We have seen in the previous module that ideal filters cannot be implemented in practice. And so, the natural question is, what is the most general, realizable LTI transformation? So, we know that linearity implies that we only use sums and multiplications. Time invariance implies that we only use multiplications by constants. And realizability implies that we only use a finite number of past and future samples. We've seen this chart before and these are the three ingredients that make up a linear time and variance system. Under this hypothesis, we can always express the input output relationship of a linear time and variance system, as a linear combination of output samples equal a linear combination of input samples. This system and this equation uses capital M input and capital N output samples. These are the limits of the summations. And the question is, given this relationship, how do we compute the frequency response of the system implemented by this equation? One way to do it, we've seen it in the past, is try to compute the inputs response explicitly and then take the Fourier transform. But for complex system, this would become quickly unwieldy. So we need a new tool and the new tool goes under the name of z-transform. Z-transform is a formal operator that maps a discrete time sequence, x[n], onto a function of the complex variable z. It is defined as the sum, for n that goes from minus infinity to plus infinity, of x[n] times z to the -n. Think of it as a formal operator or alternatively as the extension of the DTFT formula to the whole complex plane. It's very easy to see that the z-transform computed for z equal to e to the j omega, is actually the DTFT. This is a relationship that will reappear. Again if you have the z-transform and you compute the z-transform along the unit circle. Which is the locus, on the complex plane, for which z is equal to e to the j omega, then you have the discrete-time Fourier transform. In this class, we will consider the z-transform mainly as a formal operator. So we will not delve very deep into its mathematical subtleties. But there are two key properties that are absolutely essential to what follows. The first one is linearity. So if you take the z-transform of the linear combination of two sequences what do you get is a linear combination of z-transforms. And the second property is a time shift property. If you take the z-transform of the time delayed sequence, for instance this case x of small n minus big N, the z-transform is the z-transform of the original sequence multiplied by z to the minus big N. With these two properties, we can take the z-transform of a constant coefficient difference equation. So we take z-transform left and right and this CCDE here and by exploiting the linearity and the time shift property, what we get is an input, output relationship, like so. The z-transform of the output is equal to the z-transform of the input times a function H(z). And this function is defined as follows. It is a ratio of two polynomials in z to the minus 1 where the coefficients of the polynomials are the coefficients that appear in the CCDE. In particular, in the case of the numerator, you have the coefficients bk that are the coefficient's way that inputs samples in the CCDE. Whereas for the denominator, you have the coefficients ak that weigh the output samples in the CCDE. Now if you set z equal to e to the j omega, H of z become H of e to the j omega, and you have the frequency response of the system defined by the CCDE. But the reason is because, of course, the whole equation becomes like so, and you recognize this as the equation that relates input and output in a filtering operation. So H is indeed a filter and it's frequency response is easily computed by a z-transform. It should be clear now why we use the notation X of e to the j omega, although we hadn't defined the z-transform in the beginning of the class. But now this notation should make a lot more sense. Another thing that should make more sense now is the notation that we have been using for the delay operator. Indeed, the transfer function of a delay by one system is z to the minus 1. To appreciate the power of the z-transform formalism, let's revisit the leaky integrator and derive it's frequency response in a few easy steps. So, from the CCDE of the leaking integrator, which we remember very well, it's very easy to derive the z-transform. We take z-transforms left to right. Then we rearrange terms and we find out that the transfer function of the system H of z is simply 1 minus lambda divided by 1 minus lambda z to the minus 1. One remaining question that we have to answer before we start using z-transform with reckless abandon is when does the z-transform exist? For the z-transform, existence is equivalent to convergence of the power series that defines it. And so we will define for each case, a region of convergence that determines the points on the complex plane for which the z-transform exists. One property to remember is that because the z-transform is a power series, when it converges, it converges absolutely. So that means that the convergence of the z-transform depends only on the magnitude of z and not its phase. Let's look at three different cases. The first case is the simplest. When we have a finance support sequence. Z-transform is the sum of a finite number of terms. So convergence is not a problem. In other cases, the region of convergence of the z-transform has circular symmetry. In the sense that because of the absolute convergence, if the z-transform converges in one point, it will converge for all points in the plane with the same