Digital Signal Processing Reference
In-Depth Information
U
m
The mean value is the (constant) value U
m
which is part of the
rectangular area . This area has the same size and period as the
area - calculated by integration - between curvature and time axis .
The rectangular area is the same as the
area between curvature and time axis
Area: The "Integral of 0 to T" = U
m
T
Mean value: U
m
= (1/T)
·
"Integral of 0 to T"
Illustration 138:
Precise definition of the (arithmetical) mean value
As is clear from the above Illustration and the notes and definitions given there the precise procedure for
determining the (arithmetic) mean value is as follows: determine the area from 0 to T between the curve
and the time axis by means of integration. Transform this area into a rectangular area of the same size with
the same time duration T. The height of this rectangle represents the (arithmetic) mean value U
m
.
The subject „mean value“ is actually part of measuring technology. For this reason we
shall not go into all the possible forms of mean value here. The (arithmetic) mean value
is simply the average value from a sequence of numbers. If the sequence of numbers is,
for example, (21; 4; 7; -12) the mean value is U
m
= (21+ 4 + 7 + (-12)) / 4 = 5. Here it is
simply presented as the possible outcome of integration and its relevance as a technical
signal process described.
We can now see the following from Illustration 136 and Illustration 137:
• In the process of integration the mean value U
m
is only indicated precisely after 1s.
T = 1 s applies here
• The mean value U
m
for any other duration T can be determined by dividing the integral
value from 0 to T by the duration T
• In the context of signal technology the forming of the mean value describes a process
of inertia which is not able to recognise rapid changes (see Illustration 136) but only
an averaged value. This is similar to the behaviour of a lowpass filter. This calculates
a kind of „gliding mean value“ in which - similar to Illustration 52 and Illustration 68
- the mean value of overlapping „windows“ is shown continuously.
