Digital Sound & Music: Concepts, Applications, & Science, Chapter 4, last updated 6/25/2013
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sound louder. This is because the square wave is at its peak level more of the time as compared
to the sine wave. To account for this difference in perceived loudness, RMS amplitude (root-
mean-square amplitude) can be used as an alternative to peak amplitude, providing a better
match for the way we perceive the loudness of the sound.
Figure 4.3 Sine wave
representing sound
Figure 4.4 Sine wave representing a higher
amplitude sound
Figure 4.5 Square wave
representing sound
Rather than being an instantaneous peak level, RMS amplitude is similar to a standard
deviation, a kind of average of the deviation from 0 over time. RMS amplitude is defined as
follows:


Equation 4.9 Equation for RMS amplitude ,
Notice that squaring each sample makes all the
values in the summation positive. If this were
not the case, the summation would be 0
(assuming an equal number of positive and
negative crests) since the sine wave is perfectly
symmetrical.
The definition in Equation 4.9 could be
applied using whatever units are appropriate
for the context. If the samples are being
measured as voltages, then RMS amplitude is
also called RMS voltage. The samples could also be quantized as values in the range determined
by the bit depth, or the samples could also be measured in dimensionless decibels, as shown for
Adobe Audition in Figure 4.6.
For a pure sine wave, there is a simple relationship between RMS amplitude and peak
amplitude.
Aside: In some sources, the term RMS
power is used interchangeably with RMS
amplitude or RMS voltage. This isn’t very
good usage. To be consistent with the
definition of power, RMS power ought to
mean “RMS voltage multiplied by RMS
current.” Nevertheless, you sometimes see
term RMS power used as a synonym of RMS
amplitude as defined in Equation 4.9.
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