Page 50 - Fundamentals of Communications Systems

2.2 Chapter Two

x(t) is called an energy signal when E x < ∞. Energy signals are normally
associated with pulsed or finite duration waveforms (e.g., speech or a finite
length information transmission). In contrast, a signal is called a power signal if
it does not have finite energy. In reality, all signals are energy signals (infinity is
hard to produce in a physical system) but it is often mathematically convenient
to model certain signals as power signals. For example, when considering a
voice signal it is usually appropriate to consider the signal as an energy signal
for voice recognition applications but in radio broadcast applications we often
model voice as a power signal.

EXAMPLE 2.1
A pulse is an energy signal:
1
⎧
⎨ √ 0 ≤ t ≤ T p
x(t) = T p E x = 1 (2.2)
0 elsewhere
⎩

A common signal that will be used frequently in this text is the sinc function.
Definition 2.2 The sinc function is
sin(πx)
sinc(x) = (2.3)
πx
It should be noted that no standard definition exists among authors for sinc(•)
so care must be exercised in using results from other authors. This text uses
the definition of sinc(•) as adopted by Matlab.

EXAMPLE 2.2
Not all energy signals have finite duration:
sin(2πWt)
x(t) = 2W = 2Wsinc(2Wt) E x = 2W (2.4)
2πWt

EXAMPLE 2.3
A voice signal. Figure 2.1 shows the time waveform for a computer-generated voice
saying “Bingo.” This signal is an obvious energy signal due to its finite time duration.

Definition 2.3 The signal power, P x ,is
1 T m /2
2
P x = lim |x(t)| dt (2.5)
T m →∞ T m −T m /2
Note that if E x < ∞, then P x = 0 and if P x > 0, then E x =∞.