal filter cannot be realized because of requirement of multiple values at the edge of the pass band that you should have multiple values for a single frequency. So electronic circuit can only realize okay single valued functions so the practical electronic circuit realize only a specific output for a given input okay in signal processing these are the components that can be used. So now we have to do approximation so this is going to be purely mathematical aspect of filter approximation. How do we do filter approximation so that what we realize is still close to the ideal. So ideal band pass filtering function in terms of mathematics T transmission equal to 1 okay for X naught – delta X by 2 less than X less than X naught – delta X by 2. So that means actually we have the transmission equal to 1 okay for X very nearly equal to 1 and – 1 in this range – 1 to 1 then beyond this it should abruptly go to 0 transmission should go to 0. However this is impossible to realize practically so what we do first approximation is the transmission changes from 1 to 0 in a gradual manner but very fast okay this is called the pass band edge. So there the rapidity of this attenuation is very important okay and then in the pass band you are permitting certain minimal variation within the pass band. In fact today what is done is that it is made equi ripple that it is equi ripple that means if the error is there it is going to be rippling through the whole band and equal ripple but we are not so rigid about the fact that it should be equal it should be limited within certain range here and limited in terms of attenuation it may not give full attenuation okay 0 transmission but it should be attenuated considerably okay. So this is the characteristic we are trying to reach okay using the network elements that we can use for this design of filters. So what are these physically realizable functions mathematically now we look at it T is some numerator polynomial in X divided by denominator polynomial in X okay X is what is called a sort of variable which is normalized that is dimensionless. So order of DX now what are the requirement if in order to have box like approximation okay that is outside the band it should attenuate as X increases beyond 1 okay and X becoming more negative than minus 1 okay and it should be symmetric around X equal to 0 okay it is characteristic it should be symmetric around X equal to 0 and at X equal to 0 it is 1 let us see okay and therefore the slope around this that means let us draw that this is X so it should be having okay a slope equal to 0 around X equal to 0 and then maybe slope may gradually change here okay at X equal to 1 come to again 0 slope beyond this edge again slope may be something here and come to 0 slope at this point. So this is called flatness okay around X equal to 0 it should be as flat as possible so a function of this behavior okay is called a flat function this way. Let us look at first such flat function that is mathematically possible okay you can say that this kind of thing is approximated by something which is close to so it should be what is called an even function of X that is for sure because there is symmetry around X equal to 0 so whether X is positive or negative it should okay go in the same manner symmetrically. So that means it is an even function of X so I have given here T equal to H naught H naught is the transmission at X equal to 0 in this previous case we have taken it as 1 1 plus n 1 X squared m 2 X to power 4 so on up to nm X to power 2 let us say and this is 1 plus k 1 X squared plus k 2 X to power 4 so on up to k n X to power 2. So this everybody agrees is a general symmetric function around X equal to 0 that much is established because it is a polynomial in even function of X X squared X 4 4 X to power 2 m and m should be less than n this requirement is because beyond X equal to 1 it should start attenuating as X becomes greater than 1 okay all the lower order terms do not matter at all only this highest order in the numerator and the highest order in the denominator cause the variation with respect to X and obviously the denominator is the only one which is which should cause the final attenuation. So if it is X 2 m by X 2 n it will be X 2 n minus 2 m in the denominator okay. So that should exist for causing the attenuation that means actually attenuation is now caused by at high frequencies when X is much greater than 1 okay by a factor nm by k n okay X to power 2 n minus 2 m obviously this condition requires that m should be less than n okay that means higher the value of n okay compared to m over is the attenuation that is it is this n minus m which determines the high frequency attenuation. So this filter is called a nth order filter okay. So this is a nth order filter which will cause the attenuation at a rate which is equal to n minus m okay at high frequencies okay we will come to that exactly in when we actually realize filters later but mathematics of it is very clear in terms of attenuation at high frequency at low frequencies that is in the range X close to 0 in the range X close to 0 this lower order terms are the ones which dominate because okay X squared okay is going to contribute okay ahh higher value than X to power 4 and so on that means it is the lower order X dependencies which determine how it is changing with respect to ahh X equal to X close to 0 that means actually speaking at X close to 0 you can approximate it to X squared divided by 1 plus k 1 X squared and forget about all the higher order terms. So around the X close to 0 this is the variation okay and around X equal to 1 and beyond this is what the higher order terms dominate okay. So X equal to 1 and X much greater than 1 these are the ones which are dominating. So now understanding this much we can talk a lot about the filter function that becomes necessary okay in order to do signal processing okay a flat function that has all is n-1 derivatives at X equal to 0 should be 0 okay a flank function is 1 okay is definitely 1 that has all is n- derivatives at X equal to 0 okay this comes okay from basically nothing but Taylor series X bars mathematically it is understood that this can be obtained