 Anyway, as I said, let us come to one very important special conclusion of this so called multiplication principle and that is as follows. Let us look at what happens when we put omega equal to 0. So we have x 1 to n with DTFT is x 1 to omega and we know that x 1 n times x 2 n if it does have a DTFT it would be essentially 1 by 2 pi integral minus pi to pi x 1 lambda x 2 omega minus lambda d lambda. Now let us in particular take x 2 complex conjugate n you see. So if x 2 n has the DTFT capital X 2 omega, what is the DTFT of x 2 complex conjugate n we know what it is. It is x 2 minus omega complex conjugate and therefore if you choose not to multiply x 1 and x 2 but x 1 and x 2 conjugate what do we have x 1 n x 2 complex conjugate n would have the DTFT 1 by 2 pi integral from minus pi to pi x 1 lambda now here you have to write x 2 omega minus lambda but it would be lambda minus omega complex conjugate d lambda because you reverse or you change the sign of the argument and you also complex conjugate is that correct. Now in particular look at the consequence of this for omega equal to 0 look at the consequence of this at omega equal to 0 it says that the DTFT of x 1 n x 2 complex conjugate n evaluated at omega equal to 0 is 1 by 2 pi integral minus pi to pi x 1 lambda x 2 lambda complex conjugate d lambda. Isn't that true because in this expression all you are doing is putting omega equal to 0 now we have a very very interesting statement what indeed what would you mean by the DTFT of x 1 x 2 complex conjugate evaluated at omega equal to 0 the DTFT of x 1 n x 2 complex conjugate n evaluated at omega equal to 0 is simply summation n going from minus to plus infinity x 1 n x 2 complex conjugate n e raised to the power minus j 0 n which is summation x 1 x 2 complex conjugate over all n and this is known to us we are familiar with this. This is our familiar inner product of the sequences x 1 and x 2 it is a very interesting result why so interesting let us put them down together then we will see why it is so beautiful we are saying summation n going from minus to plus infinity x 1 n x 2 n complex conjugate is 1 by 2 pi integral minus pi to pi capital X 1 lambda capital X 2 lambda complex conjugate d lambda. Now notice the similarity in these two sides the left hand and the right hand side similarity in spirit if not exactly in expression what do I mean by the similarity in spirit look at the left hand side on the left hand side we are saying multiply corresponding samples but of course the second one is complex conjugated and add over all such samples. On the right hand side we are saying multiply corresponding points on the frequency axis but the second point is complex conjugated on the left hand side because we are dealing with a discrete independent variable we are saying add on the right hand side because we are dealing with a continuous independent variable we are saying integrate on the left hand side because we are dealing with an independent variable which runs all the way from minus to plus infinity we are saying add from minus to plus infinity. On the right hand side because we are dealing with periodic functions or the discrete time Fourier transform we are integrating only over the principal period but on both sides we are multiplying corresponding components or corresponding points on the two functions the second one is complex conjugated and we are integrating on the right hand side and adding on the left we are putting them together we are putting together all such sum. In other words we are bringing out an equivalence between two inner products and now we need to reflect for a moment on this. In fact we shall we have seen something very interesting in this lecture and we shall dwell more on this as we begin the next lecture. Thank you.