Page 64 - Fundamentals of Communications Systems

2.16 Chapter Two

EXAMPLE 2.24
Example 2.2 (cont.). For the Fourier transform pair of

1 | f |≤ W

x(t) = 2Wsinc(2Wt) G x (f ) = (2.64)
0 elsewhere
the 3-dB bandwidth, B 3 = W, and the 40-dB bandwidth, B 40 = W, are identical.
Integrating the power spectrum gives a 98% energy bandwidth of B 98 = 0.98W. These
bandwidths parameterization demonstrate that a sinc pulse is effective at distributing
the energy in a compact spectrum.

EXAMPLE 2.25
Example 2.3 (cont.). For the computer generated voice signal the 3-dB bandwidth is
B 3 = 360 Hz and the 40-dB bandwidth is B 40 = 4962 Hz. Integrating the power spec-
trum gives a 98% energy bandwidth of B 98 = 2354 Hz.

It is clear from the preceding examples that one parameter such as bandwidth
does not do a particularly good job of characterizing a signal’s spectrum. There
are many definitions of bandwidth and these numbers while giving insight into
the signal characteristics, do not fully characterize a signal.
For lowpass power signals similar ideas hold with G x (f ) being replaced with
S x ( f , T m ).
Definition 2.12 If a signal x(t) has a sampled power spectral density S x ( f , T m ), then
B X is determined as

10 log(max S x ( f , T m )) = X + 10 log(S x (B X , T m )) (2.65)
f
where S x (B X , T m ) > S x ( f , T m ) for | f | > B X .
Definition 2.13 If a signal x(t) has a sampled power spectral density S x ( f , T m ), then
B P is determined as

B P
S x ( f , T m )df
−B P
P = (2.66)
P x (T m )
2.2.4 Fourier Transform Representation of Periodic Signals
The Fourier transform for power signals is not rigorously defined and yet we
often want to use frequency representations of power signals. Typically the
power signals we will be using in this text are periodic signals that have a
Fourier series representation. A Fourier series can be represented in the fre-
quency domain with the help of the following result
∞

δ( f − f 1 ) = exp[ j 2π f 1 t] exp[− j 2π ft]dt (2.67)
−∞