Page 63 - Fundamentals of Communications Systems

Signals and Systems Review 2.15

not have a common definition across all engineering disciplines. The common
definitions of bandwidth used in engineering practice can be enumerated as

1. X dB relative bandwidth, B X Hz
2. P % energy (power) integral bandwidth, B P Hz
For lowpass energy signals we have the following two definitions

Definition 2.10 If a signal x(t) has an energy spectrum G x (f ), then B X is determined as

10 log(max G x (f )) = X + 10 log(G x (B X )) (2.60)
f

where G x (B X ) > G x (f ) for | f | > B X
In words, a signal has a relative bandwidth B X , if the energy spectrum is at
least XdB down from the peak at all frequencies at or above B X Hz. Often used
values for X in engineering practice are the 3-dB bandwidth and the 40-dB
bandwidth.

Definition 2.11 If a signal x(t) has an energy spectrum G x (f ), then B P is deter-
mined as

B P

G x (f )df
−B P
P = (2.61)
E x
In words, a signal has an integral bandwidth B P if the percent of the total
energy in the interval [−B P , B P ] is equal to P%. Often used values for P in
engineering practice are 98% and 99%.

EXAMPLE 2.23
Consider the rectangular pulse which is given as

1 0 ≤ t ≤ T p
x(t) = (2.62)
0 elsewhere
The energy spectrum of this signal is given as

2 2 2
G x (f ) =|X(f )| = T (sinc( fT p )) (2.63)
p
Figure 2.3 shows a normalized plot of this energy spectrum. Examining this plot carefully
produces the 3-dB bandwidth of B 3 = 0.442/T p and the 40-dB bandwidth of B 40 =
31.54/T p . Integrating the power spectrum in Eq. (2.63) gives a 98% energy bandwidth
of B 98 = 5.25/T p . These bandwidths parameterizations demonstrate that a rectangular
pulse is not very effective at distributing the energy in a compact spectrum.