 So, one welcome to the 13th lecture on the subject of digital signal processing and its applications. In this lecture, we intend to introduce a new transform but before we do that, we would like to ensure that we have understood the previous transform entirely, the discrete time Fourier transform and therefore I would like to allow a couple of few questions. You know that we clarify any doubts that might still be relevant before we proceed to discussing a new transform. So, we will take a few questions from the class. Are there any questions, doubts or difficulties? I may mention that we also have a course model page, one should make use of the technical discussion forum there to discuss concepts. Yes. So, any doubts or difficulties so far, not at all. So, the question is what can we say about the discrete time Fourier transform? What can we say about the DTFT of un? The sequence un is very important in discrete time signal processing. It is a sequence which is 1 for n not equal to 0, in fact for n greater than equal to 0 you know and 0 for n less than 0. So, essentially it is 1 for the entire positive set of integers n, positive and I mean 0 and positive integers and so it is called a unit step because it looks like that, it looks like a unit step, it looks like a unit step. Now, this unit step function is a very important function in discrete time signal processing. One importance is that it when multiply, when you multiply the unit step by any function it retains the positive side of samples and destroys the negative side right. So, it is useful in concept, it is also useful in describing what is called the accumulator system. So, an accumulator system as the name suggests, accumulates the inputs, so it keeps taking the sum of what it already has with the current input. So, if you consider this system and if you take the LSI system described by this equation where the output at a current time is equal to the output at the previous time plus the input at the current time, what it does is it keeps accumulating whatever is coming now it accumulates in itself. So, it keeps taking a sum until that point in time, this is called an accumulator system. Now, if we treat it in fact this can be looked upon as an LSI system, if you treat the accumulators having operated all the way from minus infinity right and the impulse response of this LSI system is un, that is easily seen because if you have an impulse then if it the impulse is given to an accumulator the output is 1 starting from 0 onwards. The question is one way to ask the question is would the accumulator have a frequency response that is another way to reframe the same question. The answer strictly is no that is because if you look at the discrete time Fourier transform of the or so called DTFT I mean that does not exist really DTFT of un it would be summation n going from 0 to infinity e raised to power minus j omega n which is actually divergent because the common ratio it is a geometric progression with common ratio equal to e raised to power minus j omega and the modulus of this common ratio is 1 so such a geometric progression is known to be divergent. However, some people like to write this as the sum 1 by 1 minus e raised to power you know some people write it and say it should be accepted except for omega equal to 0. Some people like to write that that is not entirely correct actually this is divergent. However I mean the way to interpret this is that if you happen to give the accumulator an input other than un so among the complex exponentials if you happen to give the accumulator a complex exponential input other than un you could possibly hope for a non unbounded or a bounded output that is the interesting part. So, although this system if you look at the LSI system its impulse response is not absolutely summable. So, it is very clear that the system is unstable Bbo unstable, but then you know it is it exhibits an unbounded input for some selected bounded outputs I am sorry it exhibits an unbounded output for some selected bounded inputs. So, you know it is not unlike some other systems that we will soon see this system is selective in its instability. So, some people like to call it marginally stable or marginally unstable of course these are all informal terms they have no real formal meaning right otherwise the system is unstable strictly speaking it does not have a frequency response in the true sense. But the question whether it produces a an unbounded output for every bounded input has the answer no it does produce bounded outputs for several bounded inputs including some of the complex exponentials, but you know which needs to be seen carefully alright. In particular when you take Un itself if Un is given as the input to the system the output is un is unbounded clearly if you give a constant 0 frequency input so to speak it has an unbounded output. But if you give it for example the highest possible frequency that you can see in discrete time where you alternate 1 minus 1 1 minus 1 omega equal to pi there actually the output is not unbounded the output is bounded by 1, but the output is divergent still right so these are the subtle points in that sense the frequency response does not exist anyway so that is one does not one should not really say that it has a DTFT does not alright any other questions.