 A warm welcome to this discussion or response session from me. In this edX course, which is offered in self-paced mode, we are trying to engage the students as much as possible with our resources and I must first apologize that we did not have enough manpower to respond to queries quickly to date, which we are now rectifying. So, now we have people on board who can answer queries quickly on the discussion forum and therefore, you must have seen that for some time the responses were a bit slow. In fact, let me begin by responding to some of the discussion that I see on the discussion forum. So, in one post, I noticed that somebody has requested that the staff should engage with the students more. So, the person is correct in saying that the more you make us engage, the more the students will like the course and the more they can benefit from it. And my principal teaching associate also explained the reason to you. It took some time for us to identify the teaching associates for the semester and this course began little before the teaching associates were assigned. And unfortunately, we did not have enough bandwidth to deal with several courses that were going on and therefore, we have been a little slow in answering the questions. Also, as I mentioned in my email to all of you, in the first part of the course we were trying to gauge the mutual responses and mutual discussion among the students. So, we were in fact for some time keeping quiet simply because we wanted you to respond to one another's queries. We wanted to see how the discussion evolves and then step in at a point where we thought it would be appropriate to give a quote and quote final answer or quote and quote decisive answer and so on. So, to some extent this silence was intentional, to some extent the silence was dictated by the fact that we did not have too many people to answer. But notwithstanding the fact that we shall be answering your queries a little more regularly in future, we do want you to discuss among yourself also that is one of the objectives of a self-paced course. So, here we would really like it if all of you participated in the discussion and I like the idea that somebody has given about community TAs, we will try and identify some community TAs if we can identify some people who will be able to give such answers. But meanwhile, we have our own teaching associates from the institute whom we will now engage a little more actively in answering your questions. This is with the background that I wanted to give to explain the reason for this response session and also to start answering some of these queries. So, we will have these response sessions once in a while to answer your discussion queries. Now, let me take up some of the queries that have been brought up on the discussion forum. So, you know let us I am going to take up queries that relate to technical questions, the questions that relate to administrative matters or even to the examinations. I would not like to take up in this response because they will be answered by my teaching associates where required and in some cases where the examination question should not be answered or discussed in a public forum, we are avoiding doing that altogether. But now let us look at this question which I have here. So, you know after referring to all the examples, Rathuraj says after referring to all the examples, can I say that if I define linearity of the system, one can easily find out the limit up to which a system works. This is from responding to discussion session one in the course. See, if you look at it, linearity is defined very clearly. Linearity has two parts, it has additivity and homogeneity. The problem is in what zone of operation systems exhibit linearity. In most cases, practical systems exhibit linearity only in a limited zone and that zone can be very different for different kinds of systems. Now, in the discussion, you saw extreme example, you saw the human system which we, you know in some sense pretended that it was linear and we found that it is a ridiculous thing to do. A human system is far from linear. In contrast, if you took a simple resistance or an inductance, it would have linear behavior in a large range of voltage and current. So, in that sense, the issue is not about the definition of linearity as Rathuraj says, but it relates to the limits or the zone in which linearity applies. I hope that takes care of his question or her question and if there are any further remarks, Rathuraj can post it. Now, of course, I can see there are several doubts about questions in the exam. Those I will not take up because that, you know, should be answered by my teaching associates, only if they think it is appropriate to answer them. Now, let me also answer another related question. So, I have Raghavad, R-A-G-H-A-V-A-D who has said, does homogeneity also become lost after a certain limit? Yes, indeed. I mean, both additivity and homogeneity operate in a certain zone. Now, in the human system, you know, which is ridiculously non-linear, I mean, it is very highly non-linear, if you double a stimulus, the response is definitely not doubled in most cases. So, we had created a parody on that, you know, to show how linearity being applied to some systems is ridiculous. But on the other hand, in many of the systems which do not have life, it is reasonable to assume, for example, take a simple resistance. In fact, Ohm's law in the resistance is an expression of homogeneity and Ohm's law is applicable in a wide range of voltage and current regions. So, yes, it does, it is lost. See, homogeneity can be lost after, take