 So, one welcome to the 14th lecture on the subject of discrete time signal processing and its applications. This lecture will continue with where we left in the previous lecture namely on the properties of the Z-transform. We have introduced the Z-transform in the previous lecture and we had started looking at some of its properties. I would like to recall a couple of points from the previous lecture before we proceed to say more about the Z-transform and its properties. We had looked at the expression for the Z-transform. Let us recapitulate that. So, if you have a sequence Xn then we said its Z-transform is given by summation over all n, all integer n I mean Xn Z raised to the power minus n and this always has a region of convergence or ROC as it is often called for short. Now the region of convergence is always a region between two concentric circles with the circles centered at the origin and we said that one of the concentric circles could possibly have radius 0 and the other one could possibly have radius infinity. So, 0 and infinity are also possible radii. It was also a point that needed to be addressed to look at whether the boundaries are included in the region of convergence. The boundaries may or may not be included and that needs to be checked in each case individually. One guideline is if there is a singularity on the boundary, a point where that quantity diverges then of course that boundary cannot be included in the region of convergence. These are about the Z-transform in general and its definition and its contents. Z-transform always has an expression with a region of convergence. Yes, there is a question. Yes, that is a good question. It says you know we had said alright, so let us come to that. We had said that Z is of the form r e raised to the power j omega. So, the question is omega seems to have a physical significance and that physical significance is the angular normalized angular frequency, is not it? However, the question is does r have a physical significance? r does indeed have a physical significance. There are two ways of understanding r. One way is you know a formal way. That is it is the rate at which the exponential should multiply that sequence so as to bring the Z-transform into convergence. That is the formal way of understanding r. So, if you look at it x Z is equal to summation over all n, n from minus to plus infinity x n and now let us write this in terms of r and omega. So, e raised to the power minus j omega n and r raised to the power minus n. So, r is the exponential parameter in a way, exponential or curtailing parameter if you may call it that. Essentially it ensures that the sequence converges on multiplication by this exponential. So, r is in some sense the rate at which the exponential must you know grow to capture the sequence or on the way of looking at it is r to the power minus n is the exponential that must capture the sequence to bring it into convergence. That is the way to understand it. If we speak informal language, if we recall the analogy of the tiger that I gave last time r is the size or the strength of the case that you need to encapsulate the tiger. So, you have the sequence which refuses to converge which you cannot train for whom you cannot find a frequency response. So, you need r to capture it to tame it so that a frequency response exists that is the physical significance of r that is a good question. Any other questions from the previous lecture? Yes, there is a question. Yes. So, the question is can we view the Z transform as the discrete time Fourier transform of a sequence. Can you think of the Z transform as indicating the frequency content of some other sequence? Yes indeed of course, if you look at it in its region of convergence X Z which is summation n going from minus to plus infinity X n r raise the power minus n times e raise the power minus j omega n is the DTFT of X n r raise the power minus n. So, in fact it is the discrete time Fourier transform or in other words it indicates the frequency content in a suitably weighted sequence X n or X n weighted by an exponential. The exponential should strongly or it should be strong enough the exponential should be quick enough to capture the growth of X n so that this discrete time Fourier transform converges. So, in a way that Z transform on any particular circle in the region of convergence take any circle centered at the origin lying entirely in the region of convergence. On that circle the Z transform is essentially the sequence is the discrete time Fourier transform of the sequence given by X n multiplied by r raise to the power minus n where r is the radius of that circle that is the way to understand. Any other questions? Any other questions before we proceed? Yes, no other questions none at all. Alright, so then we will proceed to look at a few more properties of the Z transform. Now, we had seen some properties we had seen the Z transform is linear we had seen that it is linear as an operator. Secondly, we had seen the Z transform has a property of delay the delay property or the shift property is not it that is if you shift the sequence by D then the Z transform gets multiplied by my Z to the power minus T. Of course, we had talked about the region of convergence. We also saw what happens when we differentiate this was what we were looking at last time. So, we saw that if you differentiated the Z transform right. So, if you have D X Z D Z and multiply this by minus Z it is the Z transform of n times X that is interesting and we also looked at the region of convergence in each case we had seen the region of convergence right. We saw that you know normally a Z transform is analytic in its region of convergence that we are going to restrict ourselves to that class of Z transform. So, the same region of convergence would hold for this derivative save for the boundaries the boundaries need to be checked right. Now, we look at a very important property of the Z transform namely the property of convolution that is the real you know value of the Z transform. So, what happens when we convolve two sequences. So, let X in have a Z transform X Z with the region of convergence R X and let H in have a Z transform H Z with the region of convergence R H. We ask what is the Z transform of the convolution of X with H. Now, you see the principle that we will use to arrive at the answer is very similar to what we did in the case of the discrete time Fourier transform. There is no fundamental difference the only thing is that we have to remember that Z must lie you see in principle here let us take a Z to lie in the region of convergence of both namely the intersection of the regions of convergence. Now, there are tricky issues here what happens if the intersection is null can you still convolve where we would not answer these questions. They are tricky ones, but for the moment let us take a situation where there is a non null intersection of the region of convergence and let us pick a Z from that intersection of the regions of convergence of R