 A warm welcome to the 38th session in the second module of the course, Signals and Systems. In the previous session, we had been looking at a very important theorem called Passivell's theorem, which essentially equates the inner product with signal representations in two different domains. In fact, the Passivell's theorem is profound, a deep result in its own right. Although the way we looked at it, it derived from the multiplication theorem, it has a very deep meaning on its own. Let us look at what the Passivell's theorem says specifically in the context of the Fourier transform and let us also interpret it once again and bring to our own notice the depth and the importance in that result. So, we have talked about Passivell's theorem last time in the Fourier domain and what the Passivell's theorem said really was that if you looked at integral minus to plus infinity xty bar t dt, it was equal to 1 by 2 pi integral minus to plus infinity capital X omega capital Y omega bar d omega given that xt and yt both had Fourier transforms and their Fourier transforms were X omega and Y omega. Now, we had interpreted this as a consequence of inner product. So, we noted that integral minus to plus infinity xty bar t dt is an inner product using a time domain representation xt is a combination of impulses, unit impulses and so is yt. So, what we really meant was that you could think of these I mean X lambda and Y lambda as the components and you could think of these the delta t minus lambda as the orthogonal basis. Now, in this particular case, the orthogonality is trivial because if you look at any two impulses which do not coincide they are non-overlapped. So, their inner product is trivially 0. Though of course, I must caution you that inner products of impulses need to be understood by looking at the limiting situation. So, you start from a little narrow pulse and assume that impulses are not at the same point, the pulses are also away from one another and as you narrow the pulses their overlap goes to 0. So, in that sense you should think of two impulses which are non coincidental that means they do not come to the same point as being perpendicular or orthogonal and you know it is the U unit impulse the unit part of the impulse which also makes the basis orthonormal in a more general sense. See I must caution you that I am trying to use certain intuitive explanations based on a vector perspective to make it easier to understand and appreciate the results that we have in the Fourier transform. But I am not being terribly rigorous if an expert in functional analysis or real analysis were to listen to this exposition he or she would sometimes be dissatisfied at my lack of rigor but that is intentional. I wish to make it as intuitive as I can without sacrificing on correctness. So, you know at these points when I am saying that impulses are orthogonal one must understand them intuitively and then of course, one can also make a more rigorous statement if one looks at books on functional analysis or real analysis and that I would certainly encourage you to do if you are so inclined it would give you a much more accurate insight. But right now our aim is to get an operational understanding with some level of intuitive appreciation of what is going on. So, this is a little background I needed to explain this because you know I want to be cautious in telling you that I am not being perfect from a functional analytic or a real analytic sense. Anyway with those remarks two impulses located at different points are in that sense perpendicular because they are non-overlapping trivially orthogonal. And let me once again review and give you a slightly different way of understanding that the e raised to power j 2 pi f t or e raised to power j omega t's whichever you like to call them are also orthogonal. So, let us look at that too. So, I now continue to look at the specific interpretation of orthogonality of e raised to power j omega t let us look at it from a slightly different perspective. You could begin with two different e raised to power j omega t's but spaced by a certain amount. So, let us consider the interval from minus capital T to plus capital T and we take what we call the so called fundamental frequency and we take frequency spaced by the fundamental frequency 2 pi by 2 t 2 t is the interval here. So, therefore, essentially we are talking about two different e raised to power j omega t's but spaced by this frequency spacing. So, we consider omega 1 equal to some multiple let us say k times 2 pi by 2 t and omega 2 equal to l times the same quantity. And now it is meaningful to talk about the inner product of e raised to power j omega 1 t and e raised to power j omega 2 t restricted to the interval minus t to plus t and that is going to be integral from minus t to plus t e raised to power j omega 1 t e raised to power j omega 2 t bar d t which is very easy to evaluate. In fact, it is minus t to plus t e raised to power j omega 1 minus omega 2 t d t and of course, there are two possibilities here either we could have omega 1 equal to omega 2 in which case we just get a constant independent of k and l let us do that and if omega 1 is not equal to omega 2 in other words k is not equal to l let us consider both the cases. So, let us look at this case now here if omega 1 is equal to omega 2 of course, we are talking about the case k equal to l if omega 1 is not equal to omega 2 we are talking about the case k not equal to l. So, we have the two cases k equal to l or omega 1 equal to omega 1 equal to omega 2 implies the inner product becomes simply minus t to plus t 1 d t and that is 2 t and of course, when k is not equal to l or omega 1 is not equal to omega 2 then we have the inner product becoming e raised to power j omega 1 minus omega 2 omega 1 minus omega 2 t divided by j omega 1 minus omega 2 evaluated from minus t to plus t and that is very easy to evaluate it is e raised to power j omega 1 minus omega 2 into t minus e raised to power minus j omega 1 minus omega 2 into t the whole divided by j omega 1 minus omega 2 note that omega 1 is not equal to omega 2 and of course, you can simplify this because this is essentially a sine kind of quantity. So, we have this becomes 2 j sine omega 1 minus omega 2 into t divided by j omega 1 minus omega 2 and you can multiply and divide by t. So, that gives you 2 t sine omega 1 minus omega 2 into t divided by omega 1 minus omega 2 into t divided by omega 1 minus omega 2 into t. Now, what is omega 1 what is sine omega 1 minus omega 2 into t let us evaluate it is k 2 pi by 2 t minus l 2 pi by 2 t into t minus l 2 pi by 2 t into t that is essentially k pi minus l pi and since k is not equal to l of course, the sine argument sitting on each of them and since k is not equal to l this must be equal to 0 anyway even if k 1 equal to l to be 0, but then you could not have the denominator. So, very clearly when k is not equal to l we have orthogonality. Now, you know you will notice a slight difference in the way I am constructing the program I am taking frequencies which are spaced by that fundamental spacing of fundamental frequency of 2 pi by 2 t and I take any 2 of this I have got a general expression for the inner product when I substitute the specific values of the frequency and I find they are identically 0 these inner products are all identically 0 when k not equal to l. Now, let me draw this on the frequency axis. So, what have I said here this is when k is not equal to l and the spacing is frequency spacing I mean 2 pi by 2 t. So, this is let us now look at the frequency axis what you are saying is on the time axis I took my interval to be minus t to plus t this interval 0 exactly in the middle on the frequency or angular frequency axis let me write it explicitly this is time t angular frequency axis I have a spacing of 2 pi by 2 t and I am considering these frequency which I am marking here 0 being one of them uniform spacing and we are saying that these are all perpendicular frequencies mutually orthogonality. Now, what happens as t grows to infinity as t grows to infinity this widens this way and this way. So, it starts covering the whole real axis covering all time and this spacing goes smaller and smaller. However, this continues this orthogonality continues this mutual orthogonality continues. So, now we understand what is happening as we take t towards infinity you have frequencies coming closer and closer and they are still mutually orthogonal. The bigger you make capital T the more the set of orthogonal frequencies you get and the limit of course as you go all over the real axis the entire angular frequency axis comprises of orthogonal components. So, on the other side of passable theorem we have a set of orthogonal components which we have now identified we will see more about this in the next session. Thank you.