the case of a transistor, transistor amplifier. So, if you operate, you know, in fact, you can see this. I mean, if you operate a microphone system, if you operate an amplifier system in a hall, in a certain range of operation, if you speak twice as, if you give it twice as much amplitude as the input, it will give you twice as much amplitude as the output. But if the amplitude grows too large, then there can be non-linear effects in the amplifier behavior. So, that is commonly seen in the amplifiers which are used in public systems. So, homogeneity also operates up to a certain limit. In fact, my teaching associate has answered this question. He has very correctly, Ashwit has very correctly said that if one talks about a real system, it is very rarely linear or shift invariant. Now, linearity and shift invariance is a statement of what we want for simplicity of analysis and it also applies in a reasonable zone of operation. It is also a good model for the system in that zone of operation. It makes our whole understanding of the system easier and it gives us insights. But if those insights must be used to the extent that these assumptions apply, that I must emphasize. Let me see this question about a cascaded system impulse response. This seems to be related to a quiz question. But anyway, let me answer it. You see, I must emphasize what my teaching associate has said. The question related to whether, when you convolved to signals related to multiplication, now you know convolution. In fact, later on in this course, you will see there is a parallel between convolution and multiplication. So, convolution in the natural domain corresponds to multiplication in the frequency domain. In that sense, there is some kind of a relation between convolution and multiplication. Let me take his specific question, YG Prashant specific question. Impulse response of a cascaded system is the product of impulse response of the individual systems. No, that is not correct. So, when I cascade two systems, the overall impulse response is the convolution. You must convolve the two impulse responses to get the overall response of the cascaded system. However, if you look at the frequency responses, they are multiplied. So, later on in the course, you will look at the frequency responses of cascaded systems and you will find that they get multiplied. So, that is, I think there is some confusion here. And I believe my teaching associate has answered it very well. Akshay's row day, my teaching associate has said that convolution is not the same as multiplication. And please note that when you convolve to signals X, T and Y, T, the expression looks like that integral minus to plus infinity X tau, YT minus tau d tau and the impulse response of a cascade of two LTI systems is the convolution of the impulse response of the two LTI systems. Let me look at this question on filters. What exactly are Butterworth and Chebyshev filters? How are they designed? Now, since you have raised the question, I will answer it very briefly. You know, because to answer this question, I actually need to take recourse to the Laplace transform or the Z transform which will come in the second course. So, I am hoping that YG Prashant also continues into the second course in self-paced mode that we are offering, part 2, signal systems, part 2. So, we discussed the Laplace and Z transform. But anyway, the Butterworth and the Chebyshev, you see, the ideal low pass filter is unrealizable and you can only approximate the ideal low pass filters in different ways. And the approximation tends to, when you say Butterworth filter, Chebyshev filter, they are different polynomial expressions which can approximate the ideal response that you want. To cut a long story short, the idea behind the Butterworth filter is to place the poles of the analog filter on a circle in the complex plane. Now, I know that this sounds like Greek and Latin to many of you at this stage, because the idea of poles and zeros would come up in the second course. But to make it even simpler, you know, with the background that you have at this point in time, the Butterworth filter and Chebyshev filter essentially create ratios of polynomials to describe the system behavior that you want which would approximate the ideal filter. So, you will see later that one of the issues in designing filters is to come up with rational approximation, rational. The word rational means a ratio of two polynomials. So, two polynomials in what is called the system variable. Now, why do you need a ratio of two polynomials? Because you need to have a realizable system, a system which can be realized, which can be constructed with a finite amount of hardware or software operations. And if you want to do that, then you have to come up with a rational approximation to the ideal filter which can be done in many ways. The Butterworth and Chebyshev filters are two examples of rational approximation to the ideal filter. Now, you might ask why do you need to approximate? Why cannot you just realize the ideal filter? That also we see in detail, but there are three basic reasons why an ideal filter cannot be realized. And if I want to put in simple language, the first reason is the ideal filter is non-causal and it cannot be made causal by any reasonable change. The second is the ideal filter is unstable. You can understand that with the background that you have now. And the third one is something that you may not understand at this stage and that is the ideal filter is irrational because of which it cannot be realized with finite resources. That is in short an