and R X and R H. So, let us say for the moment I am saying for the moment assume that R X intersection R H is not null is not empty pick a Z from that intersection and use that Z in the discussion that follows or use any of those Z's in the discussion that follows. Now, let us look at the Z transform let us call X convolve with H let us call it Y. So, of course we need to find out the Z transform of Y. So, Y Z is of course summation N going from minus to plus infinity Y N Z raised to power minus N and we have picked a Z Z chosen suitably chosen in R X intersection R H as we have said. Now, expand Y. So, you have Y N is summation over all k integer X k H N minus k where upon we have Y Z is given by summation N over all integer summation k over all the integers X k H N minus k times Z raised to the power minus N and of course we use the same strategy as before we put N minus k equal to another variable N and now instead of N and k we go to k and N. So, of course k is as it is k runs over you see k and N independently run over all the integers. So, for a fixed k N runs over all the integers. Now, if N and k independently run over all the integers then for a fixed k M would also run over all the integers independent of k right and therefore, we could rewrite the summation here as summation k over all the integers summation M over all the integers. And we have X k H M and M is of course k plus M that becomes summation over I am not writing the limits every time they are understood H M Z raised to the power minus k Z raised to the power minus M and now we notice something very interesting there are terms that depend only on k there are terms that depend only on M. So, we can act summation M on the terms that depend only on M and that leaves us with you see I mean what I am saying is we could take this term and this term and operate the summation on M on them that gives us a quantity independent of k. So that leads us to summation k over all the integers X k Z raised to the power minus k in bracket summation over all M H M Z raised to the power minus M and note that this is essentially H Z and H Z is of course independent of k. So, I can draw H Z outside the summation because it is independent of k and that leaves me with this but then this happens to be X Z and therefore we have a product of H Z and X Z that is very interesting. So, Y Z is clearly X Z H Z. So, in other words this is a very important theorem it says that convolution in the natural domain leads to multiplication in the Z domain. This is not surprising because in particular if the unit circle or mod Z equal to 1 or R equal to 1 is included in the regions of conversions of both or at least definitely of Y Z then you would find that if this boils down to the specific property of the discrete time Fourier transform namely when I convolve two sequences if each of them has a discrete time Fourier transform their convolution would also have a discrete time Fourier transform given by the product of these. Now you see we can also give this an interpretation what we are saying is that if you have a linear shift invariant system with impulse response H n and if you have a sequence X n we are saying that if you see as such X n and H n may not have discrete time Fourier transform but again you could encapsulate these tigers in a cage multiply them by suitable exponentials. So, you could multiply X n multiply each of X n and H n by a suitable R raised to the power minus n and now you can use the property of the discrete time Fourier transform. So, you could now treat X you see the beauty is that now the same LSI system but now the impulse response should be viewed as H n R raised to the power minus n and here we have X n R raised to the power minus n given as the input. The beauty is that you would get Y instead of Y n you would get Y n R raised to the power minus n or in other words if you took the DTFT of this DTFT of X n R raised to the power minus n multiplied by DTFT of H n R raised to the power minus n is equal to DTFT of Y n R raised to the power minus n. So, you use the same property what we are saying is you use the same property of the discrete time Fourier transform but on a suitably treated or in a suitably tamed sequence or suitably tamed input and impulse response tamed by R raised to the power minus n that is what we are saying effectively. Of course in some cases you may in fact view this not as a taming but also as a license. So, for example you know if you look at it suppose we take X n equal to half raised to the power of n U n and H n equal to one third raised to the power of n U n we can easily obtain their Z transforms it is very easy to see that X z is one by one minus half z inverse with mod z greater than half and similarly H z is one by one minus one third z inverse with mod z greater than one third very easy to see. So, of course if you take the convolution if you happen if this H n happens to be the impulse response and of course you note that an LSI system with this impulse response H n would be stable because this sequence is easily seen to be absolutely summed. In fact you can find its absolute sum but that apart the sequence gives us a stable LSI system for the impulse response and X convolved with H which is Y has the Z transform Y Z given by X z H z which is one by one minus half z inverse into one minus one third z inverse and of course now the region of convergence is indeed the intersection of the two regions of convergence. So, mod z must be greater than half mod z must greater than one third. So, in total mod z must be greater than half because that is the intersection. Now mod z greater than half is like a license I told you you know we have been talking about encapsulating or taming the sequence but here mod z needs only to be greater than half. You can take for example mod z equal to three fourth in fact you can take z equal to three fourth if you like. Now three fourth raised to the power minus n is actually an exponentially growing sequence it is 4 by 3 raised to the power of n. So, in a way you are saying even if you multiply X n and H n by an exponentially growing sequence and take their discrete time Fourier transforms yet you could apply the property that we did a couple of points a couple of minutes ago what I meant was you know here we had X n r raised to the power minus n applied to H n r raised to the power minus n giving you Y n r raised to the power minus n. So, here r could be a quantity less than 1 r could be 3 by 4 in this example which means it is actually an exponentially growing sequence. So, there is a license also there is a bit of a license to allow the sequence to grow but still you can encapsulate you can allow it to apply the discrete time Fourier transform property. Of course, this is because the sequence itself is exponentially both of the sequences are exponentially decaying. So, you know if the exponential growth is slower than this decay then on and on this leads to a decay and therefore, you can have some license that is what we are saying in a bit.