 Welcome to the 32nd lecture on the subject of digital signal processing and its applications. You will recall that in the previous lecture we had discussed the realization of rational causal system functions. We had looked at the direct form 1 and direct form 2 of realization in both of which you were essentially taking the rational causal system function as it were and translating it into a hardware software structure. We had talked about signal flow graphs and we had shown how to draw a signal flow graph corresponding to a realization. In fact, there was a very simple relationship between a signal flow graph and a translation into hardware which was almost obvious and even into software which required little more work but was not too difficult either. What we do today is to look at other forms of realization where we use decomposition in one way or the other. So far we have not decomposed the rational system function at all. We have just taken it as it is and translated it into a signal flow graph and thereby into a hardware structure if we so desire or a software program if that is more convenient. Let us tell you list the three structures that we are going to study in greater depth in the lecture today. We are going to look at a cascade structure first. In a cascade structure we are going to have a cascade as it were of system functions something like this and we did not give a hint to how you would do this in the previous lecture. We went into direct form 2 by using the idea of a cascade but a very simple cascade. The second structure that we would like to look at is a cascade parallel. So essentially a structure something like this a combination an additive combination of cascade. You will see why we need to call it a cascade parallel structure. We could have been content with calling it a parallel structure but there are situations where we must specify cascade parallel we understand why. Finally we look at what is called a lattice structure. A lattice structure is a combination of the ideas of cascade and parallel. So you could call it a cascade parallel structure in some generalized sense but it is quite different really from both of them. In fact the beauty of a lattice structure is that in addition to realization it exhibits certain properties of the system. And that is why we study the lattice structure in its own right. So it is essentially a uniform repetition of a pre-specified unit which also exhibits the system properties explicitly. We shall now look at the first of these the cascade structure. Now the idea of the cascade structure as we would expect is that a cascade implies the output of the previous stage being given as the input to the next and this continues until we reach the final output. Obviously when you connect linear shift invariant systems in cascade we have studied this right in the beginning of the course. When we connect linear shift invariant systems in cascade one of the things that we are sure is that they can be interchanged without any change in the overall input output relationship. It is very clear that if you have unequal elements in the cascade what I mean by that is if the different parts of the cascade are not identical there are at least multiple realizations just based on this fact. That means the cascade realization is not going to be unique unless of course all the cascade elements are identical. Just an interchange, just a permutation of them will give you a different structure. Let us take an example. Now let us assume that we have a system, a rational system function that looks like this 1 minus half z inverse into 1 minus 1 fourth z inverse divided by 1 minus 1 third z inverse the whole squared into 1 minus 1 fifth z inverse. Let us assume this is hz this corresponds to a rational causal system function. Now there are several different ways of realizing this in cascade. I will take one as the first example. We could choose to divide this system function keeping the repeated pole together. Now hz could be written as h1z times h2z where h1z is 1 minus half z inverse divided by 1 minus 1 third z inverse the whole squared and h2z is 1 minus 1 fourth z inverse divided by 1 minus 1 fifth z inverse. Obviously their product yields hz. In fact what I have done in this is to keep the rational system function in factored form. You know you could have of course you know how to express it in factored form. That is not difficult. All that you need to do that is not always very easy is to find the poles to take complex conjugate pairs together. So here of course we have ensured all the poles and zeros are real but that is not necessary. You could have complex conjugate poles and then or similarly complex conjugate zeros and if you want to ensure real coefficients these must be kept together. Anyway we could then realize each of these in either direct form 1 or direct form 2. So let us just do one as an example. So let us write h1z in expanded form. 1 minus half z inverse divided by 1 minus 1 third z inverse squared would become 1 minus 2 third z inverse plus 1 by 9 z to the power minus 2. And h2z is 1 minus 1 fourth z inverse divided by 1 minus 1 fifth z inverse where upon a cascade structure would look something like this. We will realize them in direct form 2. So cascade would be essentially take h2z first and then h1z and that would essentially look like this draw a signal flow graph corresponding to h2z. You have a z inverse there you need only 1 a feedback of 1 by 5 and 1 and minus 1 by 4 that is h2 for you and then we have h1. In h1 you have a feedback of 2 by 3 and minus 1 by 9 and then a feed forward of 1 and minus half and there we go. This is the cascade structure corresponding to h2z followed by h1z. We have realized each of these in direct form 2 one and this one but we could realize them in direct form 1 if we desire though that may not be economical. In fact in a way that is splitting the cascade even further. If you think about direct form 1 even direct form 1 is a split of the numerator and denominator. So we could realize direct form 1 2 there but that is not economical in terms of feelings. Now this is not the only cascade we could also of course the simplest alternative is to exchange h2 and h1. So we could also have h1z followed by h2z and that gives me a different cascade structure. So for example express hz as h3z times h4z and what we do is not to keep the poles at 1 by 3 together. So we might go back to the original system function here. I have this I do not need to keep the 1 by 3 poles together. I can take 1 1 by 3 pole and 1 1 by 5 pole and make that one system function and I can keep the other 1 by 3 pole aside. So we could for example have h3z is of the form 1 minus half z inverse divided by 1 minus 1 third z inverse times 1 minus 1 fifth z inverse and h4z in that case becomes 1 minus 1 fourth z inverse divided by 1 minus 1 third z inverse. That is another variation. Now of course here again you get 2 variations because you could order it as h3 times h4 or h4 times h3. Now you could have still other variations. You could have the following possibility. hz could be written as h5z times h6z times h7z where h5z can be simply 1 minus half z inverse by 1 minus 1 third z inverse. h6z can be simply 1 by 1 minus 1 third z inverse and h7z could be 1 minus 1 fourth z inverse divided by 1 minus 1 fifth z inverse. And this gives me other possibility. And now again you can get rearrangements of these. These are all distinct as system functions. So you could rearrange them again in any manner that you desire. So you get several more combinations. Now in fact you can see the trick that I am playing here. What I am doing is to decide on an association of zeros and poles. Of course here as I said I have all real poles and zeros. However you know the matter becomes simpler. But if I have complex conjugates let me emphasize and I want real coefficients. I must keep complex conjugates together. So that means in a cascade the smallest order the smallest number of delays that I can use in one stage tends to be 2 at times. Because when you keep complex conjugate poles together you would get a degree 2 term. What would a term look like if you kept 2 complex conjugates? You know whenever you have a term of the form 1 minus the r cos or r e raised to the power j theta times z inverse into 1 minus r e raised to the power minus j theta times z inverse. This is a pair of complex conjugate factors. It becomes 1 minus 2 times r cos theta z inverse plus r squared z raised to the power minus 2. This is the typical term that you would get when you pair complex conjugate factors together. And we do need to do it so you would sometimes not be able to go below degree 2. But you can definitely go to degree 2 that we are assured. So in a cascade if you want to be very economical in the structure of one stage you must allow for a degree 2 stage. Degree means the number of delays that are involved there. You must allow for a degree 2 stage. But of course if you have only real terms then you can even make do with a degree 1 stage. You see this is where we will later see that the lattice structure is attractive. In the lattice structure in a much wider class of rational system functions you can make do with a degree 1 stage. And so although in many ways lattice structure is also like a cascade structure or somewhat like a parallel structure it has this important advantage.