 So warm welcome to the 11th lecture on the subject of digital signal processing and its applications. We recall that the previous lecture was devoted to an introduction and a discussion on the discrete time Fourier transform. Let us quickly recapitulate one or two important conclusions that we drew in the previous lecture and then proceed to look at some more properties of the discrete time Fourier transform. We had seen that if you have an arbitrary sequence xn and if the summation n going from minus to plus infinity xn e raised to power minus j omega n converges we call it the discrete time Fourier transform of the sequence xn denoted by capital X as a function of omega. Here the dependence is on the variable omega and you will recall that omega is a continuous variable. It can take on values continuously all over the real axis however there is periodicity in x omega right. So you will recall also that we derived that x of omega is equal to x of omega plus 2 pi for all omega and therefore there is a periodicity in x omega of 2 pi. We also saw that the prime interval over which x omega needs to be considered is the interval from minus pi to pi. So what we call the principal interval. It is adequate for us to look at the discrete time Fourier transform over this principal interval of minus pi to pi right because it would then be repeated at every multiple of 2 pi. What is more is we are also seen the inverse discrete time Fourier transform. We had seen that you can reconstruct xn from x omega in the following way. Further we are also looked at a very important property of convolution. We saw that if you convolve 2 sequences each of which has a discrete time Fourier transform and if the convolution also has a discrete time Fourier transform then there is a beautiful relationship between the convolution and the 2 sequences in the Fourier domain or in the frequency domain namely when you convolve 2 sequences their discrete time Fourier transforms are multiplied all right. So we showed that if x1 is convolved with x2 and if x1 and 2 respectively have the DTFT is capital X1 and capital X2 then x1 convolved with x2 has the DTFT x1 into x2 that is what we proved the last time. We are also started looking at some of the properties of the discrete time Fourier transform. We saw that the discrete time Fourier transform viewed as an operator is linear right. So we saw that if you think of the DTFT as an operator then x1 and 2 being operated upon by the DTFT to give capital X1 and 2 implied that alpha x1 plus beta x2 have the DTFT alpha capital X1 plus beta capital X2. What is more we have also seen that if x have the DTFT capital X then if we time reverse it leads to a frequency reversal 2 or essentially frequency negation that means every positive frequency becomes a corresponding negative frequency and that is obvious because time reversal amounts to rotating in the opposite direction and therefore every omega is replaced by minus omega. Of course we also came to same conclusion algebraically. Now we take further the discussion of some properties. For example we ask what happens when we complex conjugate. So before that let me ask if there are any questions at this point before we proceed. Are there any doubts or questions that need to be clarified before we proceed to discuss a few more properties. Yes there is a question. Okay that is a good question. So the question is we had said that you know the whole idea of reconstructing Xn in fact let me go down to the inverse DTFT. The question is when we discussed this idea of inverting the DTFT we said the idea is that you take each component multiplied by a so called unit vector in the direction of that component and instead of adding now you integrate overall component because these components are a continuum. Omega is a continuous variable and the question is how could we call you know how could we talk of a unit vector in the context of e raise the power j omega n since e raise the power j omega n does not have magnitude 1 of that matter it cannot be made to have magnitude 1. Now let me clarify exactly what we are saying here. What we are saying is that when we use this idea of multiplying a component by a unit vector it is of course true for a finite dimensional space but here we are talking about an infinite dimensional space. So we can take inspiration from that idea but we may not be able to use that idea exactly right there has to be a slight modification and the modification is that here what we assume happens is that you see we are assuming all the while remember that X omega converges that means there is an infinite summation that converges that need not happen. So under convergence I mean under X omega converging what we are saying is all the information has been retained in the X omega going from minus pi to pi and the approach that you would use to reconstruct is similar to what you would do for finite dimensional space that means take the component and multiply it by a so called unit vector in that direction and add over all such components. Now unfortunately here multiplication of e raised to the power j omega n by 1 by 2 pi does not really make it a unit vector but what we are doing here is akin to what you would do in a finite dimensional space namely multiply a component by a unit vector and add over all such components. So the idea is similar but the idea of a unit vector cannot be taken exactly from a finite dimensional space to an infinite dimensional space here right. So although what he is saying is that essentially e raised to the power j omega n by 2 pi is not really a unit vector it is magnitude or if you take the sum squared of the magnitudes of all the samples it does not converge