 Welcome to the twelfth session in the second module in the course signals and systems. We now said a lot about signals and vectors. We know how to think about inner products or dot products in the context of discrete signals and continuous signals. I mean discrete variable signals and continuous variable signals. Now we are going to address a specific question namely can I think of a set of perpendicular signals which could come together and form periodic signals more generally and in fact we will ask an even more specific question. Can I come up with a set of perpendicular signals each of which is a sinusoid or a sine wave and use them to form periodic signals? Now let us understand how we could go about this whole process. So let us consider a periodic signal with period t. In other words if I take the axis of small t or the independent variable which is continuous in nature I could divide this into intervals of size capital T. So I could have minus t to 0 and then to plus t and then to 2t and then can go on and whatever you have see at this point is also present. So if it is at the point delta it is also present at the point minus t plus delta it is also present at the point t plus delta and it is also present at the point 2t plus delta and this is true for every delta between 0 and t same value at minus t plus delta delta t plus delta 2t plus delta and so on. Now suppose if it were possible we could decompose the signal. Select that periodic signal with x of t. So x t is periodic with period t. That means formally x of t plus t is equal to x of t and this is true for all reality. And suppose it were possible to express x t as a linear combination of synosites. In other words let us take a finite or well not a finite but let us take a countably infinite, a countable combination. So suppose if it were possible to express the signal x t as a countable linear combination. Now first we need to answer why are we saying countable? Why can we rest assured that that linear combination can be countable? All these questions need to be answered. Whatever it is. Suppose it were possible to do that. So suppose it were possible to write down this expression. What should these omega k's be? And similarly with these what should they be? This is the question that we need to ask one. We need to first establish that it is valid to think of a countable set of such synusoids in a reasonable set of contexts. And then we need to find out how you could calculate these contributions. Now first let me explain to you why you can be satisfied with a countable set. So let us come back to the very drawing that we had earlier. What we will do is to put these synusoids on each of these intervals t. So suppose there were synusoids sitting on the entire, you see synusoids last forever. And if you need these synusoids to come together then you probably want the synusoids to have the same property as x t does. Namely they may they need to be periodic. But let us establish that formula. Let us establish something interesting. You see what kind of synusoids would you choose? You would choose synusoids that obey the same property as the signal. Namely that if any one of the synusoids is s t then s t plus t should be equal to s t for all t. It is quite clear that this is a simple choice that you can make of synusoids. Why bring in synusoids which do not have this property? If you can make do with synusoids which have this property. What are synusoids which have this property? Those that are periodic with period t. Now we get a synusoid periodic with period t. Essentially those that means all these synusoids have frequencies 2 pi by t multiplied by an integer. In fact we can be quite happy with positive integers. There is no need to take negative integers here. So you know although I have written k equal to minus infinity to plus infinity in the previous expression here. You look at it. I do not really need this minus infinity to plus infinity here. You know I could be quite happy with 0. Now let us write that down. Let us write that down formula. So essentially we are saying x of t is of the form summation going from 0 to infinity now ak cos 2 pi by t times k of course t plus theta k. And now let us ask a question about this synusoids. So let us take 2 different k and let us take the so called dot product, inner product of cos 2 pi by t k 1 t plus theta 1 and cos 2 pi by t times k 2 t plus theta 2 but restricted, restricted to 0 to t, the interval from 0 to infinity. That means essentially t going from 0 to t only this interval. Now you see remember I am restricting the interval because if I do not restrict the interval the dot product would diverge. The inner product you see please remember if I have 2 infinite length signals the inner product may not converge. It may be a divergent integral. The integral may go to infinity. So it does not make any sense in fact for these synusoids if I were to try and take a dot product that dot product might diverge that inner product might diverge. So anyway if I have these periodic synusoids coming together to form the signal x t I might as well see what happens on one interval say 0 to t and whatever is happening on that one interval is repeated in every other interval. So I could focus my attention on one interval and satisfy myself that I know everything about what is happening on any other interval. That is why I am saying restrict these synusoids to the interval 0 to t or for that matter any other contiguous, contiguous means all coming together one after the other, contiguous interval of size t. So if you like you could take the interval from minus t by 2 capital T by 2 to plus capital T by 2. Anything that you like any contiguous interval of t but we will take it from 0 to t for convenience. Let us take that dot. So essentially what we are saying is find out the dot product of let us say y 1 and y 2 where y 1 t is equal to cos 2 pi by t k 1 t plus theta 1 only between t going from 0 to capital T and 0 else and similarly y 2 t is cos 2 pi by t k 2 t plus theta 2 for again t going from 0 to capital T and 0 else. Well that is an easy dot product to calculate all that we need to do is to multiply these two signals and integrate between 0 and t. So here we go dot product of y 1 and y 2 essentially cos 2 pi by capital T k 1 t plus theta 1 cos 2 pi by t k 2 t plus theta 2 d t only between 0 and t and you know how to split this product of cosines. See you know that 2 times cos capital A cos capital B is essentially cos A plus B plus cos A minus and therefore this integral is integral 0 to t cos 2 pi by t k 1 plus k 2 times t plus theta 1 plus theta 2 d t plus integral from 0 to t cos 2 pi by t k 1 minus k 2 t plus theta 1 minus theta 2. Now look at these two terms look at this term this term is essentially the integral of a sinusoid of angular frequency 2 pi by t times k 1 plus k 2 this is the angular frequency here and this is the angular frequency in the next one. Now it is very easy to understand these two integrals you know you have a finite number angular frequency 2 pi by t into k 1 plus k 2 or for angular frequency 2 pi by t into k 1 minus k 2 if k 1 plus k 2 or k 1 minus k 2 are not 0 that means if you have a finite number of cycles of the sinusoid you see k 1 plus k 2 and k 1 minus k 2 are respectively number of cycles of that sinusoid which are completed in the interval 0 to t and for any sinusoid the integral over a cycle is 0. So, unless one of the sinusoid of frequency I mean with k 1 plus k 2 cycles or k 1 minus k 2 cycles degenerates into a constant the integrals are going to be 0. So, let us write that down for me we are saying this integral if k 1 plus k 2 is not equal to 0 the integral is 0 and so too if k 1 minus k 2 is not equal to 0 the integral is 0. However, suppose k 1 equal to k 2 then the second integral essentially becomes integral 0 to t cos theta 1 minus theta 2 dt which is essentially a constant it is t times cos theta 1 minus theta 2. So, you see the interesting thing here is that if you have two sinusoids both of which have nonzero angular frequency and the two angular frequencies are different their inner product is 0 when restricted to the interval. What do you mean by the inner product being 0 when is the inner product of two vectors 0 when they are perpendicular only perpendicular vectors have a 0 inner product. What have we established here we have established that two sinusoids both of which are periodic with period t are perpendicular if they do not have the same angular frequency a very important result. We shall see more in the next session how we could use this to decompose a periodic signal into its sinusoidal components. Thank you.