 A warm welcome to the 36th lecture on the subject of wavelets and multirate digital signal processing. You will recall that in the previous lecture, we had hinted briefly at the idea of a polyphase decomposition. In fact, we had used that idea of a polyphase decomposition for various reasons. We have been using it time and again to construct different kinds of structures to carry out computation efficiently in a filter bank, but we have not put down formally a whole approach based on polyphase components. Moreover, we have not quite considered the alternative approaches to the question of perfect reconstruction to date. So, in the lecture today, what I intend to talk about is two approaches to the general perfect reconstruction problem namely the polyphase approach and the modulation approach. Both of these are essentially approaches to handle the question, when does a filter bank give perfect reconstruction of the input and what we are going to do today is applicable not only to two band filter banks, but also to general m band filter banks where m could be a positive integer greater than 2. Let us begin with the first of these two approaches namely the polyphase approach. Now, we have already introduced the idea of polyphase components earlier. Let me recapitulate that idea. You see, when you have a sequence x of n with its z transform capital X of z with an appropriate region of convergence, we have seen that we can separate the samples of n which the samples of x n which lie at n a multiple of 2 and lie away from a multiple of 2. In other words, we can consider the index n as broken down into 2 m and 2 m plus 1 for all integer m. So, when we let m vary over all the integers here, then all possible integers n would be covered. So, what we have talked about here is what is called a second order polyphase component or polyphase component polyphase decomposition of order 2. What we want to do first is to generalize this. So, suppose for example, we wanted polyphase decomposition of order 3. What we are then going to do is to ask that the integer m be one of these three possibilities, 3 m, 3 m plus 1 and 3 m plus 2 over all integer m and when m varies over all possible integers, then we have covered all possible integer values of small n here. In general, suppose one talks of order capital m, we are then referring to a decomposition of the index n as follows. Capital m times small m, capital m times small m plus 1 and so on up to capital m times small m plus m minus 1 over all integer small m and on so doing on letting small m vary over all possible integers, we have actually spanned the entire set of integer n. So, now let us put down explicitly the mechanism for decomposition of x z to the z transform into the z transforms of the polyphase components of order m. To do that, we shall essentially split this index capital N in the way that we have just described. So, what we are saying effectively is to decompose x z using polyphase decomposition order capital M. Essentially, we write down capital X of z in terms of the sequence and now decompose the summation or split the sum of the sum summation on n into summation n going from minus to plus infinity is the same as summation l going from 0 to m minus 1 and summation m going from minus to plus infinity and where here you would have a variation as a function of n, here you would have variations as a function of m capital M times small m plus l. So, we just change the nature of the variable, the specification of the variable. Let me illustrate concretely. So, capital X of z, capital X of z, capital Z in other words becomes summation l going from 0 to capital M minus 1, summation m going from minus to plus infinity, x capital M times small m plus l times z raised to the power minus capital M times small m plus l. And now we can split this exponential into two parts, z raised to the power minus m times small m into z raised to the power minus l and noting that z raised to the power minus l can be brought outside the summation on capital on small m. We could rewrite this as follows, capital X of z is summation l going from 0 to capital M minus 1, z raised to the power minus l and in brackets then you have summation m going from minus to plus infinity, x capital M small m plus l times z raised to the power minus capital M times small m. Now essentially this is the z raised to the power minus z transform of all those points which lie at multiples of capital M plus small l. So, for example, when l is equal to 0, essentially this refers to z transform of all those points which lie at multiples of capital M. When l is equal to 1, this refers to all those points, z transform for all those points which lie at multiples of capital M plus 1, displaced by 1 from multiples of capital M and so on. When l is equal to 2, they are displaced by two steps and so on and this can only go up to capital M minus 1 steps because when you go to capital M, you are coming back to the original case of l equal to 0. So, what we have done here is to break x z into m parts, m disjoint parts in some sense and we shall use some notation now. We shall refer to this quantity here as capital X l m z raised to the power m. So, what we are saying here is you have the m th order polyphase component and the l th of those components with the argument given by z raised to the power m here. Let me write down this explicitly. Capital X small l capital m z raised to the z, just z is essentially summation m going from minus to plus infinity m small m plus l z raised to the power minus m not. Here, I do not have a capital M occurring here because I have replaced z raised to the power m by z and this is essentially the z transform of the l th polyphase component of the sequence x plus