 Welcome to the third session of the third module of the course signals and systems part 2. Here we look at what we did in the previous session from a slightly different angle. Let us recapitulate in a couple of seconds what we did in the previous session. We were talking about the so-called conflict or the so-called complementarity of what is called the a priori information and the measurements that we make. So, for example, we said if you actually knew that the signal given to you is of the form A times e raised to the power alpha t that is it is an exponential signal then it is adequate to make just two measurements say at t 1 and t 2 of course t 1 is not equal to t 2 that is trivial. So, you have x at t 1 is A e raised to the power alpha t 1 and x at t 2 is A e raised to the power alpha t 2 and we can obtain A and alpha from here we saw that last time and therefore, characterize x t completely now let us catch the situation that we are in. So, what we are saying is we have this exponential signal where it could be a growing exponential or decaying exponential at the moment I am showing a decaying exponential. So, alpha is negative real of course remember alpha could also be complex in general, but for the moment let us make things simple that alpha be negative real and t 1 happens to be here and t 2 here. So, essentially I am saying I have made these two measurements here and now I am asking what is in between and what is elsewhere this is x at t 1 and this is x at t 2. So, what is in between now we have actually filled up all that is in between and all that is elsewhere based on a priori information, but then now let us ask the question a little differently suppose we only knew these two samples, so this is given to us suppose we did not know the a priori form. So, we do not know what the actual expression for x t is then what are the possibilities of reconstruction in fact the problem is not that there are no possibilities the problem is there are just too many of them and I will show you a few. So, you could reconstruct the signal by connecting this like this and then maybe continuing it like that or you could reconstruct the signal by connecting these maybe something like this and then continuing like this. In fact, there is an infinity of possibilities, so you see when you have just the samples here or just the measurements as you might call them, but not the a priori information. What is the difficulty? The difficulty is not that you cannot reconstruct something you know this is very typically like what we call a crossword. In a crossword if you just took one of the rows of a crossword and did not quite know about the columns and you know that there are a few letters provided there and several missing there are several the problem is not that you cannot fill up a word which would match that description the problem is there are just too many of them and you have to arrive at the right one. So, in a certain sense the problem of sampling and reconstruction from samples is like solving a crossword the clues of a crossword are like the a priori information or well you know you have letters provided to you in the partial crossword and you have clues you could think of the clues as a priori information and you could think of the letters provided as the samples and together this must give you back a unique solution to the crossword normally people have designed these crosswords very intelligently, so you do not have too many solutions to a crossword. Now, here too something similar is happening you have taken two samples if you have this a priori information is very strong a priori information that the signal is exponential and there is no choice there is only one way in which you can reconstruct that it passes through those samples. On the other hand if you know nothing at all about the signal then you can see there is an infinity of possibilities in fact now we could go a step further and ask how big is that infinity. I have been saying you know very casually tongue in cheek that infinities have different grades too for example there is an infinity of the natural numbers how many or the infinity of the integers how many integers are there that is one infinity 1 2 3 4 and so on you know minus 1 minus 2 by the way the infinity of the integers the infinity of the natural numbers the positive number positive even numbers positive odd numbers are all the same infinity and then there is one infinity next to it called the infinity of the reals and the infinity of the reals is definitely more than the infinity of the or bigger or higher in status than the infinity of the integers it is a bigger infinity of so to speak bigger in the sense that if you take the real numbers you cannot put them in one to one correspondence with the integers or the natural numbers you will always have to leave out some real number that can be shown I am not going into those detail and just trying to give you a feel of what kind of infinity we are talking about here. Now here we are talking about an infinity look at this you know here we are talking about infinity where at every point you need to define a value at two of the points which I am ticking off here you have a value defined but at every other point you have full freedom to put down the value now it turns out that this infinity is bigger than the infinity of the reals bigger infinity than the reals well you know it is not so important to worry about this infinity and the class of infinities except for the fact that you must understand that there is this degree of docking uniqueness in the problem the problem is not that you cannot obtain a solution the problem is that there are just too many of these solutions and how many is what I am trying to bring your notice to too many of them and infinity even bigger than the reals that is the number of solutions you have to the interpolation problem to the reconstruction problem. Now what is required is to be able to choose from among these so the question of whether you are losing something in sampling and whether you will be able to reconstruct the signal from the samples is akin to asking like in a crossword can I fill up that crossword uniquely from the clues and the letters inside of course in some crosswords they do not give you any letters at all there are just clues that I am talking about a crossword where some letter has been put here and there you know some people play this game called Sudoku in a way in Sudoku also you are doing something similar you are putting a sprinkling a few numbers a few numbers from 0 to 9 or 1 to 9 all over the board and that is in some sense the samples that you are given and you are given certain a priori information the a priori information is about not repeating numbers across and things like that you are given certain rules those rules form the a priori information and then the Sudoku game is to reconstruct all the numbers on that board based on the few sprinkled numbers that you have and of course if it is constructed properly there is just one solution think of sampling and reconstruction as a much bigger version of that. It is a game between a priori information and samples that are provided and you are required to reconstruct how well can you reconstruct or how uniquely can you reconstruct is essentially based on whether your a priori information completely complements the samples that you have knowing the a priori information and knowing the samples is there just one possibility left. Now we had agreed in the previous module all over module 2 that one class of representations for signals is based on sinusoids sin waves. So we also agreed that there is a wide class of signals which can be represented with the Fourier transform and typically those signals are the ones which in some sense do not become unbounded all over a certain infinite region of the real axis and so on. So you know those things are required you cannot if a signal is unbounded for an infinite part of the real axis it cannot have a Fourier transform. So with under certain circumstances the Fourier transform exists and that means you are essentially thinking of the signal as comprising of sinusoids the sine waves. So we can now naturally ask a question. Suppose I just take one of the components of a signal just take one sine wave from the signal and focus my attention upon it. If I take samples of the signal and you are thinking of the signal as a sum of sine waves obviously I am also thinking of each sine wave being sampled and these samples are added point by point. So in the next session we will now answer the following question. Suppose I took just one sine wave and I sampled it I know that means the a priori information is I know it is a sine wave. So I have sampled a sine wave. If I sample a sine wave what is the ambiguity that I create and in one more sense of the term maybe in fact taking it a step further. Can I play with that ambiguity actually to see what I would get if I just kept pulses where the samples are and nothing else where. You will understand more of this when we go to the next session but basically we are going to look at a sine wave which is going to get sampled in the next session. Thank you.