 With that program we must now see some examples of window function. We saw two in the previous lecture but we must see more in this one. So we will now look at a very common window called the hum window. We shall now take one by one expressions for different windows. So in general we shall specify a window as follows. We will specify a window to lie between k equal to minus m to plus n and we will specify the window by the expression W of k. So for example the rectangular window is specified by Wk equal to 1 and the hum window is specified by Wk equal to half 1 plus cos 2 pi k by 2n and we can easily see that at k equal to minus n we have cos pi. So this is 0 and so also at k equal to plus n plus 7 minus n. So this becomes 0. And of course what we are really doing is to place one whole cycle. You see if you look at it when k goes over an entire interval from minus capital N to plus capital N you are running over a whole cycle of the cosine taking you from an argument of minus pi to an argument of plus pi and you are putting that whole cycle of a cosine on a constant and therefore it will always be non-negative but it will follow a cos sinusoidal pattern. So we can sketch the hum window for our convenience. First before the best way to sketch is to draw the corresponding continuous function and then discretize it. So you see we have this cos sinusoidal function here. Bell shape if you want to call it that. The hum envelope. So for each window we have an envelope and the window is obtained by sampling the envelope. This idea of the envelope is important. You see how you get the envelope is very easy. You get the envelope by replacing k by t. t becomes continuous. So in this expression you can think of replacing k by t. It gives you cos 2 pi t by 2n. So t equal to minus N you can say that anyway of course it takes the value 0 as before and plus n also it takes the value 0 and then of course at k equal to 0 it takes the value half. So it rises and then you know so I am sorry at k equal to 0 it takes the value 1. I mean half into 1 plus 1. The tip of the window is important for all the windows because when we are looking for an even length FR filter we need to do a little bit of work on this. You see instead of for an odd length window of course you would include t equal to 0 among the samples but when the window is of even length you must include plus half and minus half among the samples and then displace the other samples by 1. So you will get an equal number of samples on the positive side of t and on the negative side of t. So you know when you have an even length window you need to draw the envelope and then sample it starting from plus half and minus half and then displace by the integers and the same is true for the ideal impulse response. We need to take the ideal impulse response, draw the envelope of the ideal impulse response and then resample it starting from plus half and minus half and then displace by 1. So for even length FR filter design we need to take this care that is why the envelope is important. Now let us look at a few more envelopes. Humming window. By the way hum and humming are names of researchers in this field who have suggested this window. The window is described by Wk is equal to 0.54 plus 0.46 cos 2 pi k by 2m and now of course you will wonder what sanctity these numbers have 0.54 and 0.46 except for the fact that they add to 1 they do not seem to have any other sanctity and they really do not. You see in general one can talk about a generalized humming window where you give this the value alpha and therefore this becomes 1 minus alpha that is the generalized humming window. And you see the whole game in this humming window is to try hum window or the humming window is to try and optimize between main lobe width and side lobe area. So that alpha is like a tuning parameter it allows you to tune main lobe width and side lobe area and of course you cannot you know take it arbitrarily. You cannot use any value of alpha between 0 and 1. You know you would want in fact you would normally want it to remain at least reasonably positive all over the interval but whatever it be I mean it is really a game of compromise between the main lobe width and the relative side lobe area. And again the game is that when you go to a smoother function now you know in general the hum of the humming windows do better than the triangular window in general in terms of their main lobe width come side lobe relative side lobe area performance. And the secret of this better performance is the fact that they have essentially a sinusoid or you could say a zero frequency plus a first fundamentally term. Now you can introduce one more term and perhaps introduce some more compromise and that is what leads you to what is called the blackman window. Again blackman is the name of another researcher. In the blackman window WK is described by 0.42 plus 0.5 cos 2 pi K by 2n plus 0.08 cos 4 pi K by 2n. So essentially one has introduced a second harmonic term. And just as a good dish requires just the right proportions of different ingredients a good window requires just the right proportion of different harmonics to achieve a good main lobe area and main lobe width and side lobe area compromise. And of course you could then conceive of introducing a little bit of a third harmonic and tuning it and so on you know you. So window design is as much of an art as a science. You see wherever there is no one unique answer and different answers involve different kinds of compromises. The subject also acquires the form of an art because there are beauties of various kinds. There is some kind of beauty in some window and some in some other. But a masterpiece among all these windows there are several other windows. A masterpiece among these windows is what is called the Kaiser window. Kaiser window is based on what are called the modified Bessel functions of the first kind and other 0. Let us denote it by I0x and I0x is described essentially by its Taylor series. It is difficult to describe it in closed form. It looks tentatively like an expansion of e raised to the power of x but it is not. This is a Taylor series expansion of the modified Bessel function of first kind and order 0. Now just a little bit of background you will recall that Bessel functions arise as a solution to what is called the Bessel differential equation. It is a very important equation in the whole field of differential equations. In fact the Bessel functions occur in many different contexts seemingly unconnected. There are different kinds of Bessel functions again. They occur in the context of communication in the description of frequency modulation and they also occur in other disciplines of engineering. Here of course you see they occur in window design. So they have some very interesting properties. Now Kaiser studied this modified Bessel function of the Bessel function for its spectral properties and came out with the observation that it seemed to offer an excellent compromise between main load width and side load area. What I mean by that is of course there is a fundamental limit. You cannot do arbitrarily well on both but you can probably do better on both fronts and that also can be taken to a certain degree of achievement and the Kaiser window is probably known to give some of the best results in terms of window design and the beauty about the Kaiser window is that it has two tuning parameters. Of course all windows have one tuning parameter that is the length. So let us write that down. All windows the tuning parameter given by the window length is inversely proportional. So the more the length the less the main load width. We saw that in the case of the rectangular and the triangular windows but that is true of all windows. It is not correct to say that you cannot get as small a transition band as you want. You can get as small a transition band as you desire. The only problem is what is not affected by the window length is the relative side load area and that is the tragedy of series approximations, sinusoidal series approximations.