 And one welcome to the 35th lecture on the subject of digital signal processing and its applications. We have spent several lectures until the last one to look at certain basic principles in filter design. In fact, now we are at a position where we can realize filters as well with different structures. We have talked about filter design and realization. We have not quite exhausted all the possibilities either of design or realization. After all, you know, the subject of design and the subject of realization is more than 25 years old. So, there have been several developments in the field and we cannot possibly capture all of them in a short span of time as we have. Even so, we have given some of the best known structures and approaches. And therefore, as you do your assignment on filter design and realization, you will at least get a complete knowledge of one approach to filter design and realization with the lattice structure. With that, then we now need to carry out the next step namely, if we happen to have designed a filter and realized it, we must also be able to evaluate it. And other than the question of evaluating a filter, it is also important for us to be able to depict the frequency of the transform domain on a computer. You see all this while, we have assumed that the transform domain only needs to be calculated. We have not quite thought about how it should be depicted on a computer. And if we do depict it on a computer, we have no choice but to discretize that domain. Nothing continuous can really be represented on a computer. So, what we are going to do today is to discretize the frequency axis and that would lead us to what is called the discrete Fourier transform as opposed to the discrete time Fourier transform. You see the discrete Fourier transform or the D of t as we would refer to it in the future is discrete both in time and frequency that should be clear. The discrete time Fourier transform is discrete, uses a signal or a sequence that is discrete in time but it is continuous in frequency. So, in contrast the discrete Fourier transform about which we are going to talk today is discrete in both domains. Now, the question is very simple, let us begin with the following question. We have a sequence X of n with D t of t capital X of omega. We wish to discretize X of omega and let us for the moment assume uniform discretization. What I mean by that is like uniform sampling, we sample at a uniform rate. Now, please remember sample means sample in the omega domain not in time. We sample on omega with a spacing let us say delta omega. So, yesterday we had the concept of sampling in time where we said that we would take one sample every t units of time. Now, we are saying we will take one sample on the frequency axis every delta omega spaced on the frequency axis. Now, you must remember that when you are taking the discrete time Fourier transform of a sequence, it immediately has periodicity implicit in it. So, there is really no point in taking samples all over the omega axis. You need to take samples only in one period of 2 pi and this time instead of taking the standard interval minus pi to pi, let us take the standard interval 0 to 2 pi. So, let us consider the samples taken from omega equal to 0 to omega equal to 2 pi. And let there be capital N samples in this period. Two reasons omega equal to 0 is included as we would intuitively do normally. Now, if you have N samples and if you have included omega equal to 0, what you are saying effectively is that the spacing is 2 pi divided by N, is that right? So, we are saying delta omega is 2 pi by N. First, analyze the situation. We have a discrete sequence X of N. We have taken its DTFT that gives X of omega. We have sampled with the spacing delta omega equal to 2 pi by N and we take only samples between 0 and 2 pi for obvious reasons. All the other samples are simply repetitions of these and so, as we expect we have N samples there and what we are trying to answer or analyze is the adequacy or the inadequacy of these samples for understanding the sequence and also and therefore, the discrete time Fourier transform. You see what was the question that we began with when we began this course. We said if we want to deal with signals continuous time signals on a computer, you need to sample them. Now, when you sample them, what is going to happen is a loss of information in some sense. Now, how does that loss of information manifest? We saw it manifests not by not allowing the possible signal, but allowing too many possible signals which could have had those samples. So, the loss of information is by creation of ambiguity. Now, something similar is expected here. You see here we are doing sampling twice, we are sampling in time and then we are also sampling in frequency. So, of course, we could forget for a moment that we have sampled in time and we will only look at the consequence of sampling in frequency and we shall invert the principle of duality. You see duality is a very basic principle in the Fourier transform. I shall not formally stated, but informally what duality says is that if you make a statement involving the time and the frequency domain for continuous time signals, then an equally correct statement can be obtained by