 A warm welcome to the second session of live response that I wish to give to all the participants in this course. I am very happy to see that after we had the previous response session, a lot of our participants have begun to post technical questions on the discussion forum and I am equally happy to see that there has been correspondence between participants and also correspondence between our participants and our teaching associates. So, you know there has been some very good technical interaction and I wish to point out a few good questions and technical interaction that has taken place and also respond in some detail to some of the questions that have been. So, let me now switch straight away to the interface where I am looking at some of the questions. So, I am going by order of time that means I am going from the earliest question to the most recent question. Now, one of the questions you know just around the time when I did my last response session was by Raman 111 and I am showing his post. It says I would wish to share a little bit of the three questions by Professor Gadre that I could evaluate. So, basically he is talking about what happens when you add the original sinusoid to the imposter terms, to the unwanted terms and he is tried to show that there is constructive interference at the points of sampling and destructive interference at the other points. Now, I can see that he is proceeding correctly, but I am a little sorry to see that nobody has responded to his post. So, I would like some of you to respond to this post. You know look at what he is doing or share your experience of summing the imposter terms with the original sinusoid and tell us what you get. Now, I have to some extent shown you what happens, how you sum those terms. In fact, I will give you a hint of what you are going to see. What you are going to see is as you sum more and more of those imposter terms, you will see a concentration of the sum around the points of sampling and other than at the points of sampling you will see a slow decay. Of course, this decay will not be monotonic it will be oscillatory. So, there will be a lot of oscillations in between that is because you are summing sinusoids. So, you will have a lot of oscillations in between, but the oscillations will keep dying down in amplitude and finally, you will see impulses being reached at the points of sampling. So, I have given you in some sense what you expect, but you must prove it formally by using the sum of the imposter terms. I would like more of you to respond to this post by Raman 111. In other words, I would like you to actually work out what these sums of imposter terms will give you and then we will discuss it further. You know, I am taking the questions which have technical content. Of course, there are questions based on the quizzes and the tests and my teaching associates are actively looking at whatever inputs you have as far as the quizzes and tests are concerned. I do not think I should spend time in responding to those here, but let us come to the next question which was technical in nature. In fact, there are some very interesting posts which have come in response. So, you know Yadunandana says Professor Gadre in the a prior information lecture explained that for the signal 5 cos 4 t plus 5, we do not need samples at all for reconstructing the signal. Now, it has nothing to do specifically with 5 cos 4 t plus 5. What I was trying to say there is that the moment you specify the signal completely or in other words, in particular if I took a sine wave and specify the sine wave completely, there is nothing to be sampled about the signal. You know the signal in its entirety, so no sampling is required. So, you know what is not often brought out so clearly in some references and signals and systems is that sampling is a game between a prior information and the need to measure. So, you know typically it is about band limited signals or band pass signals that people talk about and then you evolve a sampling strategy for them, but band limitedness or band pass character are one kind of a prior information. The deeper issue is that when you have a prior information or if you have more of a prior information, you need correspondingly less of additional measurement and sampling is a kind of measurement. So, this point was being brought out. I am very happy to see that my teaching associates have answered this very effectively Pratik has answered it Pratik Sathe and then my other teaching associate Sujaath has also given a response and I am also glad to see that Yadunandana has also understood you know what was being discussed by the teaching associate and he has given a very correct response. He says I want to compare this with the practical situations when he takes a system where we are using an analog digital converter and if you know exactly what you are acquiring there is no need to sample it at all that is correct. So, if you know exactly what you are acquiring there is no need to acquire it that is one way to look at it. So, and Sujaath and Pratik have given a response to that. In fact, Sujaath has said that in a signal generator if you know exactly where to tune suppose you put your knob at so many hertz of frequency or so many kilo hertz of frequency in this amplitude and phase then you know exactly what you are generating. So, there is no problem anyway this was a good response. Now, let us see some of the other technical questions. Well, this is of course, not so technical question, but the interesting response that I got D Kohut 1 he just started watching some of the lectures of week 4 and we talked little bit about electronics and circuits. Now, in fact, I did