 A warm welcome to this first live interaction between me as the instructor and all of you as students and teaching associates. I am very happy to welcome all of you once again in person in the second course on signals and systems into module 3 where you are looking at sampling and reconstruction and you are going to bring together ideas from discrete and continuous systems. So, in fact, you must have slowly gotten a feel of what we are doing. We are looking at what we can do to reconstruct the signal from its samples. So, in effect what we are saying is in practice it is unlikely that we can really look at a signal all the time and try to study its characteristics and particularly if you are trying to deal with a signal using computing devices I can only sample the signal at certain chosen points and I can try and infer characteristics of a signal or what would happen to the signal when it goes through a system from these samples. Now, in keeping with that objective we have built up some ideas in the first week and I am very happy to see that some of you have responded with some interesting questions. In fact, let me now point out to some of the questions one by one that have been raised on the discussion forum. I have a question here I am going in order in chronological order of the questions that have been raised. So, I will begin with of course, there are several points of discussion that have been there on the discussion forum. I am focusing on the ones that pertain to technical questions or where people have tried to propose the answers to technical questions. So, the other administrative issues related to the examination and so on I am very happy to see that my teaching associates are doing an excellent job of sorting them out with you and I think that should be good enough. But then let us come to the technical questions. So, I see that a few days ago Raman 111 that is his name, this is email name has posted a question. What prior information do we have when we sample an audio CD and why do we choose 44100 hertz the sampling rate of an audio CD? Is it because of the prior information that we have? And in fact, you are lucky to see a very beautiful response from my teaching associates. First Pratik Fegde has responded explaining in some detail how one chooses that sampling frequency. In fact, he is very correct in saying that typically audio is restricted in its frequency content the Fourier transform of audio seldom goes beyond 20 kilohertz. So, in fact, even 20 kilohertz the rather high frequency in audio it is only the very high frequency note in audio that goes up to 20 kilohertz. Now, if there were 20 kilohertz then you will see that you require at least two times that as the sampling rate. So, you need at least 40 kilohertz the sampling rate and my teaching associates Sujath has said this in so many words that you would be studying this in greater detail and I strongly encourage Raman 111 to listen to that part of the video carefully and I am sure he will be able to respond with a better understanding after he looks at the reference which Pratik has given and after he listens to the lecture which Sujath refers to. Now, there are some other interesting questions let us look at to the question on a priori information. So, Texan Jimbo has posed a very interesting question what kind of a priori information is typically known and or determinable in real world situations. So, professor Gadre has hinted that the signal may be bandwidth limited which is relevant to the Nyquist sampling theorem this bandwidth limitation seems to be one kind of typically relevant a priori information, but what other kinds of a priori information would typically be of interest. My teaching associates Sujath Nair has very beautifully answered the question to some extent. So, he is talked about the example of images it is two-dimensional information. So, there is some part of the image which is smooth. So, in fact, if it is smooth enough to be a constant you need just one number to represent the whole region of smoothness that is an extreme case of a priori information in a certain region. So, what Sujath has also tried to point out is that you might be able to give different kinds of a priori information for different parts of an image the same could be true of a signal. So, you may not have a priori information all over the signal, but you might be able to say this part of the signal is say constant in which case one number represents it completely or this part of a signal is a segment of a sinusoid a sine wave. If you know the amplitude the frequency and the phase of the sine wave then you can draw the sine wave completely. So, three parameters constitute a sine wave and therefore, if you have that a priori information you need to ask for only three kinds of information further. Now, let me you know in fact ban limitedness is of course one kind of a priori information, but let me now take another question which another student has asked and I will relate this question by Texan Jimbo to the question asked by the other person who has made a query. So, you see one person has asked a question this is P K Sharma 143. So, he has asked a question in one of the video lectures. In fact, in week one a priori information and measurements simple examples. I have a question that in this video lecture there was an example of an exponential signal is the equation of a signal as given or the graph of the signal as given. If the equation of signal is as given then we cannot he thinks we cannot evaluate the value of a function at different intervals at t 1 and t 2 because there are two unknowns in the equation and for reconstructing we are evaluating those unknowns with the help of these samples. So, please tell me more about it Pratik Fedre my teaching associate answered this question and he is given a very good answer. In fact, let me say a little more about now I am going to relate these two questions the one asked by P K Sharma 143 and the one asked by Texan Jimbo earlier what other kinds of you know. So, for example suppose let me take the example of an RC circuit. So, suppose I knew that I am getting my signal from an RC circuit you know. So, I have a very short pulse which was applied to the RC circuit a very narrow pulse you could almost call it an impulse and you are observing the output of this RC circuit the output is going to be a decaying exponential. So, you can actually sketch the output here and you could model the output in the form if this is the time axis t some a e raised to power minus t by tau of course, you should be writing u t because it starts from t equal to 0. Now, these RC values typically have to do with certain parameters related to that circuit. So, you know this RC circuit could be a good model for a practical system it could be a small filter it could be some part of an audio system you know. So, my teaching associate Pratik has given a very interesting example you know he said that you can have a speaker and some part of a speaker could in a crude approximation behave like an RC circuit could have resistive and capacitive elements. So, you know that part