reversing the roles of frequency and time. Now, let us use that informal understanding of duality. In fact, let us state first that informal understanding of duality and informal statement of duality. For every statement, for every correct statement involving the continuous time and continuous frequency domains, an equally correct statement where the role of time and frequency has been reversed. See, this is for continuous time and continuous frequency, but with some skill we can also adapt it to discrete time and continuous frequency as we have here. You see, we can for the moment treat discrete time as a train of impulses, train of pulses moving towards the train of impulses and even though you see of course, we have always focused between 0 and 2 pi at 0 and 2 pi of minus pi and pi, but we need not do so necessarily. We could consider the entire frequency axis and keep the periodicity and then work with that for the requirement of duality. So, what I am saying is we invoke duality for the DTFT and how do we invoke it, analyze the effect of sampling. You see, what did we have when we sampled in time, when we sampled in time with a spacing of t, the consequence was to take the original spectrum, translate it by every multiple of 1 by t and then add up each of these translates. So, we took the original spectrum, the original Fourier transform, moved it by every multiple of the sampling frequency that is 1 by t or if you are talking about the angular frequency axis, it would be 2 pi by t. So, move it by every multiple of 2 pi by t or 1 by t and add up these carbon copies and if there is an overlap between these added spectra then you have aliasing. If there is no overlap, there is no aliasing, there is no distortion of the original spectrum if these carbon copies do not overlap with the original. Now exactly similar is the consequence of sampling in omega. You see, sampling in omega with spacing theta omega equal to 2 pi by n mean create carbon copies of the original signal or sequence X of n shifted to every now every be careful. Here we are talking about angular frequency, so 2 pi by delta omega that means n. Since you are talking about angular frequency, you need to take the 2 pi by the spacing. If you were talking about Hertz frequency or the normalised Hertz frequency then you would just take the reciprocal. So that means you need to create carbon copies of the original sequence shifted to every multiple of 2 pi by delta omega that is X n plus r times n for all integer r those are the carbon copies. And the second is to add these carbon copies means the resultant is essentially a sum of these carbon copies including the one at r equal to 0. So if you want to be very careful in delineating the copies from the original you could say for all r belonging to the set of integers minus 0 if you want to be very, very fussy you could write it like that. The resultant would be P n let us call it X P n a periodic version of X n r going from minus to plus infinity X n plus r n. So we have the original sequence translated by every multiple of capital N and add it. This could be multiplied by a constant it is called kappa 0. You see a constant arises out of sampling so you see when you sample there could be a multiplication of each sample implicitly by a constant the process of sampling could multiply the whole train by a constant but that is a minor issue. We shall either neglect that constant or assume that we have taken care in the process of sampling that it is made one if not include the constant it just multiplies the whole signal by a constant it is not a very serious issue anyway. So this is the situation. So it is very simple like we had aliasing in frequency we would have aliasing in time. So if these carbon copies of the original sequence overlap with the original sequence there is time domain aliasing. If any of the X n plus r n r and integer r not equal to 0 overlap with X of n we have time domain aliasing it is not we do not but you see what is very clear from here is that if a sequence is of infinite lengths and if you sample the omega axis you cannot avoid time domain aliasing and that is not surprising it is perfectly dual to what happens in frequency. If a signal is not band limited if you sample it in time its spectrum has to be aliased there is no choice. So aliasing is inevitable when you have an infinite length sequence time domain aliasing I mean is inevitable when you have an infinite length sequence and time domain and frequency domain aliasing is inevitable when you have a signal which is not band limited. Just like the case of band limited signals if a sequence be a finite length then it is possible to avoid time domain aliasing and the rule is very simple if you have sampled with a sampling rate of 2 pi by n on the frequency axis you can avoid time domain aliasing if the number of the range over which the sequence is non-zero is less than or equal to capital n. So very simple we can avoid time domain aliasing for finite length sequences xn which have lengths which have length now when I say length range of non-zero sample values less than or equal to capital N what it means is if the sequence be shifted by capital N there is no overlap either backward or forward as simple as that yes.