 Anyway, so much so for license and taming. Now let us get back to this example of YZ here. You see here you have a product of 2Z transforms and the beauty is this product of the 2Z transforms can also be expressed as a linear combination of the 2Z transforms. So suppose I wish to find the sequence Yn from here, how would I go about doing it? Now one would use what is called a partial fraction expansion? How would we obtain Yn? Decompose Yz. Now Yz which is 1 by 1 minus half z inverse 1 minus one third z inverse can be decomposed as something on 1 minus half z inverse and something upon 1 minus one third z inverse. What upon each? And the idea is very simple. When I want to find this factor, I multiply Yz by 1 minus half z inverse and put z equal to half. So essentially I am multiplying by 1 minus half z inverse and making this factor 0. So multiply by 1 minus half z inverse put z equal to half. So I get 1 minus one third into half inverse that is 2. So I have 1 by 1 minus 2 by 3 and here I would use the same idea. I would multiply this by 1 minus one third z inverse and put z equal to one third. So in other words z inverse or yeah so z equal to one third that is right. So z inverse is 3. There we are 1 by 1 minus 3 by 2. Now let us verify indeed, let us verify. What we are saying is 1 by 1 minus 2 third that is 1 by one third. So 3 by 1 minus half z inverse and 1 minus 3 by 2 that is minus half. So minus 2 by 1 minus one third z inverse is very easy to combine. So 3 minus z inverse minus 2 plus z inverse and that indeed gives you back 1 by exactly the Yz that we had. Now what I just illustrated is what is called a method of partial fraction expansion. It is a method of expanding a rational Yz in terms of factors where there are where there are denominator terms all acquiring the value 0 only at one point in the z plane. Now we often deal with this class of z transforms. Let us introduce this class of z transforms. We call them rational z transforms. Rational z transforms are z transforms of the form finite series in powers of z divided by finite series in powers of z. There is a numerator and denominator each of which are a finite series in powers of z not infinite. And of course the word rational comes actually from the idea of rationality in the integers. Let us take a rational number as we understand it for integers. 116.72 divided by 21.68 is a rational number. In fact even if I put 689 it will be a rational number. Why is this a rational number? Because I can multiply the numerator and denominator by 1000 and that leads me to 116.720 divided by 216.89 which is a ratio of integers. So it is rational. Now let us expand this quantity 116.72 by 21.689 in terms of powers of 10. So in fact we can see that 116.72 divided by 21.689 can be written as 10 raised to the 2 into 1 plus 10 raised to 0 into 6 plus 10 raised to minus 1 into 7 plus 10 raised to minus 2 into 2 divided by similarly for the denominator. You can easily write the denominator as 2 into 10 raise to power 1 plus 1 into 10 raise to 0 plus 6 into 10 raised to minus 1 plus 8 into 10 raise to minus 2 plus 9 into 10 raise to minus 3. So, essentially a rational number is a finite series in the base here the base is 10 of course, you could write it you could write the numbers to the base 8 or you could write the numbers to the base 3 or you could write it in binary whatever you like. So, what is a rational number? Rational number is a ratio of 2 series each of which is finite in the base with respect to which those numbers are written. The same idea is being used here except that the base is an indeterminate z. So, it is a generalization of the idea of rationality as understood from the context of integer the numbers. Now, this generalization does not stop there. There are certain important properties that rationals have and systems whose impulse response has a rational z transform are called rational systems. So, now let us define rational systems. Rational and of course, the word rational system immediately refers to an LSI system otherwise it has no meaning. So, rational LSI system is an LSI system whose impulse response has a rational z transform and of course, the LSI system that we saw a few minutes ago is an example. Rational systems are very important. In fact, almost all through this course in the sequel we shall be talking about rational systems. We shall of course, take examples of irrational systems, but we shall actually design only rational system. There is a good reason for that. Today it is only rational systems that have a meaningful realization. What is mean? What is meant by realizing a system? Realizing a system means translating a system into a hardware and or software structure. Implementing a system with concrete components and operations and of course, they must be finite in number. You can always conceive of a system that requires an infinite number of operations to implement. That is of course, of no practical significance. Unless a system can be implemented with finite resources, the system is not very useful in practice. It is only rational systems that can be implemented with finite resources as far as LSI systems go. That is why rational systems are of such great importance and relevance to us. So you see the system that we saw a minute ago is indeed an example of rational system. And now of course, let us complete the job of inverting the Z transform for which we had set out. You see, we had set out to invert y Z equal to 1 by 1 minus half Z inverse times 1 by 1 minus 1 third Z inverse. And we had seen that this is essentially 3 by 1 minus half Z inverse minus 2 by 1 minus 1 third Z inverse. And of course, the region of convergence is mod Z greater than half. And now you see, this is a sum of 2 Z transforms. Take this term, when mod Z is greater than half, 3 by 1 minus half Z inverse from the linearity of the Z transform must correspond to the sequence 3 times half to the power of n u n. And since mod Z is greater than half, mod Z is of course, greater than 1 third. So, for this