TABLE OF CONTENTS The objective of this appendix is to provide the reader with the necessary mathematical basis for understanding the characteristics of signals which are related to their analysis and transmission through communication channels. Concepts and equations involving the Fourier transform, which constitute powerful tools for spectral analysis, are presented (Alencar, 1999).
The direct Fourier transform is a mapping from the time to the frequency domain,
it is sometimes denoted in the literature as F ( ?) = 
[ f ( t)].
The inverse Fourier transform can be defined in a similar manner, as
A Fourier transform pair is often denoted as f ( t) ? F ( ?).
Fourier transforms of known signals are presented in the following (Haykin, 1988).
The pulse function is used to aid in the simulation of digital signals, and is defined by the expression p T ( t) = A[ u( t + T/2) ? u( t ? T /2)], or
in which u( t) denotes the unit step function, defined as
The pulse function is illustrated in Figure B.1. The Fourier transform of the pulse function can be calculated as
which can be put in the form
and finally
in which Sa ( x) = sin( x)/ x is the sampling function. This function converges to 1, as x goes to zero. The sampling...
