Digital Sound & Music: Concepts, Applications, & Science, Chapter 2, last updated 6/25/2013
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any periodic sinusoidal function, regardless of its complexity, can be formulated as a sum of
frequency components. These frequency components consist of a fundamental frequency and
the harmonic frequencies related to this fundamental. Fourier's theorem says that no matter how
complex a sound is, it's possible to break it down into its component frequencies that is, to
determine the different frequencies that are in that sound, and how much of each frequency
component there is.
Fourier analysis begins with the fundamental frequency of the sound the frequency of
the longest repeated pattern of the sound. Then all the remaining frequency components that can
be yielded by Fourier analysis i.e., the harmonic frequencies are integer multiples of the
fundamental frequency. By “integer multiple” we mean that if the fundamental frequency is ,
then each harmonic frequency is equal to for some non-negative integer
n
.
The Fourier transform is a mathematical operation used in digital filters and frequency
analysis software to determine the frequency components of a sound. Figure 2.17 shows Adobe
Audition‟s waveform view and a frequency analysis view for a sound with frequency
components at 262 Hz, 330 Hz, and 393 Hz. The frequency analysis view is to the left of the
waveform view. The graph in the frequency analysis view is called a frequency response
graph or simply a frequency response. The waveform view has time on the x-axis and
amplitude on the y-axis. The frequency analysis view has frequency on the x-axis and the
magnitude of the frequency component on the y-axis. (See Figure 2.18.) In the frequency
analysis view in Figure 2.17, we zoomed in on the portion of the x-axis between about 100 and
500 Hz to show that there are three spikes there, at approximately the positions of the three
frequency components. You might expect that there would be three perfect vertical lines at 262,
330, and 393 Hz, but digitized sound is not a perfectly accurate representation of sound. Still,
the Fourier transform is accurate enough to be the basis for filters and special effects with
sounds.
Figure 2.17 Frequency analysis of sound with three frequency
components
Aside: "Frequency response"
has a number of related usages
in the realm of sound. It can
refer to a graph showing the
relative magnitudes of audible
frequencies in a given sound.
With regard to an audio filter,
the frequency response shows
how a filter boosts or
attenuates the frequencies in
the sound to which it is applied.
With regard to loudspeakers,
the frequency response is the
way in which the loudspeakers
boost or attenuate the audible
frequencies. With regard to a
microphone, the frequency
response is the microphone's
sensitivity to frequencies over
the audible spectrum.
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