l. So, here there are two things that are important in general in polyphase decomposition. The order of decomposition and the component number, there would be as many components as the order. So, when capital M is equal to 2, they are going to be 2 components 0 and 1. When capital M is equal to 3, they are going to be 3 components 0, 1 and 2 and so on. In fact, we can write down a very simple relationship between the z transform of the original sequence and the z transforms of its polyphase components. So, capital X of z becomes summation l going from 0 to m minus 1, z raised to the minus l, x l m, z raised to the power m. This is the manifestation of polyphase decomposition in the z domain. Now, this is just to explain and define the idea of polyphase decomposition in general. What we would now like to do is to see how this polyphase decomposition works when you have an analysis and a synthesis side. We would like to write down a general relationship for analysis synthesis polyphase components and how they interact to give perfect reconstruction. So, you see let us look at the analysis side. In fact, now for some variety and also to generalize, let us look at the general case, general analysis branch in an m-band filter bank. Now, you know, so far we have been talking just two bands and slowly my objective in this lecture is to go from 2 to m-band. So, I would like to bring in m-band first and then put m equal to 2 as a special case. So, what is firstly the m-band filter bank? Essentially, it is an analysis synthesis structure with the following nature. On the analysis side, you have a typical let us say l-th branch where you have a filter or let us use k-th branch to avoid mixing up with the polyphase component index. So, capital H kz followed by a down sampling by m. This is a typical analysis branch and in general, we could have b, branches here. Now, the number b could be different from the number capital M that should be stressed. They do not have to be the same. A typical synthesis branch would look like this. You would have an upsampler. We earlier had upsamplers of factor 2. Now, we would have upsamplers of factor m in general followed by the so-called synthesis filter, the k-th such. So, g kz and there could be b such branches too. Again, I must stress that b need not be the same as capital M. Needless to say, in a given m-band filter bank, the number of analysis branches and the number of synthesis branches must be the same. In fact, there is a one-to-one correspondence. When there is an analysis, there is also a synthesis on the same branch. So, let us draw the overall structure. So, this is the structure we are talking about. Now, notice that you have the same b on the analysis and synthesis side, but b is different from m and there are therefore, three possibilities. b could be equal to m, b could be less than m, b could be greater than m. Let us write down these three possibilities here. Notice, of course, that an analysis filter is always followed by a synthesis filter. So, you have an analysis and synthesis filter coming in cascade with the down and up samplers in between. So, b could be equal to m. Such a filter bank, such an m-band filter bank is called a critically sampled m-band filter bank. b could be less than m, in which case we call it an undersampled m-band filter bank. And on the other hand, b could be greater than m, in which case we call it an oversampled m-band filter bank. Whether the filter bank is critically sampled or undersampled or oversampled, the essential mechanism of analysis, whether by the polyphase approach or the modulation approach does not change. The essential idea remains the same. What needs to be checked is on analysis, whether we get the desired conditions, whether we get perfect reconstruction, whether we get something else. Sometimes we may not want perfect reconstruction. We might want something else from a filter bank. We might want the overall filter bank to perform a certain overall filtering operation. That is possible. It is not necessary that we always want perfect reconstruction. Whatever it is, the overall analysis of what a filter bank does can be done by the polyphase approach or by the modulation approach and that holds whether or not it is critically sampled. That is whether it is critically sampled, undersampled or oversampled. This is a point which I want to make before I begin to discuss the approaches in depth. Now, the polyphase approach says decompose the filters both analysis and synthesis into polyphase components and decompose the input and the output also into its polyphase components of order m. Note the order of the polyphase decomposition is the same as the down and up sampling factors. So, the polyphase approach says, let us write it down clearly, decompose the input, output, analysis filters and synthesis filters into polyphase components of order m. And what we shall do here is to take one of those B branches and see what we get. And once we understand what happens on the kth of the B branches, we shall understand what happens overall. So, consider the kth branch. So, we have the input subjected to the action of h k z followed by down sampling by m followed by up sampling by m and then subjected to the action of g k z. And we wish to analyze what this is doing in the polyphase domain. So, let us write down the input z transform capital X of z here. And let us see what z transform emerges here. So, you see x z can of course, be written in terms of its polyphase components and that gives you summation l going from 0 to m minus 1 capital X l capital M z raise the power m multiplied by z raise the power minus l, where you have capital X l m as the l th polyphase component of order m of the sequence