 a warm welcome to the 23rd session of the third module of signals and systems. We have seen something very interesting in the last few sessions, namely that sampling essentially amounts to multiplying an input signal by a periodic waveform, so simple. In fact, we wonder why we did start the whole discussion on sampling with such a simple interpretation. Well, there was an intent behind it, because sampling is deeper than just ideal sampling or sampling with pulses. Sampling essentially is a philosophical question of whether you can represent a signal more efficiently than in its raw form by representing it as a function of every point in time or every point on the independent variable. And we did begin the discussion that way and carried it out to a certain extent and finally, looked at what is conventionally understood to be sampling or sampling and reconstruction in the context of signal tossing. Now, one thing that we would now like to bring in is a little beyond sampling, see after you sample would you like to just let the sample waveform drop to 0 or would you like to hold on to it. So, the idea of holding on or smoothly joining samples is a new idea that we wish to bring in. So, you see we have been talking about sampling and reconstruction, in a way we need to talk about sampling and interpolation, let us write that down clearly here. So, sampling and interpolation, interpolation really means interpolate means to fill up the region in between to complete the waveform in between samples. Now, what approaches can we take to interpolate the waveform, let us see there are several different possibilities. So, let me illustrate graphically, let us take the smooth waveform that we have and let me choose to sample you remember we are always going to write sample that every multiple of T s and we should choose a few negative multiples to otherwise it gives the impression sampling starts at 0 which is not quite correct. So, minus 2 T s minus T s and then several positive values of T s. So, you have this waveform x of t that I have shown you with t as the time passes and now I start taking samples at every point, every point of sampling. So, here we are here we have this sample here. Now, the issue is you have this sample here and you have this sample here, what do we want to put in between that is the question of interpolation. What is it that we want to put in between samples? Do we want to hold on to the previous sample until we reach the next and then change over to the next, if we do that the waveform will look like this let us show that. So, here if we do what is called a sample and hold of course, this is just you know. So, this is this is the waveform. So, if we do a sample and hold we get the waveform which I am now showing in green you know in bold green. So, the bold green waveform that we have here is essentially a sampled and held output or the output of a sample and hold circuit a sample and hold device. Now, you could have many other possibilities. I shall show one more possibility in red on the same graph. So, we could have this. So, the red waveform essentially what is called a linear interpolate. I am drawn you know you can think of them as straight line joining the points that you have obtained by sample. Of course, you could have other kinds of interpolation and when you interpolate it is not necessary that you should just consider two at a time. So, you could for example, want to look at one sample and the next two samples and make a smooth joining of all of them and keep doing this for you know the next three samples 1 plus 2, 1 plus 2 and so on. So, you know you could have more advanced forms of interpolation. At the moment what we want to do is to analyze what would happen if we sampled and held because this would then give us an idea how to handle this whole idea of interpolating of filling up the gaps between samples. So, let us now analyze a sample and hold circuit or a sample and hold mechanism from a spectral perspective. Can we think of this whole business of sampling and holding as a set of systems. So, systemically what is it? Essentially we could think of a sample and hold as a cascade or as a series connection of two systems. So, you have the signal to be sampled, you have an ideal sampler here and then you have a linear shift invariant system. The impulse response of this linear shift invariant system is essentially a pulse, but remember here a pulse lasting all through the sample interval. So, here the pulse that we are talking about is not that short pulse which is going towards an impulse. We are talking about this pulse which begins at 0 and lasts until t s. Now, let us analyze this cascade that you saw and let us confirm, let us verify or convince ourselves that indeed this leads to what we expected to namely it leads to a sampling and held waveform. So, let us mark the outputs everywhere. Let us call this signal x t. Let us call the output of the of course, the ideal sampler let us say the sampling interval is t s. Let us call this x s t for sampled and let us call the output here x h t h for held. Let us draw this for each specific sample point by assuming some random x t. So, we have a simple signal let us draw it like this. Let us mark the sampling points. Let us draw the output now of just the ideal sampler. So, here you must understand ideal impulse is coming in. Now, you know whenever you hold, so whenever you pass this x s t through the linear shift invariant system that we have shown with impulse response h t what happens? Each impulse here is replaced by that pulse that we showed. So, on passing x s t through h t it is very simple. A linear shift invariant system when you say the impulse response of a linear shift invariant system you are essentially saying that the system response to each impulse with its impulse response. So, this impulse here at minus t s is going to result in a pulse from here to here. Similarly, this impulse here results in a pulse of course, scaled appropriately and so to this and so on. So, the green one is the output x h t. So, you see the good thing is we now have a systemic description what really is meant by sampling and holding. And now a frequency domain analysis is very easy. Let us do it. We already know how to handle sampling. Now we need to find out what happens when you pass through the LSI system. So, in the frequency domain passing through that linear shift invariant system essentially means multiplying the Fourier transform of x s t with the frequency response of that system. That system is clearly stable because its impulse response is absolutely integrable. So, let us write that down. Let us find the frequency response. This is a very easy frequency response to evaluate. Essentially, it would be a sine x by x kind of function or a sine by variable kind of function, sinc function as we know it. Let us evaluate it. As usual, we can make a slight replacement here rearrange the terms as they say. So, you know this part can be rewritten in terms of sine and we could now simplify it. Let us do that. So, it will be easiest for us to multiply by T s by 2 and multiply by 2 by T or T s by 2 in the numerator as well and we can now simplify this. So, we will take the 2 from here and put it along with this. And thus we get T s raised to the power minus j omega T s by 2 sine omega T s by 2 by omega T s by 2. So, here if we put omega is 2 pi f, then we get essentially pi f T s here. You will recall that we have seen this before. It is not entirely new to us. Now, this is the frequency response. So, it is clear. We are now well set to do the frequency domain analysis. We have the ideal sampler for which I know the Fourier domain analysis. I have this linear shift invariant system for which I again know the analysis. Let me put them together and we will put them together in the session that is to come. We will meet again in the 24th session. Thank you.