but the idea that you can multiply components by vectors in different directions suitably normalized is being employed here and what we did later was to take inspiration from that idea but to derive the factor 1 by 2 pi exactly by algebra right. So later on we used algebra to come exactly to the conclusion that you needed a factor of 1 by 2 pi there. So it is not correct really to call it a unit vector rather it may be more appropriate to say that we take inspiration from that idea of multiplying components by unit vectors and adding over all components but the conversion of a vector to a unit vector essentially involves normalization by a constant. We allow that normalization by a constant using the factor 1 by 2 pi here that is the way to understand this right. So when you go from finite to infinite dimensional spaces there are certain patches of this kind which you need to deal with right. So for a greater or deeper understanding of this I would recommend going into functional analysis but that is not the objective of this course. What we are doing here is to take inspiration from that idea and we know that you probably would require a factor kappa 0 here. We saw from algebra that factor turns out to be 1 by 2 pi we later on prove this relation exactly right we later on derived this you know inverse exactly and there we saw that the factor of 1 by 2 pi was required right that is the way to understand this. So in general it is a good idea to use geometric insights to understand several signal processing ideas but when using geometrical insights one must be careful to distinguish between certain things that happen obviously in finite dimensional spaces which do not generalize exactly to infinite dimensional spaces but what you know in finite dimensional spaces gives you an idea of what to expect in an infinite dimensional space and you then have to tune or to correct what you expect by looking at the basic algebra okay is there any other questions alright. So in that case we will proceed then with the discussion of some more properties of the discrete time Fourier transform namely let us look at what happens when we complex conjugate. So if you know that the discrete time Fourier transform of Xn is capital X of omega what is the discrete time Fourier transform of X bar n is the question that we would like to ask. Now that is easy to do let us calculate the discrete time Fourier transform of this it will be summation n going from minus to plus infinity X bar n e raised to power minus j omega n and of course we can rewrite this see what we do is essentially we want to take the bar all above here. So you know how would we do that you could of course remove the minus sign and take the conjugate here so this is the same as e raised to power j omega n complex conjugated and therefore I have now of course when you take a sum of complex conjugates it is the complex conjugate of the sum so I have Xn now here I can rewrite this as e raised to power minus j minus omega times n and the whole complex conjugate which turns out to be X minus omega the whole complex conjugate. Complex conjugating without time reversal amounts to taking a negation on the frequency axis then complex conjugating once again and this leads us to a very interesting conclusion. If I take Xn to be a real sequence then X bar n is equal to Xn that is what you mean by real and that means X minus omega bar is X omega. So what we are saying is if you take corresponding frequencies omega n minus omega they are related by complex conjugation that means on the omega axis if I take corresponding positive and negative frequencies the discrete time Fourier transforms are the complex conjugates of one another. Now what is the physical interpretation of this? You see if you look at how the signal is formed so to speak if you look back at the inverse DTFT it says that X of n is 1 by 2 pi integral from minus pi to pi X omega e raised to power j omega n d omega. Now of course we reinterpret this as before as saying that we take so many different omega actually all the continuum of omegas for omega going from minus pi to plus pi. For each of these omegas you can think of a pair of omegas 1 plus omega 0 the other are minus omega 0 they come together so X at omega 0 contributes X omega 0 times e raised to power j omega 0 n minus omega 0 contributes X of minus omega 0 times e raised to power minus j omega n. So you have a corresponding complex rotating phasor one rotating clockwise and the other rotating anticlockwise with respective frequencies omega 0 and minus omega 0 and what we are saying is that the corresponding components are complex conjugates they have the same magnitude but the opposite phase or the opposite starting angle. In other words what we are saying is if you plot X omega 0 and X minus omega 0 they would look like this they have the same magnitude and they have exactly the opposite starting angle this is the symbol for angle or starting angle of phase you can visualize if there are two vectors if there are two rotating phases like this beginning here at t equal to 0 and here and if this one rotates in a counter clockwise direction and this one in a clockwise direction they will always add up to a sinusoid a cosine wave then imaginary parts will always cancel and their real parts would enhance or be doubled that is what we are saying here. So every property of the discrete time Fourier transform also has a corresponding interpretation that we must understand right it means every pair of corresponding positive and negative frequencies have the same magnitude but opposite starting angle and that is illustrated very clearly from how we see them coming together to form a real signal here.