answer. But you know, if you want a more detailed answer, I will try and see if some of my teaching associates can give you a link to look up in more depth. Let me look at this question. I see this is a question related to the practice problem. I think my teaching associated answer it relates to the independent variable. Let me see this question. Yes. So, now this relates to finding the intervals for a continuous time convolution. You see, one very simple. In fact, my teaching associates, Akshay Sarode has answered the question very beautifully. But I will try and give a little more insight to the student here, Monica Sahay. You know, the idea is very simple. If you have two finite length continuous time signals that you want to convolve. So, let me actually write it down. I have two signals to be convolved. X of t and H of t. X of t is non-zero from capital T1 to capital T2. And H of t is non-zero from capital T3 to capital T4. Now, you know, when you take the convolution, you are going to evaluate the integral. And obviously, you would want tau to be between capital T1 and capital T2. And you would want t minus tau to be between capital T3 and capital T4. Just add these two. And there you are. You get t needs to be between T1 plus T3 and T2 plus T4. Now, this also applies to the parts. What I mean by that is, if you have a signal x t, if you break it up into non-overlapping intervals and you convolve each of these intervals of a signal with h t, that is also a correct way of calculating the output y t, except that you must then add up the contributions of each of these intervals. You can exercise this principle to find the length or the region of the output of the convolution for each of these intervals. And there could be overlaps. So, although the initial intervals in x t are non-overlapping, when you convolve with h t, the resultant intervals can be overlapping. You should be careful about that. Let me see some of the other questions now. Now, there are, yeah, lecture four question one. Let me see what this is. How would you prove analytically? So, DNA Dallas says, how would you prove analytically that the system is shift invariant? And in fact, I already see an answer to this. You see analytically what you need to do is to check what happens when you shift a signal. When you shift a signal, see what happens to the output. Is it undergoing no change except a shift by the same amount? If so, the system is shift invariant, this should happen for all possible such shifts. It is not enough that it happens for a particular shift. You must see a general shift in the input signal and check whether the output signal has no other change except being shifted by the same amount. And by the same amount is important too. It should not be shifted by a different amount. My teaching associates, in fact, Kirti Ravel has also answered the question very well. So, I do not need to go into more details there. Now, let us see what this question is. This relates, so DNA Dallas again has a question related to the superposition, proving the superposition property with implicit equations. You know, in fact, Sunil Madhekar has answered the question. And I think the emphasis of the question relates to what happens when you have implicit relations. Like here, there is in some sense an implicit relation between y t and x t, which you can see here. And I believe the question is, how do you prove superposition when you have such implicit relationships? Now, what you need to check is whether the sum also obeys. So, you know, does the sum y 1 t plus y 2 t obey the same implicit relation with x 1 t n plus x 2 t being put in in place of x t? If it does, then obviously, the additivity property is satisfied. Similarly, does here, for example, in place of y t if you multiply, so you put alpha times y t is alpha times x t and alpha times y t consistent in this equation, that would allow you to check homogeneity. Now, question about continuous, all right. In discrete convolution, Bharath 12 has the question. In discrete convolution, we flip the time axis. Can we do the same in continuous case? For example, if you want to convolve an impulse function with sin t, can we invert the sin t function, which is minus? See, inverting the sin t function. Well, you see, in any case, see, when you convolve with an impulse function, my teaching associates, Akshay Sarod has answered the question well. So, anyway, convolving with an impulse gives you back, impulse is the identity of convolution. So, convolving with an impulse anyway gives you back the same function. Now, in general, I think the question that is being asked is, if I replace t by minus, so if I invert in the process of convolution, you need to fix one of the signals and you need to shift and invert the other signal. So, can you do it universally? Yes, in principle, you can do it universally. There is no problem. You can do it, but you have to be careful. So, I believe, example by example, one should come up. In this case, of course, the answer is simple and Akshay has given it, that when you convolve with an impulse, no matter how you do it, you would arrive at the same, the impulse is the identity of convolution. But maybe if the, if Barath 12 has some other example where he is finding trouble, he can post it on the discussion forum and we will try and answer it. So, in this case, there was no problem. Now, yes, you know, this question about stitching together a function with narrow pulses. I can see that, you know, some people have had trouble, Tara Bishi has had trouble. You know what I suggest you do? Take the discrete case here. So, I