that because I want to show you that you can relate a lot of what you are studying in signal systems to actual circuits and electronics and of course, he is encourages to give us give a course on electronics. So, we will think about that it is a very interesting suggestion we will think about it seriously. Now, a very important question by Abhisad. So, which domain is best suited to see the sampling process what aspects make time and frequency different and how do we choose which domain is best suited for a particular situation elaborate with an example. Now, I had in fact, given a partial response myself to the question I had told Abhisad that if you really wish to understand which situation which domain is better it also has to do with the context in what you want to understand. So, you know I had given in the example of a fan now you know you must have noticed that when you switch on a fan and when it rotates slowly it seems to rotate in the correct direction and after a while when you start increasing its speed and you keep looking at it it seems to actually even if you are increasing the speeds it seems to rotate slowly in the opposite direction. Now, this is directly a consequence of aliasing and to understand this it is better to use what we call the natural domain or the time domain. So, let me now put this down in terms of you know drawing. So, let us take for example, a wheel like this and let us put spokes on the wheels from 0, 1, 2 and so on. So, I put 24 spokes on the wheel. Now, let us assume that you are causing this wheel to rotate in the anticlockwise direction or other it is clockwise here that is all it does not matter really. Now, you see suppose the wheel is rotating very slowly and you have fixed your gaze here. What would you see in sequence? We would see 0 followed by 23 followed by 22 and then 21 and that keeps going slowly. Now, suppose we rotate it faster. So, you know remember when we fix our gaze at a point our eyes do not look at that point continuously. In fact, our eyes in a way all the time sample and this is why we have what is called persistence of vision. So, when a movie is recorded you only need a certain number of frames per second because in between those frames the eye interpolates. So, this is already sampling taking place in our visual process. So, here too in this case if we fix our gaze at that point and we start rotating faster maybe we will see 0 followed by let us say suppose you skip 2 and then see the third. So, 21 skip 2 and see the third. So, 18 and then skip 2 and then it comes to 15 and so on 12 and then so on. Now, suppose we go even faster you can go faster and faster and faster and then you will come to a point where you see 0 followed by the one exactly opposite to 0. So, let us take that situation again let me show that in blue even fast you see 0 followed by let us say 12 and then followed by 0 and 12 and so on you know. So, you see this. So, what are we seeing here this is a point where confusion begins because you see 0 followed by 12 followed by 0 followed by 12 and then you do not quite know whether the wheel was rotating in the direction that I have shown you or whether it was rotating in the opposite direction in both cases you would see 0 followed by 12. So, it is a point where confusion just begins and now if you rotate the wheel even faster. So, I am going to show that case in black even fast and I am going to write inside we will see 0 and then all this will get skipped all this you see 11 next and then all this will get skipped and we will see 22 next you know. So, it gives the impression that you are seeing 0 followed by a number on the right half side and then followed by a number on the left half side and the I would tend to feel that the wheel is rotating in the opposite direction with a lower velocity rather than in the correct direction with a higher velocity. In fact, if you go even faster perhaps you might see 0 followed by one of the numbers here let us say 5. So, you have gone through all this between two strokes of the I you know. So, you have just skipped out 6 steps. So, after the 24 you have skipped out 6. So, you have covered 18 steps now you go 18 more steps here. So, when it rotates like this you know you are seeing 0 followed by 5 and then you go 18 more steps from 5. So, you see something so you know you can now visualize this when the wheel rotates faster and faster it actually appears to bring numbers from the opposite side as if the wheel were rotating in the opposite direction at a much lower velocity you understand. So, this is best understood in the natural domain this is an instance of aliasing best understood in the natural domain. When would you like to understand aliasing in the frequency domain and now I will go back to the question that Abhisar had raised when would you like to go back to the frequency domain when you wish to analyze non-idealities. So, you know I have said so I have the fan rotating faster and faster I have explained here now. Now, I told this is the situation b. Now, the situation is you have used a whole circuit after sampling and I have said that there are advantages of using the time domain there are advantages in using the frequency domain. If you use the time domain you understand the nature of the waveform that you expect, but you are not clear what is happening in the frequency domain what frequency components are being created, what synersoids have come into the picture. When you go into the frequency domain it is very difficult to visualize what waveform you can expect in that simple sample and whole circuit, but you understand what is happening which frequencies are being brought in which synersoids