Page 52 - Fundamentals of Communications Systems

2.4 Chapter Two

x(t)
1
…
τ T T + τ
Figure 2.2 A periodic pulse train.

Throughout this text, signals will be considered and analyzed independent
of physical units but the concept of units is worth a couple of comments at this
point. Energy is typically measured in Joules and power is typically measured
in Watts. Most signals electrical engineers consider are voltages or currents,
so to obtain energy and power in the appropriate units a resistance needs to
2
be specified (e.g., Watts = Volts /Ohms). To simplify notation, we will just
define energy and power as earlier which is equivalent to having the signal x(t)
measured in Volts or Amperes and the resistance being unity (R = 1 ).

2.1.2 Periodic versus Aperiodic
A periodic signal is one that repeats itself in time.
Definition 2.4 x(t) is a periodic signal when

x(t) = x(t + T 0 ) ∀t and for some T 0 = 0 (2.8)

Definition 2.5 The signal period is
T = min(|T 0 |) (2.9)

The fundamental frequency is then
1
f T = (2.10)
T

EXAMPLE 2.6
A simple example of a periodic signal is

n 1
x(t) = cos(2π f m t) T 0 = T = (2.11)
f m f m

Most periodic signals are power signals (note if the energy in one period is
nonzero, then the periodic signal is a power signal) and again periodicity is a
mathematical convenience that is not rigorously true for any real signal. We
use the model of periodicity when the signal has approximately the property in
Eq. (2.8) over the time range of interest. An aperiodic signal is defined to be a
signal that is not periodic.