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Laplace Transforms


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
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