 So, if there are no other questions, we will then take this very example further. In fact, it was good that this question was asked, because it leads us to the next limitation of the discrete time Fourier transform. The discrete time Fourier transform, as you see, does not exist for all sequences. In fact, the discrete time Fourier transform exists when the summation corresponding to the discrete time Fourier transform converges. What if the summation does not converge, obviously, does not exist. So, then what do we do? Do we have some other transform domain where we can study the same sequence or do we have to make do with natural domain studies or natural domain processing. It turns out that we do have a more general transform, which allows us to deal with sequences which are not aptitude to summable as well. Let us take an example. So, let us take a more troublesome sequence. I say more troublesome because, you know, as we said, the un sequence, the unit step sequence at least does not produce an unbounded output for all bounded inputs. But the sequence that we soon see, if it becomes the impulse response of an LSI system, it more or less always wishes to produce an unbounded output, except for some very chosen bounded inputs. That is the interesting thing. You know, in its, the system is, you know, so there are, I mean, instability also has different grades. The accumulator is an unstable system, but unstable, you know, it is like saying the person is insane, but you know, there are only certain points or certain situations where he exhibits insanity. But here you have a system which normally behaves insanely. It is very rarely the system exhibits sanity and that system is, let an LSI system have the following impulse response, h n is 2 raised to the power of n u n. Now, this LSI system would, at the slightest provocation, exhibit an unbounded output. In fact, it is very clear that this impulse response is not absolutely summable. So, it is an unstable system and you see, I mean, when you have an unstable system, what do you do? You want to study its frequency response. Now, you know, I liken the frequency response of a system to the behavior under training. So, when you give a complex exponential, what does it do? You know, you have trained the system and you, so, you know, I would like then to take an analogy. Here, you could think of a system like this, like a ferocious tiger and a system where the DTFT converges, maybe even u n perhaps at the extreme or definitely half raised to the power of n u n, like a little dog. Now, it is not too difficult to train a dog. You know, you can just hold a dog in open space and open, you know, in front of you and slowly by using rewards, train it. On the other hand, trying to train a ferocious tiger is dangerous both for the trainer and for people around. Is that right? And therefore, the only way to train a ferocious tiger is to encapsulate the tiger in a cage and then, of course, use rewards or punishments to train. Now, this system that we have here is like a tiger and we need to encapsulate such systems first in a safe cage and then train it. Train it means see what happens when you give it specific inputs. How do we encapsulate the tiger in a cage? We encapsulate the tiger in a cage by multiplying it. You know, what is a cage? A cage is stronger than the tiger. So, we encapsulate the tiger in a cage by forcing an even more troublesome sequence on that system. So, multiply h n by r raised to the power minus n. r is a number greater than 0. So, for example, here if you happen to multiply this sequence by any such exponential where r is greater than 2, it would get encapsulated, isn't it? Here r greater than 2 would encapsulate the tiger and now, you can of course, you can take the DTFT. So, we can take the DTFT of h, the encapsulated tiger, you can train the encapsulated tiger. That is obvious. For example, take r equal to 2.5, then h n r raised to the power minus n would certainly be, you know, the DTFT of that sequence exists. Now, you see that is what we are now going to do in our effort to deal with such unstable system. We are going to encapsulate. We are going to first capture their naughty behavior and therefore, instead of multiplying by e raised to the power j omega n, we multiply. So, sequences like this, you see sequences like this need to be multiplied by r raised to the power of n or minus n, e raised to the power minus j omega n and then summed. r is like a radius and e raised to the power j omega, so r e raised to the power j omega is the complex number that we are talking about here. Now, you know the role played by r and e raised to the power j omega, put z equal to r e raised to the power j omega. So, essentially we are saying, consider summation n going from minus to plus infinity x n z raised to the power minus n, we call this the z transform of x n and we denote it by capital x of z for obvious reasons. So, of course, a z transform unlike the discrete time Fourier transform is a function of a complex variable. The discrete time Fourier transform is a function of a real variable omega. In fact, in a way there is a relationship between the z transform and the discrete time Fourier transform. If the z transform can be evaluated, if the z transform can be evaluated for r equal to 1, then that gives its DTFT. So, z transform for r equal to 1 is the DTFT, if that can be evaluated, it may not be possible to evaluate it, it may not converge.