x. Similarly, we can take the kth filter here and decompose this as well into its polyphase components of order capital M. So, we could write now here you will have a three fold index. So, filter number, polyphase component of order of the polyphase component z raise the power m. And now let us multiply. So, x z k z is what emerges after the filtering operation. Now, you know we are not really interested in x z times h k z. So, let us short circuit you know the solution to the first step. What we are really interested in is this subjected to down sampling. We are not interested at this point, we are interested at this point. So, what we want to get at this point is only the 0 th order polyphase component sorry the 0 th polyphase component of order m. We are not interested in the other polyphase components the 1 th up to the m minus 1 th. After we have taken this product, we just wish to understand or bring out or pull out that component which is of order 0 and how will we get that. So, you know we have this product x z times h k z is of the form summation l 1 going from 0 to m minus 1 summation l 2 going from 0 to m minus 1 z raise to the minus l 1 times z raise to the minus l 2 times capital X minus l 1 m z raise to the m and capital H k l 2 m z raise to the m. So, you know as far as pulling out the component of order 0 is concerned what we need to see is that there must only be z raise the power of m terms. There must not be a hanging power of z in whatever we choose when will that happen that is going to happen when l 1 plus l 2 you know when this essentially contributes a power of m. So, what we are saying in effect is the 0 th polyphase component results when z raise to the power minus l 1 z raise to the power minus l 2 which is equal to z raise to the power minus l 1 plus l 2 contributes z raise to the power m times some integer. So, let us say l 0 and that is easy to document we can easily see when this is going to happen. So, let us make a table in fact we can make a table of l 1 and l 1. So, when l 1 is 0 there is no other possibility, but that l 2 be 0 when l 1 is 1 there is no possibility other than that l 2 be m minus 1. And finally, when l 1 is m minus 1 there is no possibility other than that l 2 be 1 there is a very easy association. So, in fact there is just one pair that needs to go together with each l 1 we can choose only one unique l 2. So, that we can pull out the 0 th order polyphase component. So, in fact we have a very simple expression for this you know all that we are saying is consider m minus l 1. So, you know when you take m minus 0, but take it more to low capital M you get 0 m minus 1 will of course, give you m minus 1 as expected m minus m minus 1 gives you 1. So, in other words what we are saying here is that l 2 has been chosen as m minus l 1 modulo m. So, this is modulo m. So, with this choice what we are saying is x z h k z done sampled by m is essentially going to give you the following. It is going to give you x 0 z or if you like x 0 m z h k 0 m z plus summation l going from 1 to m minus 1 x l m z h k m minus l m z and there is a multiplication by z in 1. So, this is going to give you the following words here. Please note there is a multiplication by z inverse, because except for l equal to 0 or in other words except for this case where l 1 is equal to 0 and l 2 is equal to 0. For all the other cases you have l 1 plus l 2 summing up to capital M. So, when you done sample by capital M you get a z inverse power a z raise the power minus m term would have remained originally and on down sample it becomes z inverse. A simple, but elegant and beautiful expression that we have right here. So, x 0 m I will repeat it x 0 m times h k 0 m plus z inverse times summation l going from 1 to m minus 1 x l m h k m minus l m. In fact, we can now write down a matrix for each of these branches. So, you know what we are saying in effect is I can put down the outputs of these b branches a typical one emerging from the say kth branch at this point and we could arrange them in the same way. So, let us arrange them in the form of a vector. So, let me write down the analysis vector and let me find out the typical kth row of the matrix here. So, typical element in the analysis vector would correspond to the kth row times a vector of poly phase components of x. So, let us write that down capital X 0 m capital X 1 m up to capital X m minus 1 m and I will write down the kth row the kth row would look like this. So, note in the kth row we have taken care of the z inverse here. We have taken care of the fact that you know if you go back to the matrix vector product the number or the expression in z which should get multiplied by x 1 m is going to be h k m minus 1 m. The number getting multiplied by x m minus 1 m is h k 1 m. So, there is that so called inversion. This structure which we obtain here where you have b rows and the vector of poly phase components of x is called the analysis poly phase matrix. So, in general we have what is called the analysis poly phase matrix with kth row as described and the size of the analysis poly phase matrix is obviously going to be as many branches times the order of poly phase decomposition. Upsampling every z is going to be replaced by z raise the power m that is simple. So, let us consider what happens on the kth branch after upsampling by m followed by filtering by g. So, you know essentially upsampling would replace z by z raise the power m and then filtering with g k z would again produce m poly phase components here. So, let us call this output capital Y k z and if we happen to decompose capital Y k z into its poly phase components that is if we were to write capital Y k z is summation l going from 0 to m minus 1 z raise the minus l Y k m