 So much so then for design, we now come to the next part of our endeavor in building discrete systems namely realization. And of course, we have talked about realization before, we are not entirely new to the subject. Realization means translating a given system function into a set of components and their interconnections. So, let us write that down. Realization means translating a system description or a system function into a hardware or software and say hardware or software or a combination of the two. Because in today's world, it is largely software that is used. Typically what is, in fact that is the whole advantage of discrete time systems. You see, you could set up one hardware system and then use the software to realize different filters that is the beauty of discrete time signal processing. You cannot do that with analog processing. In analog processing, you could perhaps conceive of a generic structure which would realize a few classes of filters. But they are going to be very restricted in the class that they realize. In contrast, for discrete time systems, one hardware setup, what would that hardware setup really comprise of, let us see, typical hardware setup. So, you have the analog signal and analog to digital converter or a sampler, they are really synonymous for all practical purposes which gives us a sequence or stream of samples, sample train. Typically, the sample train is given as an input to a discrete time signal processor or a DSP as they call it, it is also called a digital signal processor. The output of a digital signal processor is then given to what is called a digital to analog converter and there we get the processed signal out. Now, you see the beauty of this is that the essential setup is the same. It does not matter what we want to do here. We could be doing a discrete time filter here, we could be doing some non-linear operations here, we could be doing a combination of the two, but we are essentially working with this setup all the time. This setup works very well for us and this is really the typical hardware setup and the software inside this tells, gives us full flexibility on what exactly we wish to do to the discrete time sequence that we obtain after sample, is that right? So that is the reason why discrete time processing is attractive. Now of course, although there is flexibility and you know there is also versatility, so the same thing can do many different operations, but versatility is not without structure. So we need to put down a systematic process even though for translating the given system description to a realization and we will now do that. We will begin to do that today and we shall continue to do that as we proceed in the subsequent lectures. Now let us take a typical system function, so let us take the system function given by yz by xz is equal to summation m going from 0 to capital M, bm z raised to the power minus m divided by 1 minus summation n going from 1 to capital N an z raised to the power minus n. This is a typical system function, rational system function which we would obtain. Now of course if the system is FIR then we have no denominator, so all the a's would be 0 and moreover if the system you know we are assuming the function is also causal, we are not assuming it is stable of course, but we are assuming it is causal. So if it is causal then you can always put it in this form, I leave it to you to prove that. If the rational system function is causal then we can always write it in this form. Now you have to realize this, so one simple and straight forward way is to just cross multiply. So we have yz into 1 minus summation n going from 1 to N not 0, 1 to N an z raised to the power minus n is xz summation m going from 0 to capital M bm z raised to the power minus m which translates in time, this translates to yn is equal to summation n going from 1 to capital N an y, now maybe we should use a different symbol because in time n can get confused. So we will write l here instead, a l yn minus l plus summation m going from 0 to capital M bm xn minus m because now the time index can get confused, so we will use a different symbol l. Now it is a very simple way in which we can create this system, in fact what we could do is use what is called a signal flow graph, now I shall just introduce the idea of a signal flow graph. A signal flow graph is a way of representing a realization, in fact we have seen a little bit about signal flow graphs before but we will now formally introduce signal flow graphs for this course. You see a signal flow graph is a collection of what are called nodes and directed edges. The directed edges should be thought of as trucks which have some processing machinery inside and then a depositing machinery on them. So each edge is like a truck with some machine located on the truck and a delivery mechanism. So it starts, so you know when you have a directed and nodes are like go downs. So you know a truck takes something from the go down, processes it in some way as specified on the edge and then delivers it to the place where it ends. The processing can be as simple as multiplication or it could be delaying. Now what it means for example is that if you have these two nodes, node 1 and node 2 and you have node 3 there and you have two directed edges like this. On this directed edge we write Z inverse and on this directed edge we write 3. The meaning of this is you have two trucks moving from this station to this station and from this station to this station. This truck carries whatever is present on this node, delays it, Z inverse means a delay. Z inverse is a Z transform of a system delay by one sample. So the truck carries whatever is there on node 1, delays it by one sample and deposits it on node 3. The second truck which corresponds to this edge carries whatever is on node 2, multiplies it by 3 and deposits it on node 3. Now the beauty of the signal flow graph is that no matter how many trucks take away the material from a station, the material at that station is unaffected. And moreover although what is present at a given station is the sum of all the trucks which deposit at that station, no effect is felt for as many trucks as take away from that station. So there is no law of conservation there. There are some stations which are permanent sources, that is they have no trucks coming to them and there are some stations which are permanent sinks, so they have no trucks going away and there are some stations which have some trucks coming in and some trucks going away. We shall see more about signal flow graphs in the next lecture. We shall see how to evolve a general philosophy for realization of discrete time system starting from the system function.