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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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
College Definitions

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