okay directly from this that this is a flat function if n1 is equal to k1 n2 is equal to k2 so on that means all these coefficients are equal until nm is equal to km and numerator polynomial is same as denominator it is very flat okay now beyond that okay the that is beyond nm equal to km km plus 1 to k n-1 coefficients should be 0 okay so that they do not cause a variation okay for X less than 1 and X greater than 1 okay this coefficient kn into X to power 2n is the one that causes attenuation. So that is mathematically put in this manner n1 this requires mathematically n-1 derivatives at X equal to 0 to be 0 so slope is 0 slope of slope is 0 slope of slope of slope is 0 right until we have nm is equal to km reached right and beyond that km plus 1 km plus 2 all should go to 0 until kn minus 1 also is 0 and it is only the last one that exists the coefficient of kn X to power 2n which is kn should be not be 0 in order to cause the attenuation that is what is called a Butterworth function okay that is a maximally flat function okay there is no numerator polynomial in X then only the denominator polynomials all the other coefficients other than coefficient of kn should go to 0 that is called a Butterworth function that is a maximally flat function okay t will have value between 1 plus epsilon 1 and 1 minus epsilon 2 where epsilon 2 are small okay much less than 1 positive values within the band defined by minus 1 to plus 1. So that means you might permit certain amount of variation within the pass band if you do not strictly adopt this okay you might permit certain amount of variation within the pass band. Now we will explain about the band okay later because fx is equal to fx0 plus f dash x0 by factorial 1 into x minus x0 so on right up to the nth order let us see for nth order polynomial in X okay. Now all of these slopes should be going to 0 if fx should be same as fx0 that means it is flatness right f at any x within the band should be same as fx0 in this case x0 is equal to 0 so it should be same as f0 which is h0 in our case so that is what the Taylor series says and if you do that differentiation you can prove that what we said about maximum flatness is valid n1 is equal to k1 n2 is equal to k2 and nm equal to km and km plus 1 up to kn minus 1 should go to 0 this is depicted in terms of these curves here you can see here we have taken 1 plus epsilon 1 by 1 plus epsilon 1 into X to power 4 so let us say until it remains at X equal to 1 you will see that it is 1 okay and at X equal to 0 it is 1 plus epsilon 1. So we are permitting certain variation this is necessary to define what is called a bandwidth bandwidth is normally defined in terms of what is that as far as the voltage transfer function is concerned it is called 3 dB bandwidth as far as power is concerned it is half power bandwidth okay so if this is a power thing that is plotted with frequency then we should go up to 0.5 to define a bandwidth but it can be done also for any epsilon 1 in general right. So let us for mathematical sake treat it as a bandwidth which is having a variation in magnitude occurring by only epsilon 1 okay so we start with 1.2 we go up to 1 okay so the permitted variation is only point 2 in another case we start with 1.1 okay and we go up to 1 so this is for comparison sake so if this is X to power squared X squared okay it is something okay and therefore here we have plotted it for some specific order n equal to 2 X to power 4 is taken and different values of epsilon 1 okay so you can see the curve changing this way okay. Now for first order there is 1 by 1 plus epsilon y by 1 plus epsilon X squared okay you can see this is for the first order okay then when it is changed from X squared to X to power 4 you notice that the attenuation is more rapid and it is going to be flatter than the first order this is the way and when it is X to power 6 okay you will see that it is flatter still than these two within the passband and attenuation more rapidly so it is better to go for higher order in order to approximate it to closer to the ideal characteristic so these are all called Butterworth functions so higher and higher order Butterworth functions have better passband response as well as higher rates of attenuation in the stop band so rate of attenuation close to the edge of the passband okay this may not always be acceptable so in order to improve the rate we can resort to other means of transfer function selection so we come to that let us see we have a second order transfer function so n is equal to 2 so X to power 4 is coming picture so one type is 1 by 1 plus K1 X squared plus K2 X power 4 K2 is greater than 0 that is a basic requirement. Now another type is 1 plus N1 X squared it has both numerator and denominator divided by 1 plus K1 X squared plus K2 X power 4 N1 can be greater than K1 okay and K2 is always greater than 0 this is the requirement let us take these that will how this kind of thing will improve the passband edge characteristic is what we want to demonstrate mathematically. So when we have this 1 by 1 plus K1 X squared plus K2 X power 4 K2 being positive you can now have K1 equal to minus K2 what happens this to this is something that is interesting as far as this function is concerned it is a maximally flat function alright but if K1 and K2 both are positive it can go on like this okay so for example if K1 is 0 if this is the case that is earlier been drawn if K1 is positive it starts okay decreasing earlier okay whereas if K1 is negative it can go up because when K1 is negative for X close to 0 it will start increasing beyond 1 because 1 by 1 minus some value let us say okay X squared let us say K1 as 1 okay but positive. So it is going to increase and then obviously it will come to let us say we can make it come to the same 1 here 0.5 I assume that this is the attenuation of 0.5. So how can we do that let us say if we select K2 as let us say K1 is 1 so X equal to 1 this 1 gets cancelled with 1 so if you want this K2 equal to let us say 1 so this becomes equal to 1 at some point if it is 2 it becomes equal to 0.5 at this point. So K2 equal to 0.5 it goes like this and becomes equal to 1 and then becomes equal to 0.5 so that means