 Welcome to the 13th session in the second module in the course Signals and Systems. In the previous session, we had embarked upon a slightly difficult subject, slightly difficult I say because it required us to appreciate several different ideas and what I would like to do is before I proceed to build more ideas now, I would like to recapitulate some of the ideas that I brought out maybe with a little speed in the previous session. The ideas are as follows. Let me put them down one by one. One, we assume that we wish to look at either meiotic signal or signals over a finite interval. Let that interval be t and again without loss of generality 0 to t. You know please understand the meaning of this notation here. This notation means that we are talking about the open interval from 0 to t. Secondly, 0 is our choice, where we put 0 is our choice. That is why I am saying without loss of generality. Point number 2, we have assumed that such signals are spanned by all sinusoids of angular frequency 2 pi by t times k. Now, we could say k is an integer or we could say k is only the set of positive integers. It does not make a difference because when you take positive and negative integers, it really gives you the same sinusoids. So, we do not need to distinguish positive and negative integers here while we are talking about sinusoids. Now, I must make one point clear here. You know, if you talk about all possible signals that lie in the interval 0 to t, we will see in due course that they are not spanned just by the set of all sinusoids with this set of angular frequency. What I mean by that is there is a reasonably big class, but not the full class of signals which can be constructed by putting together sine waves with these angular frequencies. And how did we arrive at this specific choice of angular frequencies? We said there are two possibilities. Either the signal is periodic, in which case of course I want whatever comes in this interval between 0 and t to be repeated in every successive and preceding interval of length t. And therefore, you need these sinusoids if they were to come together and construct the signal, you need them to be periodic with period t too. And if they are periodic with period t, that means they have a frequency which is a multiple of 1 by t on the herd scale or on the pure frequency scale cycles per second scale, which can also be understood that the angular frequency is a multiple of 2 pi by t. That is how we arrived at 2 pi by t times k, where k is the set of all positive integers. Well, 0 needs to be included. So, we will need to include 0 because we have to consider the possibility of a constant term also requiring to be a part of the signal. We will see that in due course. Do not worry, we are going to do several exercises to understand this, but I am trying to get our coordinates right in our understanding. Now, if the signal is not periodic, but you have chosen to observe it only in the interval 0 to t, you are not bothered about what happens outside this interval. Again, more generally you are saying that you want to observe it in an interval of t. And you have said without any loss of generality, I have the freedom to put my 0 where I want to and therefore, I could say 0 to t. Now, you see, if you are interested in the signal only between 0 and t, there is no harm in putting anything that you like outside that interval if it makes your analysis easier. So, one way to make your analysis easier is to make what is called a periodic extension of this signal. So, whatever is in 0 to t is repeated in every subsequent and preceding interval of t. And you construct a pseudo, a false periodic signal, the original signal may not be periodic, but you have paid attention only to what happens between 0 and t. So, you have just filled up the rest of the time axis with what you want. And one way to fill it up is to make the overall signal periodic. Now, the advantage of doing that, the advantage of filling up the signal in that way periodically is that you could use this discrete set of sine waves to come together and form the signal at least in the interval 0 to t, you do not care what happens outside. This combination of sine waves may not reproduce the signal outside the interval 0 to t, but we are not worried about that. We are happy that it does what we wanted to do in the interval 0 to t, so much so. Now, what is the next thing we said? The next thing we said was that if you look at these sinusoids, let us write these sinusoids down explicitly. So, we wrote an expression for these sinusoids. A typical sinusoid, typical such sinusoid is ak cos 2 pi by t times kt plus phi k or theta k, whatever you want to call it. This is the amplitude of that sinusoid. And this is the phase. And this is the angular frequency. Now, we must consider k equal to 0, 1, 2, etc. k equal to 0 is essentially a constant wave form, a sinusoid of frequency 0. And the others are all what are called harmonics. First harmonic with k equal to 1 or the fundamental k equal to 1 is often also called the fundamental or the fundamental frequency. And the others are called true harmonics. Now, what we also saw was that all these sinusoids are orthogonal, meaning if I take the dot product of one of these with any other, it is equal to 0. In fact, we proved that last time. I must mention there was just one detail which I would like to correct in my derivation last time. It is not a serious issue. You recall that I had written down that when I integrate, well, all that I need to do is to integrate between 0 and t. I do not have to consider the remaining 0 part. So, integral 0 to t Ak cos 2 pi by t times kt plus phi k into Al cos 2 pi by t times LT plus phi L is equal to 0 if k is not equal to L. That is what we had written down. And the little detail that I wanted to correct was that we had decomposed cos 2 pi by t kt plus phi k into cos 2 pi by t LT plus phi L as half or rather if you put a 2 here, you could write this is cos of the sum plus the cos of the difference. Now, I had missed out this 2. This was missed in the derivation in the previous session. I mentioned this because I had paid attention there to essentially the main term. I had not paid attention to this constant. That is not a very serious issue. It still gives you the same result. But I do hope some of you noticed. I was just trying to see if some of you would notice this little detail and I am now pointing out that that was there. So, you know, we do intentionally introduce some of these little, maybe you could call them errors or inadequacies in discussions or some of you should be alert and say, well, you know, we did this and there was this little, perhaps a little deviation from what should be correct. But by enlarge the derivation was correct. So, here was a little trick to keep you attentive. Anyway, that was besides the point. Now, we saw that these 2 are orthogonal if we take any 2 sinusoids. So, what we have established here is that we have an orthogonal set and it is not clear that this orthogonal set spans all such signals. In fact, it does not. What it means is that there are some signals which cannot so be constructed. But those signals are what I might call pathological. That means they are beyond the realm of the kind of discussion that we want to undertake at least at a first level in our understanding on signals and systems. Most reasonable signals that you might encounter in an interval of t are going to be so decomposable. So, this little issue of spanning, we are kind of hand waving and moving it aside. We are saying that we are content with dealing with those signals which are spanned. Now, we will understand which these signals are, where these limitations are as we go along. So, I do not want to preempt too much of the discussion. For the moment, we will confine ourselves to those signals which could so be spanned which is which you can believe or take from me at the moment is very large. So, very large class of signals which can thus be spanned by these sinusoids. That means you could bring together sinusoids of all these angular frequencies and form a very wide class of signals which covers many of the signals that we wish to deal with. Now, the next question that we are going to ask and answer in the next session is how do I use this idea that we have built here to actually get these a k's and phi k's? That is a question which you know we have not answered though we have given all the foundations that are required to give the answer. So, let us come to the next session and answer that question. Thank you.