Classification of Signals - Signal Processing of Random Physiological Signals - page 16

second moment. The definition is stated in a statistical sense, which is especially useful
for signals that do not have a definite starting or stopping time.
In the case of the Time Duration of a signal, the moments are normalized by
dividing the moments by the energy associated with the signal [the generalized energy
equation (3.9)]

∞

x 2 dt E = (3.9)

−∞

Then, the normalized first moment is shown in (3.10):

−∞

tx 2 ( t ) dt

t 0 =

∞
−∞

(3.10)

x 2 ( t ) dt

And the normalized second-central moment of the square of the time function is given
by (3.11):

∞
−∞

1
2

t 0 ) 2 x 2 ( t ) dt ( t −

∞
−∞

= (3.11)

T e

x 2 ( t ) dt

where T e defines the measure of the time duration, and t 0 is the normalized first moment
or mean about the origin. Therefore, the time duration ( T e ) is in reality the normalized
second moment about the mean, t 0 .

3.2.1 Bandwidth
What is the bandwidth of the signal? Most engineers are very familiar with the definition
of bandwidth, but let us review the various definitions. Bandwidth is intended to indicate
a range of frequencies within which most of the energy lies. Engineers use the half power
or 3 dB point, if a frequency response plot is available (Fig. 3.1).
However, the bandwidth of a general signal is defined in terms of the derivative
of the time function as shown in (3.12).

⎡
⎣

⎤
⎦

dx ( t )
dt

2

1
2

∞
−∞

dt

1
2

∞
−∞

W e = Hz (3.12)

π

x 2 ( t ) dt

Caution : This definition has problems with discontinuities in the signals.