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4.3. RANDOM SIGNAL ANALYSIS
The RMS value (t/1) of a random signal x(t) is defined by
lim
T+oo T
T
f
x
2
(r) dr
0
( 1 )
where Tis the analysis averaging time, which for 1 00% accuracy must tend to infinity.
In practice, this is impossible, and T must be restricted to a finite value. This restriction
leads to an error in the RMS measurement. A commonly used expression for the normal-
ized standard deviation (or error)±
e
of an RMS measurement is
€
1
2
0ff
for BT
>
5
( 2)
where B is the analysis bandwidth in Hz and T is the averaging time of the analysis in
seconds. A common parameter in analyses of this type is the number of statistical de-
grees of freedom (k) which is given by
k
=
2 BT
(3)
It is common practice to work with a value of 50 for k, this being an efficient analysis
value.
Thus, if
2 BT =50
BT = 25
Substituting back for BT in equation 2 gives
€
=
±10% for BT
=
25
The value of BT will be taken as 25 in the following analysis.
4.3.1. Selection of Averaging Time
The optimum scanning rate for an analysis is that which gives the lowest total analysis
time over a chosen frequency range. From the considerations above, the scanning rate
may be connected with the averaging time required by specifying that one bandwidth
must never be scanned in less than two averaging times -
one averaging time being al-
lowed for full response of the RC integrator and a further averaging time for settling of
the Recorder pen to its steady reading. The time taken to scan one bandwidth is given
the term "dwell time" To.
For
€
<
±10% (section 4.3)
25
TAV~g
(4)
Once the minimum averaging time has been calculated from equation 4, the Level Recor-
der PAPER SPEED setting can be calculated for a particular analysis from the required fil-
ter dwell time. This follows directly from the above considerations and from equation
4
as:
45
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