 Welcome to the 29th lecture on the subject of digital signal processing and its applications. In this lecture we continue with the subject of FIR filter design by using the window approach and we need to recapitulate a few ideas that we have learned in the couple of lectures that have passed on the subject of FIR filter design. The most important idea that we have seen in the previous lecture is the role of the spectrum of the window function. You see it was a fundamental question why we should choose a window function other than the very simple option of a rectangular window and the rectangular window seems easy to do is retain a few samples and reject the rest from the impulse response. Why should we multiply that set of samples that are retained by some other sequence which has some quote unquote desirable properties and what are those desirable desirable properties. We have seen a little bit of this in the previous lecture and we need now to substantiate by actually giving examples but let us recapitulate what we have seen in the previous lecture. We have seen that the window spectrum has an effect on three things, the passband tolerance, the stopband tolerance and the transition band or transition bandwidth and the window approach we do not have separate control on the passband and the stopband tolerance. In fact, the passband and the stopband tolerance are equal in the windowed approach and we have seen that in the previous lecture. The difference of passband and stopband tolerance is the area under the side load. The passband and stopband tolerance influenced by side load area for this is simple the tolerance is caused by side bands entering and leaving the ideal passband in the periodic convolution between the ideal DTFT or the ideal frequency response and the window spectrum is the region where the main load has completely entered the ideal passband and now the side lobes are entering and leaving. The stopband is a region where the main load is completely outside and the transition band is caused by the main load actually entering and the whole process of the main load entering the ideal passband and ultimately going entirely inside is the cause of the transition band. And therefore the main load width influences the transition band, we have seen that in the previous lecture. Transition band is influenced or is caused by actually influenced by the main load width. It is the main load width and the side load areas that play a role the main load area has only a peripheral influence and the side load widths again have only a peripheral influence. The influence of the main load area is only to determine the average height of the resultant passband. Now that is not a terribly important issue because the height can be changed by simply multiplying all the coefficients by a constant. Now even when we say the passband and our stopband tolerance is influenced by side load area we should qualify this by writing relative side load area that means side load area relative to main load area. So it is a relative side load area which really plays a role not the absolute side load area. By relative side load area we mean the ratio of the side load area to the main load area because the main load area is what determines the average height of the passband and super posed on this height are the oscillations caused by the side load area. And if you multiply the whole set of impulse the resultant impulse response coefficients by a constant both of these are going to get multiplied by the same constant for the relative side load area is not affected by multiplication by a constant. So you see these two things play a very crucial role and we have seen two examples in the previous lecture the rectangular window and the triangular window. We saw that in the triangular window we have less relative now we did not show that but we gave an intuitive argument for it. We said that you know in the triangular window we definitely have a reduction of side load height that we showed relative side load height but what you now put a challenge before you show by reasonable approximation by reasonable asymptotic approximation by asymptotic approximation I mean approximation as n tends to infinity that the triangular window less relative side load area than the rectangular window to solve this challenge is that you can treat the side lobes asymptotically like rectangles where the base is equal to the width of the side lobe and the height can be determined by an asymptotic approximation. So you may treat them very close to triangles and find out the area under the triangle and then see how the area asymptotically changes as the length of the window grows to infinity. Of course I told you this comes with a cost the cost is that the main load width expands so it is not for free although the oscillations in the past and stop and decrease they do so at the cost of an increased main load width which means a larger transition back. So the next point that we had seen in the previous lecture is that there is a compromise between main load width and relative side load area among windows. We also remember that this compromise though generally present can also be to some extent optimized in the sense that one can do well on both fronts to an extent how to what extent is a very very basic and very fundamental question. How we determine how best we can do on both fronts together there are two conflicting things happening here the main load width and the side load area relative side load area but you see there is a fundamental limit on how well you can do on both fronts together and although I shall not go into the genesis of this fundamental limit or even try to explain it mathematically I shall state the general idea behind this fundamental limit. This fundamental limit stems from something called the uncertainty principle on how well we can do on both fronts essentially governed by what is called the uncertainty principle governed by the uncertainty principle of signal processing. I shall just say a couple of sentences to put this in perspective but to understand the relationship better or even to understand the principle better one would need to take a higher level course on signal processing. The principle says that there is a basic conflict between the time and the frequency domain representation of a signal. So if you try and narrow a signal down in time the tendency for the signal is to expand in frequency which of course we do not need a principle or uncertainty principle to understand because we know that you know if you take the Fourier transform of a signal and then if you contract or expand the signal there is a respective expansion or contraction in the Fourier transform that is the basic property of the Fourier transform. So that does not require too much of exposure. What requires a much deeper understanding is that there is a fundamental limit on how you can narrow you can be in both domains. You can be arbitrary narrow or broad in one domain but to what extent can you narrow down in two domains together is what the uncertainty principle addresses. And in fact it is very closely related to this whole question of how much you can narrow down the window in terms of main low width and side low barrier. Now I shall not go entirely into the details here safe to say that there is a there is a relationship and for those of you who are interested you may want to look up more about the uncertainty principle or take a higher level course or also try and reason out how this in other words expresses a basic conflict between expansivity in time and frequency.