 Welcome to the eighth session in the second module on signals and systems. Let me begin by answering the exercise that I had posed to you in the end of the previous session. I had put before you a sum of two sine waves of different frequencies omega 1 and omega 2 and of course, you know appropriate amplitudes and phases. So, I made all of them quite free. There were two different frequencies, two different amplitudes and two different phases. And I said the system is linear and shift invariant, note it is linear. So, when I use the additivity part of linearity, I can recognize that the output is going to be the sum of the output to each of those sinusoid. So, let me put that down. So, answer, I have the system S, it is linear shift invariant stable. It has an impulse response h t and I can define the frequency response of the system. Let us call it h omega defined by minus infinity to plus infinity integral h lambda e raised to the power minus j omega lambda d lambda. And I have the input 2 times a 1 cos omega 1 t plus phi 1 plus 2 times a 2 cos omega 2 t plus phi 2 and the output is very simple. The output is simply 2 a 1 mod h evaluated at omega 1 cos omega 1 t plus phi 1 plus the angle of h evaluated at omega 1 plus twice a 2 mod h omega 2 cos omega 2 t plus phi 2 plus angle of h evaluated at omega 2. Simple enough, to each sinusoid its own point on the frequency response. For the angle of frequency omega 1, use the amplitude at omega 1 and the phase change at omega 1. For the angle of frequency omega 2, use the amplitude change at omega 2 and the phase change at omega 2 and make those changes and add those outputs and there you are. Now, in fact, look at this expression. Look at this expression again carefully. You see the beauty is you can predict what happens when you have 2 sinusoids to each its own so to speak. We could do this with more sinusoids. If you have 3 sinusoids, you would essentially have 3 term. If you have 10 sinusoids, you would have 10 terms and you can go on carrying that argument further. So, there is some attractiveness in decomposing an input if we could into sinusoidal component. That means, if I could think of any arbitrary input that I had as a linear combination of sinusoids, I am well placed as far as finding out the output is concerned. There is one convenient way in which the output could be found by looking at individual sinusoids comprising the input. Now, you know what we are getting to slowly is a change of paradigm. You will recall that this whole module was meant to introduce a change of paradigm, a change of the way in which we look at linear shift invariant systems when it is meaningful to do so. Now, you know recall we have been talking all the time about stable linear shift invariant systems and why stable? Because unless the system is stable, I am not guaranteed that the frequency response exists. So, the frequency response let me look at that frequency response clearly once again. Let me write down the expression for the frequency response and highlight where stability comes in. The frequency response H omega is minus to plus infinity H lambda e raised to power minus j omega lambda d lambda and the modulus of H omega is bounded is finite because of this because of the absolute integrity of the impulse response and that is why stability comes. Now, of course, you can see that stability is a sufficient conditions for the frequency response to exist and now I am going to put a big challenge before you. It is not an easy question and I do not even intend to answer this question immediately, but for some of you who are mathematically inclined you know who have done a course on real analysis or who have some exposure to functional spaces, advanced learners I might call them that. I have a challenge for you let me put that challenge down. Is stability a necessary condition for the existence of a frequency response? In fact, I will give you a hint it is not that means the integral may converge. So, it is very interesting later on we will see that even certain and see not all unstable linear shift invariant systems may not have a frequency response. There are a few linear shift invariant systems which are unstable, but which still do have a frequency response that is interesting. Now, what does that mean? That means that stability is of course a sufficient condition if a linear shift invariant system is stable a frequency response is guaranteed. If it is unstable it may or may not have a frequency response and the challenge before you is give examples of linear shift invariant system which are unstable some of which have a frequency response and give some examples of those which do not have a frequency response. Now, as I said this is a question for slightly advanced learners those who want to take a slightly difficult challenge if you do not want to do it at this stage do not worry. Implicitly you will find the answer will come in subsequent discussions in this course, but I wanted to just ignite your mind at this phase anyway. Now, as I was saying you could carry this argument to two synosoids you could carry this argument to 10 synosoids and then you could even think of a continuum of synosoids. So, for example, you could think of synosoids going all taking frequencies all the way from say 1 kilohertz to 2 kilohertz the whole continuum of frequencies from 1 to 2 kilohertz could have corresponding amplitudes and phases associated and you could think of combining all these synosoids and forming a waveform and you could query what happens when that waveform goes into a linear shift invariant system. Now, that is a slightly difficult situation to visualize. So, at least let us visualize something simpler. Suppose I could decompose one class of waveforms into a sum of discrete synosoids you know. So, you have a discrete collection of frequencies here I had to well we could have 3, 4, 10 or we could go to infinite, but discrete you know countably infinite. Now, in fact that is what is called a Fourier series expansion. So, many of us are familiar with what is called a Fourier series expansion of periodic waveforms. Now, when I have a Fourier series expansion of periodic waveforms I am essentially thinking of that periodic waveform as a combination of synosoids with frequencies which are multiples of the frequency of the periodic waveform. So, suppose I have a periodic waveform with a frequency of 1 kilohertz I am saying that periodic waveform has synosoids of frequency 1 kilohertz, 2 kilohertz, 3 kilohertz, 4 kilohertz and all multiples of 1 kilohertz. So, now I have a very convenient way of dealing with periodic waveforms going into a linear shift invariant system. In principle I could see what happens to each of these synosoil components of the periodic waveform and since the system is linear I could combine at the output the output corresponding to each of those components of the periodic waveform and thus generate the periodic waveform again at the output. So, let me complete one little detail here. I shall show you that if a periodic input goes into a shift invariant system the output has no choice but to be periodic. Let us prove that it is a simple proof and we can do it in a couple of minutes. Lemma a periodic input to a linear to a shift invariant we do not even need linearity produces a periodic output. Well, that is very easy. Let the input be x of t and let the period of the input be t, whereupon we have x of t plus capital T is equal to x of t for all t by definition. Now, let x t result in y t as the output of the linear shift invariant system. Now, from shifted variance what do we infer? x t plus capital T should result in y t plus capital T because of shift invariance and of course, x t results in y t and therefore, these two must be equal since these two are equal for all t, these two must be equal for all t. Therefore, we have proved the statement y t plus capital T must be equal to y t for all t and we have proved the lemma that the output is also periodic. In fact, the output is also periodic with the same period. So, now we have a clear situation that we deal with. If I give a periodic input to a linear shift invariant system, the linearity is not important. In fact, we are guaranteed the output is also periodic with the same period on account of shift invariance. And now on account of linearity, if I could take an arbitrary periodic input and decompose it into a combination of synosoids, then I could find the output periodic waveform by seeing what happens to each of the synosoids individually and we shall take this up in much greater depth in the next few sessions. By recapitulating the idea of a Fourier series decomposition and generalizing it to what we call a Fourier transform. Thank you.