Page 54 - Fundamentals of Communications Systems

2.6 Chapter Two

2.1.4 Continuous Time Signals versus Discrete Time Signals
A signal, x(t), is defined to be a continuous time signal if the domain of the
function defining the signal contains intervals of the real line. A signal, x(t), is
defined to be a discrete time signal if the domain of the signal is a countable
subset of the real line. Often a discrete signal is denoted by x(k), where k is an
integer and a discrete signal often arises from (uniform) sampling of a contin-
uous time signal, e.g., x(k) = x(kT s ), where T s is the sampling period. Discrete
signals and systems are of increasing importance because of the widespread
use of computers and digital signal processors, but in communication systems
the vast majority of the transmitted and received signals are continuous time
signals. Consequently since this is a course in transmitter and receiver design
(physical layer communications), we will be primarily concerned with contin-
uous time signals and systems. Alternately, the digital computer is a great
tool for visualization and discrete valued signals are processed in the com-
puter. Section 2.4 will discuss in more detail how the computer and specifically
the software package Matlab can be used to examine continuous time signal
models.

2.2 Frequency Domain Characterization of Signals
Signal analysis can be completed in either the time or frequency domains. This
section briefly overviews some simple results for frequency domain analysis. We
first review the Fourier series representation for periodic signal, then discuss
the Fourier transform for energy signals and finally relate the two concepts.

2.2.1 Fourier Series
If x(t) is periodic with period T , then x(t) can be represented as

∞
∞
j 2πnt
x(t) = x n exp = x n exp[ j 2π f T nt] (2.18)
T
n=−∞ n=−∞
where f T = 1/T and

1 T
x n = x(t) exp[− j 2π f T nt]dt (2.19)
T 0

This is known as the complex exponential Fourier series. Note a sine-cosine
Fourier series is also possible with equivalence due to Euler’s rule. Note in
general the x n are complex numbers. In words: A periodic signal, x(t), with
period T can be decomposed into a weighted sum of complex sinusoids with
frequencies that are an integer multiple of the fundamental frequency
(f T = 1/T).