or for that matter you know this even if you do not take the context of audio it could be some other context where you know you have typically you know a first order kind of response this is called a first order kind of response where you have essentially one exponential emerging. So, that one exponential situation is not impractical you know it is often true that you can model a system as being close to a first order system first order system is one with one time constant and that is what I mean when I say an exponentially decaying or exponentially growing waveform whatever it is. So, in that situation what I meant when I said use the a priori information is that you could just observe this at two points. Let us say you could choose two points any two points and you must of course record where you are observing them. So, you have two measurements now. So, a e raised to the power minus t 1 by tau and a e raised to the power minus t 2 by tau and we know t 1 and t 2 we also know these measurements. So, if I divide x 2 divide by x 1 I would get e raised to the power minus t 2 minus t 1 by tau where upon we could get log natural x 2 by x 1 or x x 2 minus x a log natural x 2 minus log natural x 1 that would give us minus t 1 minus t 2 rather minus t 2 minus t 1 by tau and you could take a negative sign. So, you see this is equal to minus t 2 minus t 1 by tau where upon tau is essentially minus t 2 minus t 1 upon log natural x 2 by x 1. So, you have an explicit expression for tau this gives you the time constant explicitly. Now, once we have the time constant with knowledge of tau we can now find a from here because you know t 1 and you know x 1 and now you also know tau from here. So, this is the way in which you would obtain the two parameters the amplitude and the exponentially decrease the time constant. Now, what I said in that part was that if I knew that the signal was of this form then these two measurements would immediately allow me to reconstruct the signal. And this is an example of a priori information that I might have if I know the circuit is an RC circuit or if it is a composite RC circuit is more than one stage I would know how many exponential parameters it has. So, for example, it could be a sum of two exponential or exponentially decaying functions even then I can formulate a generalized approach starting from this idea and I could reconstruct a signal from a few samples. So, this answers the question you know what other kinds of a priori information might I have you know where I might and this is not atypical you see when you have systems that are closely modeled with closely modeled by LCR like circuits, circuits which have inductances resistance and capacitances. Then essentially you would expect responses which are either exponentials or exponentials modulated by sinusoids. So, this is not imprises a situation which can occur in practice or at least approximately be true in practice. So, that is to do with this example that we talked about. If it is still not clear I would be very happy P. K. Sharma and Texan Jimbo posed further queries or put further comments and observations on the discussion forum. Now, I also see to my very pleasant surprise that Raman 111 has posed a very important solution to the question that I have given. You know in the first week I have talked about adding the original sinusoid to the impostors and I believe Raman is going in the right direction. Raman so what you should try to do is now to take this further the reasoning is correct you must take it further and in fact you know you try and add more and more terms and see how the summation comes at a finely chosen grid. You know what I mean is take the sampling points and take a large number of points between the sampling points and evaluate the sum of these original sinusoid and imposter terms all together at a very finely chosen grid. So, large number of points between samples and choose a few sampling intervals. So, you will see what is happening and you are going in the right direction. If there are small errors they will get corrected as you go along. So, well done it is good to see that and I am hoping that other people will also contribute to this discussion here. Well, I believe for this week these are the questions at the technical level which have been posed. I am very happy to see that this technical discussion is taking place. Now, there is one which I am going to take up in some depth yes here it is. So, ETK 150549 has brought out a very important issue and of course, I have traced the whole discussion there. The discussion starts with the you know what has been raised here by ETK 15049 549 is the issue of what happens when you take all sinusoids with the same samples. There was some confusion you know about the idea of measuring the difference between two samples in terms of the phase change and measuring it in terms of time. Now, of course, to cut a long story short my teaching associates have actually given a very thorough explanation and I am glad to see that ETK 150549 has understood the answer, but I also wanted everybody to look at this discussion carefully because if you remember in the email I sent in the beginning of the course I said you must critically review you know though the general idea is correct you must be convinced about the nitty gritties of what is going on. So, which are the sinusoids which all have the same samples the same sampling points and why are these imposters created? This should be appreciated. I am insisting on your appreciating this idea of imposters because otherwise very often you know in a formal proof of the sampling theorem this insight is omitted. So, I would be happier if all of you obtain this insight of how several sinusoids actually have the same samples at the points of sampling and all of them when they come together they create impulses of the points of sampling and all other points they have destructive interference that is all of them together destroy one another that is interesting. If you just take a few of those sine wave they may not come to destructive interference between the samples, but all of them together become destructive between the samples. This needs to be appreciated and you know I would strongly recommend that you see that discussion because it could be a confusion in the minds of many students. So, I am very glad that ETK 150549 has taken it up and discussed this whole question of which sinusoids and how you measure the what is the interpretation of measuring the phase difference between sampling points and so on. So, well done. I believe that is enough for this discussion session. I am sure that with this discussion that we have had this live interaction that we have had many of you will be inspired to post more queries and more comments and more discussions and observations and more responses to on the discussion forum and I look forward to that and to talk into you again after a few days. I hope you are all enjoying the course. We are all very happy to have you and goodbye. See you in the next response session. Of course, see you in between in the regular videos. Thank you so much.