recommend that, take the discrete case to understand this properly. So, I would recommend, I would recommend you carry out this exercise. Let me look at the question again. So, you know, they were not able to understand what is the, what is the reason for flipping? You know, you wrote x t into delta delta t minus t 0 d t is x of t 0 and then, you know, we said delta delta of delta capital delta of t 0 minus t. You see, so I mean, let us take a situation. So, what we are saying is you, let us take the discrete case. So, you have x of n and delta n minus n 0, delta n minus n 0 as a function of n is an impulse at n 0. So, it is equal to 1 for n equal to n 0 and 0 else. So, you multiply x of n by delta n minus n 0 amounts to just picking x of n 0. This gives you x at n 0. x at n 0 now is a constant value delta n minus n 0. So, it leaves you only with one sample. This is what is being illustrated. So, you know, you can now change the variable here. You can exchange the roles and you can write x of n 0 into delta n minus n 0 as x of n into delta n 0 minus n. So, that will make it clear. You know, so maybe many of you might find it easier to understand the discrete case here first and then go to the continuous case in that discussion. Now, let me see. There is an interesting question which I have from Marcelo Borja. What does it mean? An abstraction into the context of mathematics for engineering. Very nice. So, I have a beautiful answer by Hetal Rao. In mathematical context, you see abstraction is a tool which allows you to deal with many situations at once using the same principles. So, there are many different forces. There are many different kinds of masses and every time force acts upon a mass, it produces acceleration. This is the general principle. So, force is equal to mass times acceleration Newton's law. Now, this is an abstraction in a way. The idea of mass is an abstraction. The idea of force is an abstraction. There are many different kinds of forces in nature. There are many different kinds of masses in nature. But this principle applies everywhere. In this course also, we have tried to build some such abstractions. When we assume linearity and shift in variance or rather when it is reasonable to assume linearity and shift in variance in a range of operation, then that leads to several consequences in system behavior and that is what we have been trying to do. Now, let me see some more questions. I can see problem choosing intervals for convolutional sum. So, in fact, my teaching associate has already answered Monica Sahay's question there and I have already given an answer before this to this as well. So, winch 1, 2, 2, 9 to ask, I am sorry, but I cannot get how non-shift in variant example winch. I am not sure which one he is talking about. He or she is talking about here. Maybe he or she should explain. So, winch 1, 2, 2, 9, 2 should explain what example he is talking about. He or she is talking about. Then we can answer the question better. Thank you so much. So, I am so happy to know that 29 minus like the introduction to the course. So, I have an anonymous post here which says linear systems can be analyzed with very simple math that is mostly intuitive. Non-linear systems can be broken into small linear ones with variables chosen are segments of continuous functions. He is given the example of a diode. So, it is a switch and step function. See a diode, I am not sure how meaningful it will be to call it a linear system. Anyway, I think my teaching associate has answered the question well. So, he talks about how circuit designers use linearization all the time. So, I do not think there is too much to add at this point about this question. Now, here is an interesting question. In fact, I also responded to that. I have a question from Devesh Monga which says, is a system with cascaded non-linear systems always non-linear. And perfet 010 has answered very beautifully. And I actually appreciated perfet 1010. There we already have dealt with this question. So, there was this question about how you make a system not shift invariant system like this. So, x t into x t minus 1 by x t minus 2. And beautifully Oleg Haak has answered the question very nicely by saying one of the simplest ways to make it shift invariant is to multiply by t. The moment you look at the time and then respond, the system becomes shift variant. It is not shift invariant because the response automatically depends on time. That is a very good answer given by Oleg Haak and it has been endorsed by my teaching associate. So, that was you know. So, what I did here was to take of course, I have not answered all your questions that would take a long time, but I have picked up a few questions. Do not think that if your question has not been picked up, it is not important. I am sure all questions are important. I have just picked up a few questions which were representative and where I could provide answers on this forum, but we will try and answer more questions in the discussion forum future. Also please try and answer one another's question because that excites us as well into giving you further answers or giving you further comments. If you only see questions posted normally responding to them, we also feel that people are not sufficiently engaged in the course and we look forward to your engagement in answering one another's questions. Though we will also on our own answer more questions in the discussion forum in the coming weeks. So, we will end this session here now. Thank you so much and we look forward to more live interaction with you of this kind future. Thank you.