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.