10: Fourier Analysis- 1
10:
F
OURIER
A
NALYSIS OF
C
OMPLEX
S
OUNDS
Amplitude, loudness, and decibels
Several weeks ago we found that we could synthesize complex sounds with
a particular frequency,
f,
by adding together sine waves from the harmonic series
with a fundamental equal to the frequency
(
f, 2f, 3f, 4f,…).
We can reverse the
process: a complex sound with particular frequency can be analyzed and
quantified by the amplitude spectrum: the relative amplitudes of the harmonics.
Our goal today is to understand how the pitch, loudness, and timbre of a
sound are represented in a spectrum. However, rather than use analog Band Filter,
like last week, we will use the
P
RAAT
program, which has powerful analysis
software built in to calculate and graph the spectrum of a recorded sound.
For the last couple of weeks we have used the amplitudes to represent the
different waves.
We can represent these in a table (the choice for the amplitude of
the fundamental wave to be 2V is completely arbitrary):
Another way to represent the relative amplitudes, which is used by
P
RAAT
, is
to graph the power for each harmonic.
Since the sound power is proportional to
the square of the amplitude for each harmonic, these numbers often get quite small
for the higher harmonics.
For this reason, the power is often expressed as a decibel.
The power for each harmonic in
decibels
(dB) is:
Relative Power
(dB)
=
P
n
= 10 log
10
[ (
A
n
/A
1
)
2
] = 20 log
10
[ (
A
n
/A
1
)
]
Since the logarithm of zero is negative infinity, the Relative Power (dB) for
the even harmonics is "Error".

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