sequence 2, with this expression with mod Z greater than half can also be thought of as the same expression with mod Z greater than 1 third. Mod Z greater than 1 third is of course, true. So, therefore, the inverse Z or the inverse the sequence corresponding to this or we might call it the inverse Z transform of y Z which is y n is 3 times half raised to power of n u n minus 2 into 1 third raised to the power of n u n from the linearity of the Z transform. Example brings a very important point to light. You notice that in this example, the output Z transform is the product, it is the convolution of the sequences x n and h n. It is the convolution of the input with the impulse response, but it also happens to be a linear combination of the input and the impulse response. This is something very peculiar to rational Z transforms. In fact, you also see why this happens. You notice that we inverted the Z transform by decomposing into partial fractions. When you decompose into partial fractions what you are saying is each of the terms in the denominator give rise to one term in the partial fraction expansion. And therefore, a product of various terms in the denominator leads to a sum of the same or a linear combination of the same terms. Now, we introduce one more idea here. In a rational Z transform, denominator equal to 0 gives us what are called the poles of the system. We will explain in a minute why they are called the poles. Well, I should not call it the system necessarily, system or sequence as the case may be. We say system if that rational Z transform is the Z transform of the impulse response. And we say sequence if we are talking about the Z transform of a sequence. Now, the numerator equal to 0 gives us the 0s of the system or the sequence and we will explain in a minute why this nomenclature. Let us visualize the magnitude of the Z transform. You see the magnitude of the Z transform is always non-negative. So, you can visualize the magnitude of the Z transform plotted on the Z plane. So, I draw a three dimensional or a seemingly three or pseudo three dimensional picture here. I have the Z plane and the vertical. Remember the vertical is only in one direction. It is mod HZ. Now, wherever there is a 0, you can visualize mod HZ now as a tent on this Z plane. That is easy to visualize because you see a tent means it must be only you cannot have a tent under the ground. So, a tent must be only above the surface and that is true because mod HZ is non-negative. So, this tent, what happens to the tent wherever there is a 0, the tent joins the ground at that point. What happens to the tent wherever there is a pole, the tent rises upwards. You see, goes towards infinity. That is why it is called a pole. The point where the denominator becomes 0 leads to the tent reaching the sky by virtue of a pole and the point where the numerator equal to 0, the tent goes and touches the ground. So, that is why this terminology poles and zeros. Anyway, the poles are very important. If you have a linear shift invariant system, now we define what a rational system is. So, in a rational system, we call the pole the front stage actors and the zeros, the king makers or back stage manipulators. You see, when you make a partial fraction expansion of a rational Z transform, it is the poles which tell you what terms you should have. What do the zeros do? The zeros influence the coefficients of these terms. So, zeros do not explicitly figure in the partial fraction expansion. They implicitly figure. They are king makers. They say, who should go forward, who should go back stage. The actual actors which you see on front stage are the poles. The poles are what determine what kind of response that system has, impulse response. In fact, if you look at the poles of the input and the poles of the system, together they tell you the poles of the output. What role do the zeros play? The zeros determine how important, how unimportant each pole is because they influence the terms, the numerators of the partial fraction expansion. Is that right? So, zeros do not explicitly figure in the nature of the response. But the zeros are important. Just as in a drama, the front stage is important. The actors on the stage are important, as are all the people manipulate. See, if they do not, if the people at back stage suddenly turn the lights off, the actors are of no good. That is what the zeros do. Sometimes the zeros kill one of the poles. They do that, turn the lights off on that pole. So, the zeros are back stage manipulators. Anyway, the zeros and poles together determine the nature. In fact, here we can see that, you see, when you have an input and an impulse response, both of which have rational z transforms, the poles of the input and the poles of the z of the impulse response together determine the poles of the output. That is easy to see. Because in fact, the poles of the output are a union of the poles of the input and the poles of the impulse response. That is very easy to see, because the z transform of the output is the product of the z transform of the input and the z transform of the impulse response. And therefore, it has to be the poles have to be the union of the pole. Of course, but the zeros can play a role. They can, they might come in, as I said, turn the lights off. They might cancel some of the poles. That can happen. If they do not cancel, all the poles will be there. And that also tells us. Now, you know, when you do a partial fraction expansion, it also tells us that the output is going to have terms which come from the input and which come from the impulse response. In a way, in a rational situation, in a situation where both the input and the impulse response are rational, the output can also be thought of in some sense broadly, not, do not, do not take this literally, but broadly as a linear combination of the input and the impulse response. So, convolution also leads to a linear combination. Broadly, do not take this literally. We shall see more of this in the next lecture. Thank you.