Introduction to Z Transform Submitted By RC 00016Team

Introduction to Z – Transform Submitted By: RC 00016_Team 688 Ms Gauri Gupta(team leader) Ms Vibha Bhatnagar(1 st member) Ms Maya Makwana(2 nd member) 1

Frequency domain vs Time domain • Frequency domain is a term used to describe the analysis of mathematical functions or signals with respect to frequency. • (communications point of view) A plane on which signal strength can be represented graphically as a function of frequency, instead of a function of time. • control systems) Pertaining to a method of analysis, particularly useful for fixed linear systems in which one does not deal with functions of time explicitly, but with their Laplace or Fourier transforms, which are functions of frequency. • Speaking non-technically, a time domain graph shows how a signal changes over time, whereas a frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. 2

Cont: • A frequency domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal. • The frequency domain relates to the Fourier transform or Fourier series by decomposing a function into an infinite or finite number of frequencies. This is based on the concept of Fourier series that any waveform can be expressed as a sum of sinusoids (sometimes infinitely many. ) • In using the Laplace, Z-, or Fourier transforms, the frequency spectrum is complex and describes the frequency magnitude and phase. In many applications, phase information is not important. By discarding the phase information it is possible to simplify the information in a frequency domain representation to generate a frequency spectrum or spectral density. A spectrum analyser is a device that displays the spectrum. 3

The Direct Z-Transform The z-transform of a discrete time signal is defined as the power series (1) Where z is a complex variable. For convenience, the z-transform of a signal x[n] is denoted by X(z) = Z{x[n]} Since the z-transform is an infinite series, it exists only for those values of z for which this series converges. The Region of Convergence (ROC) of X(z) is the set of all values of z for which this series converges. We illustrate the concepts by some simple examples. 4