 A warm welcome to the 26th session of the third module of signals and systems. Now enough of just looking at sampling and reconstruction. We should be able to do something with the samples that we have created and we should be able to bring variations into the reconstruction process which will be of use to us. In fact, what we are going to embark upon in this session is a whole new branch of signal and system theory where we bring two seemingly disparate ideas and signals and systems together and therefore, we bring continuous and discrete systems together that is the theme that we wish to address. What do I mean by that? Notionally, I want to do the following. I have a continuous time system or a continuous independent variable system. It takes an input x t and produces an output y t. Now, this could be anything. It could be an analog filter, it could be some analog processor, it could be an equalizer in the analog domain, whatever you desire. We wish to construct here an equivalent system where the processing that the continuous system does. So, we want an equivalent system where the processing that you are doing can equivalently be done in discrete independent variable by using a discrete system. We do not want to rely entirely on continuous variable system. There are good reasons for it. You see continuous variable systems although they do deal with a wider range of possibilities also are much more inflexible. They are sometimes less robust, they are not easy to build upon and therefore, what we envisage is a situation like this which I am now going to show you. So, I take a continuous, I take the same continuous signal x t. When I say continuous signal, I mean continuous independent. So, let me write continuous variable signal that may be better. I pass it through a sampling mechanism and remember the sampling mechanism needs to be practical. It cannot be ideal. Now, I need to convert these samples into numbers which can be stored in a digital memory. Now, the output of this which is essentially a digital memory whereas, store all these samples in digital format can be given to a digital processor. What is called a discrete time or digital? We will see more of this you know as we go along. Here what I mean by a discrete time signal processor is something that takes the sequence that is coming. Now, remember there is a sequence created. You had a continuous independent variable signal, you have sampled it and therefore, you have now produced a sequence, a sequence of samples which have been stored in digital memory. And you can do what you like with the samples because now all the flexibility that a computational device can offer is at your doorstep. So, come back to this. So, you have a discrete time or a digital signal problem. Of course, there is a little difference by the way between discrete time and digital. Discrete time really means as the name suggests that you have discretized or you have made the independent variable discrete. Digital means you have also made the dependent variable discrete. We will come to that distinction shortly. We will note it down. But assuming that you have a sequence here now, you know you can think of this as a sequence. In fact, you can call this an input sequence. This discrete time or digital signal processor would now work on this input sequence and produce an output sequence. And now, this output sequence can be subjected to the action of a reconstructor, reconstructor from samples. And we have seen several reconstructors. This gives you back the analog output or as we said earlier the continuous variable output. Now, there are standard names for different parts of the system which I am now going to identify. So, here you have a sampling mechanism. Mind you a practical sampling mechanism. And you take these samples and put them into a digital memory. This whole thing is captured by what is called analog to digital conversion. And this reconstructor from samples that you have here is termed digital to analog conversion. In fact, there are also abbreviations. So, analog to digital conversion is often abbreviated by ADC and digital to analog conversion by DAC. So, you know you must have often heard about these things. Nowadays they are quite common. You must have heard that the ADC is not doing its job so well or the DAC you know in all these digital devices these terms are well known and well understood. But now, we are looking at it from the signal systems perspective. Sampling and reconstruction has a very important whole dimension of practical applications. In fact, in a way now go back to this drawing. What you have here is a fully new paradigm of how you are going to process signals. So, instead of doing the processing in continuous independent variable format, you are doing the processing in the discrete domain. Now, of course, all this assumes that you can keep fidelity in the system. That means, by doing all this business you have not lost what you would originally want to do with the continuous time processor. So, of course, what would you assume then what you would of course, assume that your signal is band limited. So, we can only do this with band limited signals. Now, we must also bring out the clear distinction between discrete time and digital. I have already said that briefly, but let us write that down. Discrete time is essentially discrete independent variable and digital is both discrete independent variable and discrete dependent variable. So, what it means is the memory has a limited resolution or a limited accuracy with which you can represent samples. This is not too difficult to understand. You see, we talk about 8 bit memory, 16 bit memory. Essentially, you are representing each sample in a certain finite number of bits. So, it is called finite precision representation. Well, in a calculator or in a computer, you all know that numbers are ultimately represented with finite precision. You really cannot have infinite precision of representation. You do not notice it, because that precision is so large. It is not difficult today to think of 16 bit or 32 bit computers with 32 bits. We can hardly see the inaccuracy caused by finite precision representation. And therefore, today when people talk about digital signal processing as we are doing here, they are not too careful to distinguish between discrete time or discrete independent variable signal processing and digital signal processing. That is because that loss of accuracy in finite precision representation of the samples is negligible. But you know, digital signal processing has been a revolution in the realm of signals and systems. Being able to do with a digital signal processor, what you would otherwise have to do with an analog processor is not only a great flexibility, it is a great power in terms of what you can do. So, let us put down the points. Why should we be doing all this? Why digital signal processing? Why should we process? Why should we do this equivalent discrete domain processing? Many reasons. Reason number one, you know, we had talked about flexibility or versatility. The same setup can do many different things. Second, robustness. The setup does not degrade quickly. You know, for example, analog systems sometimes are prone to degrade, components degrade in time. They lose their accuracy of values and so on. Now, this is not so common in digital systems. Computers can retain their ability to compute much longer than some analog components. And you know, you can protect them much more easily. You can also ensure. You can also, you see, it is easy to replace. I mean, after all, what you need there is just a computing device. What you are calling a digital signal processor is essentially a computing device, a device which can run programs, which would take in those samples or those numbers which you have stored in the memory and can give out appropriate numbers from the output. So, you know, the only weaker parts of the system are the analog to digital conversion and digital to analog conversion. But the beauty is that you can replace these things with relative ease as compared to trying to redo the whole analog system. In an analog system, of course, one can always argue you can replace components. But you know, this system is inherently modular. It is easier to replace a few non-working parts of this system rather than trying to tweak an analog system where you might have soldered components together or where you might have made intimate analog connections and so on. That is more difficult to change or to replace. So, there are very good reasons why we want to do this. Robustness, as we said, is another. The third is that there are certain things you can do here which you just cannot do with analog systems. So, of course, there is a difference between flexibility and this point that I have set. Abilities beyond analog system, we should call them. Some of those esoteric responses that you wanted, for example, with equalization are very difficult to achieve in the analog domain. Esoteric frequency responses, funny looking frequency responses that you want to achieve. If you want specific ones, I mean, you know, analog systems by virtue of their inaccuracy, of course, can give you funny looking responses, but that is not what I am talking about. I am talking about funny looking responses that you specifically want to design energy and achieve. Now, those are not easy to do with analog systems. On the other hand, in the discrete domain, they are not too difficult to achieve. At least, you can get as close to them as you desire without too much of difficulty, essentially with a little more programming work. Well, we are clear that there are many reasons why we should be doing what we have been talking about in this session and we will talk more about it in subsequent sessions. Thank you.