z raise the power m. Then it is very easy to see that you know if you look at it here what comes here is anyway a function of z raise the power of m. So, to get hanging powers of z inverse you must take the corresponding hanging powers of z inverse appearing from this filter. So, in fact if you take the l such well I am sorry this should be l here is a little correction. So, you have Y k well you know actually we should write Y k l because there are three indices here. So, let me rewrite this summation l going from 0 to m minus 1 z raise the power minus l capital Y k l m z raise the power m there are three indices and we can make a very clear specification of what Y k l m shall look like. What we are saying is g k z would itself be decomposed as summation l going from 0 to m minus 1 g k l m z raise the power m multiplied by z raise the power minus l and essentially Y k l m z raise the power m is g k z raise the power m. So, l m z raise the power m times essentially the output of k th up sampler as is because that is already up in terms of z raise the power of m. In fact we can write down explicitly what that is the output of the k th up sampler as is essentially the k th row of m analysis poly phase matrix multiplied by the input poly phase vector. So, capital X 0 m capital X 1 m and so on up to capital X minus X m minus 1 m. The only catch is z has been replaced by z raise the power m that is the only change. In all this which are all functions of z z has been replaced by z raise the power m. Now, this is what is called the poly phase approach to analyzing the overall m band filter bank. Now, we have seen the k th branch the output is of course summation k going from 1 to b Y k z there are b branches remember and therefore, Y l m z the l th poly phase component of order m of the output is easily seen to be let me just recapitulate for you here the expression here all that we need to do is to sum over k here. So, we have summation k going from 1 to b G k l m z raise the power m times the output of k th up sampler as is. So, now we could write down a vector of output poly phase components and input poly phase components and relate them. So, we could construct a vector of output poly phase components here Y 0 m Y 1 m and so on up to Y m minus 1 m and we note that essentially this can be obtained now you know if I take any one of these. So, take Y 0 m for example, it involves G k 0 m summed over all k going from 1 to b. So, what kind of situation are we talking about we are talking about a vector of size m here and a matrix of size m cross p of essentially poly phase components of the synthesis filters poly phase components of the G k's and I could write down the l th row here this is called the synthesis poly phase matrix and let us write down the l th row. So, of course, note that l goes from 0 to m minus 1 the l th row of the synthesis poly phase matrix essentially G 0 l m z raise z raise to the m if now that depends on whether you know whether it is z whether the argument is z raise the power of m or z depends on the argument here for convenience let us assume the argument here is z raise the power of m. So, it would be z raise the power of m and this would go to b such branches. So, b minus 1 l m z raise to power m. So, now we have m such rows and each row has b elements look at the analysis poly phase matrix it has b rows with m elements in each row. So, overall the output poly phase matrix or poly phase vector as a function of z raise to the m is equal to the poly phase poly phase synthesis matrix of course, as a function of z raise the power m times the poly phase analysis matrix again as a function of z raise to the m times the input poly phase vector again as a function of z raise to the m sizes. This is well number of rows m cross 1 this is m cross b this is where it has b and again cross m and then m cross 1. So, of course, all this is going to be finally m cross 1. Now, what we have shown here is an overall structure an overall analysis of an m band filter bank with b branches in terms of the poly phase components. I would now like to spend a few minutes on discussing the modulation approach, but we shall deal with it in depth in the next lecture. So, we just wish to contrast the modulation approach with the poly phase approach and leave it at that for this lecture. So, we have seen the poly phase approach in detail. You see in the modulation approach what we do is as follows down sampling as a sum of modulations. So, in other words let me take the example of m equal to 2. Remember that down sampling by 2 followed by up sampling by 2 you know this process of down sampling by 2 followed by up sampling by 2 was treated as equivalent to multiplication by a sequence which is 1 at all multiples of 2 and 0 elsewhere. So, you have a sequence which is 1 0 1 0 alternately like this. So, together this was equivalent to multiplication by this. Now, in general for other m the corresponding multiplying sequence would be 1 at all multiples of capital M followed by m minus 1 0s in between again a 1 and then m minus 1 0s and so on. So, this periodic sequence in the modulation approach the idea is instead of decomposing the sequence in time we essentially treat the sequence as a sum of modulates and we combine the down and the up sampler when we treat it thus as a sum. Having done so we put down a different kind of matrix for the reconstruction treating it as a sum of modulates. In the next lecture we shall go further and delve deeper into the modulation approach and contrast it with the polyphase approach bringing out the differences and the similarities between the two and then we shall proceed to establish conditions for perfect reconstruction based on both of these approaches. Thank you.