B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

Figure 3.32
Interpretation of
the energy
spectral density
of a signal.
(a )
-Jo
H(f ) -j t-- 1:! f
fo
3 .7 Signal Energy and Energy Spectral Density 105
,,..- -.___/ I G (f )
1 2
lfo) 1 2 ____ '_, IYUJl 2
--- ---
-fo
0
I
fo
and Eq. (3.65) can be expressed as
Eg = L: Wg (f) df
(3.69a)
From the results in Example 3.16, the ESD of the signal g (t) = e-atu(t) is
2 1
Wg (f) = I
G(f) I
= (2 nf ) 2 + a 2
(3.69b)
3 .7.3 Essential Bandwidth of a Signal
The spectra of most signals extend to infinity. However, because the energy of a practical signal
is finite, the signal spectrum must approach O asf ➔ oo. Most of the signal energy is contained
within a certain band of B Hz, and the energy content of the components of frequencies greater
than B Hz is negligible. We can therefore suppress the signal spectrum beyond B Hz with little
effect on the signal shape and energy. The bandwidth Bis called the essential bandwidth of the
signal. The criterion for selecting B depends on the error tolerance in a particular application.
We may, for instance, select B to be that bandwidth that contains 95% of the signal energy.* The
energy level may be higher or lower than 95%, depending on the precision needed. We can use
such a criterion to determine the essential bandwidth of a signal. Suppression of all the spectral
components of g (t) beyond the essential bandwidth results in a signal g(t), which is a close
approximation of g (t).t If we use the 95% criterion for the essential bandwidth, the energy of
the error (the difference) g(t) - g(t) is 5% of E g
. The following example demonstrates the
bandwidth estimation procedure.
* Essential bandwidth for a low-pass signal may also be defined as a frequency at which the value of the amplitude
spectrum is a small fraction (about 5-10%) of its peak value. In Example 3.16, the peak of IG(f) I is 1/a, and it
occurs atf = 0.
t In practice the truncation is performed gradually, by using tapered windows, to avoid excessive spectral leakage
due to the abrupt truncation. 5