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.