it is going to peak at this point this was for K1 equal to 0 this corresponds to K1 positive this corresponds K1 equal to minus 1. So one can therefore see that by proper selection of K1 you can have a ripple in the pass band and this one will attenuate faster than even the situation of Butterworth that means the attenuation can be increased by permitting certain ripple in the pass band. So this is what is depicted here for different values of K2 and K1 okay. So we have permitted a ripple in the pass band and if you permit more ripple then the attenuation is going to be steeper this is with little ripple this is with more ripple this is still more ripple in the pass band this is mathematically understood very clearly without any sophisticated mathematical theory for it just a common sense understanding of the whole thing. Now these functions with ripple in the pass band are called Chebyshev functions the earlier functions were called Butterworth functions maximally flat and these with ripple in the pass band are called Chebyshev functions that kind of ripple can arise even if you have an emulator polynomial okay. So N1 has to be greater than K1 that is all okay effectively okay the coefficient here is going to be for X more close to 0 will be K1 – N1 and it will become negative okay N1 equal to K1 is maximally flat situation. So see how simply you can understand most of the sophisticated mathematical approximations that I mean given out okay. So we are now considering here the various cases of K2 coming into picture okay and the if you permit certain amount of ripple okay you can see that it is going to be steeper. Now we come to another approximation which is called inverse Chebyshev functions here it is going to be similar to this but what we introduce is there is another way instead of permitting a ripple in the pass band let us say we do not want to permit that so we will make maintain it maximally flat in the pass band okay and then we introduce what is called a 0 in the stop band this is very useful because this is the best technique of preventing your neighbouring frequency from interfering with your signal like the neighbouring FM station from interfering with your selected FM or neighbouring cell phone okay transmitter from affecting your transmitter frequency or receiver frequency. So this can be easily done by introducing the 0 of transmission in the numerator the 0 has to occur beyond X equal to 1 that is the thing if 0 has to occur beyond X equal to 1 then we know that N1 has to be ahh less than 1 okay. So let us say N1 equal to minus 1 by 4 will introduce a 0 at X equal to 2 that means at X equal to 2 you have the transmission 0 so it is a very simple procedure that we know that X equal to 2 is the transmitting frequency of my neighbour okay neighbouring cell phone so I just put a ahh sort of 0 of transmission okay then N1 is made 1 by 4 so X equal to 2 I have this 0 okay so you can actually therefore locate this kind of thing at any frequency you want okay. So once you make this choice for N1 and if this ahh is same as K1 N1 is equal to K1 it becomes maximally flat okay. So N1 is equal to K1 N1 is negative and is fixed according to where you want the 0 to be located and that kind of filter is called inverse JBCU function right. So this is the ahh difference in design so we can have JBCU function which will introduce a ripple in the pass pan okay or inverse JBCU function okay which is going to introduce a 0 in the stop pan this is what it is so if you actually permit the variation epsilon 1 okay to be different okay you can actually have this kind of thing then obviously it permits more of the out of the band noise okay if this is narrow it permits less of the ahh noise here beyond white noise but the specific noise component is removed very efficiently by both these schemes it is compared with the Butterworth filter of the ahh same order. So now if you permit the ripple in the pass band and ripple in the stop band then it is called an elliptic function that means instead of making it maximally flat I will simply select not N1 equal to K1 I will make N1 greater than K1 okay so that there is a ripple in the stop band and a 0 in the stop band. So that is called elliptic function this in simple terms deals with all the important filter approximations that become necessary in filter design okay so so inverse JBCU function elliptic function and now finally we come to the white band using staggered narrow bands it is ahh very simple to understand how white band can be achieved by using narrow band but stagger tuned. So let us say this is the first ahh order okay narrow band function we simply shift it to plus 1 and minus 1 okay so it just simply gets shifted X plus 1 and X minus 1 so if you now multiply this with this that means if you cascade this filter with this filter then it is possible to get what is called maximally flat function with white band. So in the next class discuss more about how the mathematics of how this is achieved so most of the white band amplifiers are designed as narrow band amplifiers connected in cascade with staggered tuning this kind of approach is also adopted in transfer function or trans conductance realization linearization of the trans conductant a typical trans conductors of a pair looks symmetric like this okay and if you offset that trans conductor and use 2 such trans conductors together then you can achieve a linear trans conductor which is okay having a wider range of signal that can be handled okay ahh without any problem without any distortion okay. So this topic of ahh filter approximation is very important even in amplifier designs. So in conclusion we have discussed a lot about filter approximation in this class the mathematical theory of approximating it to a box like ahh ideal value okay and that we have shown that higher order filters approximate closer the ideal characteristic that is required. There are 3 types of filters whatever filters Chebyshev inverse Chebyshev and then the final one elliptic okay. So these are the important filter approximations that are commonly used whether you are designing a low pass high pass band pass on band stop. The prototype is low pass.