Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 462

7.2.5.1 The Root-Mean Square (RMS) Bandwidth, B rms

Consider a low-pass process X ( t ) with a zero-mean value. We found that a PSD has
certain important properties (e.g., it is a nonnegative and a real function). Now we
will recall that a PDF has the same characteristics. Therefore, there is a similarity
between a PSD and a PDF with exception that the area under a PSD is not
necessarily unity. Next, if we normalize a PSD by dividing it with its area, the
normalized PSD will have the unity area, and thus equivalent behavior to that of
the PDF, which can be useful in different analyses.

S XX ð f Þ

S XX norm ð f Þ¼

S XX ð f Þ d f :

(7.57)

Ð 1

1

Knowing that the standard deviation is a measure of the spread of a density
function, we can define the analog quantities for the normalized PSD, just changing
x for f , and PDF for PSD, as shown below ( m ¼ 0):

t

1

t

1

p

s 2

STD ¼ s ¼ ¼ x 2 f X ðxÞ d x ! f 2 S XX norm ð f Þ d f : (7.58)

1 1

A parameter of a PSD which is equivalent to a standard deviation of a PDF is
called a root-mean squared bandwidth and is denoted as B rms

t

Ð 1

t

1 f 2 S XX ð f Þ d f

0

B rms ¼ f 2 S XX norm d f ¼ : (7.59)

Ð 1

S XX ð f Þ d f

1

0

Note that, as mentioned before, the bandwidth B rms is defined only for positive
frequencies, so the limits of the integrals are between 0 and 1 (i.e., only the right
part of the PSD).
The above concept can also be extended to a band-pass process. For more
details, see [PEE93, p. 205].

7.2.5.2 Half-Power Bandwidth

A half-power bandwidth is a bandwidth in which a PSD is greater than or equal to
half of its peak value S XX max , as illustrated in Fig. 7.13a for a LP process. This
quantity is also called a 3-dB bandwidth because

10 log ð 1 = 2 Þ¼ 3dB : (7.60)