 Now, coming back to the next possibility, so we have understood what a cascade structure is. We have seen the various possibilities, it is not one structure, it is several different structures. We now move on to what is called the cascade parallel structure and I will explain to you why we need to call it cascade parallel and not just parallel. Now, the idea is simple, in a cascade parallel structure you are trying to decompose the system function by addition. So, in the cascade structure you have become posted by multiplication, you express the rational system function as a product of several rational system functions each of which is causal. In a parallel or in a structure which a is to be parallel, you are trying to express the system function as a sum of different system functions. The idea is express H z equal to summation say q equal to 1 to capital Q H small q z. How we have to do this, in this particular let us take the same system function again here. So, let us go back to a system function that we are using, H z is 1 minus half z inverse into 1 minus 1 fourth z inverse, 1 minus 1 third z inverse square into 1 minus 1 fifth z inverse. And let us try and see in what ways we can decompose this as a sum. So, we could express this in one way as sum thing divided by 1 minus 1 third z inverse, now here is where the problem comes. You see you could either have this possibility 1 minus 1 third z inverse squared sum a 1 plus a 2 z inverse if you please actually you know you well let us see we may need to allow this possibility a 1 plus a 2 z inverse you know you need to have the possibility of 1 degree less. So, you know if this is of degree 2, you must allow a degree 1 term here and of course a 3 divided by 1 minus 1 fifth z inverse and this decomposition can be done by partial fractions. So, incidentally there are multiple forms of partial fraction decomposition, for example, the same system function could also be decomposed in the following way by partial fraction expansion. H z could have been written as sum a 1 divided by 1 minus 1 third z inverse the whole square plus a 2 divided by 1 minus 1 third z inverse plus a 3 divided or maybe we will call a 1 tilde they are not the same you see we should not write the same symbols a 1 tilde by so essentially only constants if you are insisting only on constants and this would be the form of the decomposition. But anyway what you notice is that we do have a degree 2 term here too we cannot avoid that degree 2 term I am assuming that all of us are now familiar with how to decompose a rational system function into its partial fraction terms I will spend time on that here one could look up a standard text on complex analysis or series solution. But the prior to be noted here is that whether we take this decomposition or we take this decomposition either of them this one or this one we cannot avoid this degree 2 term here. So in a way we do not gain too much by this structure we do have to realize a degree 2 term anyway and that is why we call it cascade parallel because this degree 2 term here it is degree 2 because this power is 2 if the power was higher the root had a higher multiplicity if for example suppose the root had a multiplicity of 4 then you would have a degree 4 term you could not do a degree 4 you see there is always this problem when you have repeated roots when you have repeated poles you have no alternative but to have a degree term of that degree in the partial fraction decomposition you cannot avoid it and that particular term can either be realized in cascade form or direct form 2 or a combination of them. So for example suppose we took this form of the decomposition a1 plus a2z inverse we kept the entire polynomial on the degree 2 term as we have here and this then we could realize it as fellow you see we could realize it as a sum of two functions and on this branch we could of course put the 1 minus 1 third z inverse squared here. So you have 2 third and minus 1 by 9 and on the forward branch of course you have a1 plus a2z inverse you have 2 z inverse here a1 and a2z inverse and the second term of course is easy to realize there is a one fifth here in feedback and nothing in feed forward except a3 so these can be combined and there we go to get y so you have x there and y here. So this is a cascade parallel form corresponding to this expansion here so a1 plus a2z inverse divided by 1 minus 1 third z inverse the whole squared if we call this the h tilde 1z has been realized here if we happen to call this h tilde 2 then we have realized h tilde 2 here and they have been added so that is simple. Now of course it is obvious if you want to realize this you could of course realize it in two ways one is realize this as a cascade structure you know. So here you know the main cascade parallel form is not quite clear why are we calling it cascade parallel because this is really a parallel form it is a parallel combination of two system functions but it is in this case that the cascade parallel idea will become clear you see in fact what you can do is to realize this term as a cascade of two terms which will help you then realize this term so let us do that. So we realize a cascade you know z inverse two such terms of the form 1 minus 1 third z inverse and now this is multiplied the output here is multiplied maybe you can put a node in between that will make it easier to understand we tap this off and multiply this by a 2 tilde and tap this off and multiply this by a 1 tilde and sum them and then of course the upper realization is as it is so we have a 3 tilde as it is in fact a 3 tilde is the same as a 3 in the previous expression that is not difficult to see. So the a 3 in this expression is the same as the a 3 tilde in this expression so there we are we have z inverse 1 by 5 and this and we have a 3 here so now we could add these two and that completes y for you. So this is where you need a cascade that is why we are saying there is a cascade and a parallel combination this is cascade this is parallel. So we could also realize it using a cascade you know here this would really in the true sense be a cascade parallel form you know and even though in the previous case we were able to make do with a sum of direct form two terms here we are actually using a cascade parallel form. So in general we use the term cascade parallel because when you have higher degree terms you could either realize them as a cascade or in direct form two. Anyway those are some of the decomposition realizations these are what are called obvious decomposition realizations. Now we have seen the disadvantage of these obvious decomposition realizations one of the main disadvantages is neither of them can guarantee a fixed unit degree structure being repeated. In the cascade form or in the parallel form the moment we have complex factors you need to pair complex conjugates and therefore degree 2 is what you need to be prepared for. In the parallel form if you have multiple poles and zeroes then you have no choice but to use a cascade parallel form or to use a direct form two term in the parallel realization in the partial fraction decomposition which is of higher degree than maybe even 2 to be degree 3 degree 4 depending on the multiplicity and of course where the tide if you have complex poles with multiplicity then for each complex pole the complex conjugate makes a degree 2 factor and the multiplicity multiplies at number 2 by whatever multiplicity there is. So where the tide is the same.