Date: 10/15/97 at 16:54:53 From: Jose Nunez Subject: Laplace Transforms Can you explain me the overall concept of changing from Time-Domain to Frequency-Domain with a Laplace or a Z Transform? I think I understand its practical use for solving differential equations, but how are Complex-variable-transforms related to Frequency domain? Is there an easy way to understand it? What is the interpretation of the Zeroes and Poles?
Date: 10/15/97 at 19:23:39 From: Doctor Anthony Subject: Re: Laplace Transforms The Laplace transform is an aid in solving differential equations of a continuous-time system, and the z transform performs the same task using difference equations for a discrete-time system, e.g. digital transmissions. There are tables of z transforms used in the same manner as the Laplace transforms for finding the inverse functions, and there are many other analogies. The essential difference is that the Laplace transforms deal with continuous functions and the z transforms with discrete-time systems and difference equations. In studying the response to an arbitrary input signal you have a z transfer function (similar to the Laplace transfer functions) for a discrete-time system modelled by a difference equation. The z transfer function G(z) is given by b(m).z^m + b(m-1).z^(m-1) + .... P(z) G(z) = ---------------------------------- = ------- a(n).z^n + a(n-1).z^(n-1) + ... Q(z) P(z) is the z transform of the input sequence and Q(z) the z transform of the output sequence. Q(z) = 0 is called the characteristic equation of the discrete system, its order, n, determines the order of the system, and its roots are called the poles of the transfer function. The roots of P(z) = 0 are called the zeros of the transfer function. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.