are being brought in when you put a synersoid into the sample and whole circuit. So, the answer is that both the domains are revealing in some sense you know they reveal different kinds of information and it is by experience that one can understand which domain one should use to understand a certain phenomenon better. I am sure Abhisar might want to post a little more on the discussion forum in response to this and I will be happy to see that. Well, I saw a question related to the magnitude response of filters again coming from Raman 111 and I think you know I had answered that question. Well, in summary the question related to what when we make filters reconstruct the sample signal what do we generally prefer in the magnitude response? We prefer a good gain or do we prefer to have approximately unity gain and put the amplifier stage separately and you know the answer which I gave was that a good gain is not always a primary concern what is important is the magnitude shape and phase response. So, it is a relative gain between different frequencies that is important not the absolute gains and I think that was understood. I also had a very interesting discussion on the windows with R Mirage and I answered Mirage's question with some explanation of the windows you know in fact the whole subject of if I are a finite impulse response filter design is a very rich subject in discrete time signal process and if we have an opportunity to build a course on digital signal processing in future we would spend quite a lot of time discussing the whole question of designing different windows and their merits and demerits. I have given the answer in a nutshell in this post. Now, there are several questions about the exams those I think my teaching associates are taking care of I do not need to go into those, but I see here a very interesting question on the normalized frequency Sandeep Unna has asked the question why is the maximum normalized frequency less than or equal to pi. Now, you see the pi is on the normalized angular frequency axis representative of the maximum frequency component that could be present in the original signal if you do not want allies and my teaching associates Sujata has answered the question very well. I also see a very interesting question posed by Soumya Dipto Bhattacharya and in a nutshell the question says is it that you know a signal has to be limited in time or in frequency. So, you know there are there are there are subtle points here a signal may be unlimited in both. So, the Gaussian signal for example is unlimited in both domains a signal could be limited in one, but then it cannot be limited in the other. So, what cannot happen is the signal is limited both in time and frequency that is not possible, but it could be limited in one and unlimited in the other that is possible and then since this question has been raised if you evolve a measure of the spread of the signal in both the domains in time and frequency it can be shown by something called the uncertainty principle that there is a fundamental limit lower limit to the product of these spreads the product because you can always squeeze a signal in one domain it expands in the other, but the product is lower bounded if the signal has finite energy. This is beyond the scope of this discussion, but I thought I mentioned at this point. Now, there was a question about digital signals and discrete signals from Dinesh Papula. You see digital signals are discretized in both domains in the independent variable domain and in the dependent variable domain discrete independent variable signals are discretized on the independent variable, but not on the dependent variable. Now, there is a very interesting question. There is a question relating to the region of convergence. Now, Abhisar has again posed a question on the region of convergence. So, he asks why is there no region of convergence for the Fourier transform? The Fourier transform as my teaching associate very correctly answers Pratik Feigde has answered the question very nicely. He said that the Fourier transform is actually a special case of the Laplace transform. So, the Fourier transform is meaningful if the Laplace transform converges on the imaginary axis for the case of continuous time and if you are talking about discrete time when the Z transform converges on the unit circle. So, there is no concept of region of convergence in a Fourier transform. The region of convergence needs to include a certain contour in the Laplace transform or the Z transform if the Fourier transform exists. Well, let me see are there any other questions that we have not answered here? Well, I would certainly you know please do not misunderstand if I have not answered some question. It is not because I want to ignore you or something. It is just that you know I might have missed some technical point which was raised even then I am just looking at it once to make sure that I have not missed something of importance. Well, yes I think I believe that we have taken up most of the technical questions. Now, one general remark I am very happy to see that all of you are participating wholeheartedly in technical discussion in this course. This is a second course in signals and systems and I know the material is a little more difficult than the first course and that is why discussion is even more important. So, I am looking forward to more discussion with you on the discussion forum and otherwise when you post technical questions and when you raise queries and also answer other people's technical questions. With that looking forward to even more enthusiastic participation from you. Thank you so much and we will conclude this session now.