DISCRETE-TIME SIGNAL ANALYSIS USING THE Z-TRANSFORM AND INVERSE Z-TRANSFORM VIA THE CAUCHY RESIDUE THEOREM: APPLICATIONS TO DIFFERENCE EQUATIONS AND RECURSIVE SEQUENCES

Abstract

Z-transform has been used as an analysis tool for years. Our motive is to deeply study this technique and all its basics to solve any type of discrete-time signal. In addition, we are focused to solve the discrete domain difference equations.Z-transform has a specialty that is it can modify itself according to the nature of signal or system. First, we have studied basic elements and the crucial aspects of inspecting a system. The important element of mapping a real domain function into a complex domain is to view it in a region of convergence. This region defines that the function is analytic in it. That is how we can create an outline of that transformed function. The introduction, significance and applications of Z-transform are briefly discussed in Chapter 1. While we further studied the properties of Z-transform along with their proofs. The inverse Z-transform is the main step in order to solve difference equations. Although there are different techniques of calculating the inverse Z-transform, but we have applied the methodology of Cauchy residue theorem. The solution of famous difference equation is detailed in Chapter 3. We have concluded that for discrete domain Z-transform is best for analysis and inverse Z-transform is advantageous for developing the compact forms of many recursive relations like Legendre polynomial, Fibonacci sequence and Chebyshev difference equation.