Introduction
Audio signals describe pressure variations on the ear that are perceived as sound. We focus on periodic audio signals, that is, on tones.
A pure tone can be
written as a cosinusoidal signal of amplitude , frequency
, and phase angle
:
We have written the frequency in units of radians/second, with t in seconds, and the phase angle
to be in radians. An alternative is to express
the frequency in units of Hertz, abbreviated
Hz, given by
.
The perceived loudness of a pure tone is proportional to . The pitch of a pure tone is logarithmically related to the
frequency. Perceptually, tones separated by an octave (factor of 2 in frequency) are very similar. For Western
music, an octave is divided into 12 notes
that are equally spaced on a logarithmic scale. The ordering of notes in the
octave beginning at 220 Hertz is
shown in the applet below. Clicking on a note will display the corresponding
signal
and play the
note.
Additional Tones
A more complicated tone can be represented by a Fourier series — a
sum of pure tones whose frequencies are integer multiples (harmonics) of a fundamental
frequency :
The pitch of the tone is related to . The higher harmonics affect the richness or harshness of
the tone.
Typically the frequency components making up a tone are
displayed by the amplitude spectrum
of the tone, which is a plot of the coefficients vs. k. For example, tones from different
musical instruments have different amplitude spectra. This is primarily what
gives each instrument its unique sound.
In the applet below
you can select a tone to view the corresponding and amplitude
spectrum as well as play the tone.
Effect of Phase
We have shown only the amplitude spectrum because in many
situations the tone you perceive is relatively insensitive to the phase angles,
, of the terms in the Fourier series. To illustrate this, in
the applet below you can listen to the first 10 nonzero harmonics of the
Fourier series representation of a square wave with fundamental frequency 200 Hz and also listen to the tone after
randomization of the phase angles. Even though the waveform can be much
different after randomization of the phase angles, the perceived tone changes
little.
Harmonic Contribution
It is interesting to examine the contribution of individual harmonics in a tone. In the applet below is a tone made up of 20 equal-amplitude harmonics with fundamental frequency 200 Hz. By clicking on the amplitude spectrum you can insert or delete individual harmonics and listen to the effect on the tone. Filtering is the operation of removing selected frequency components from a signal. You can also listen to a low-pass filtered tone (5 highest frequency components removed), a high-pass filtered tone (5 lowest frequency components removed), and a band-pass filtered tone (both the 5 lowest and 5 highest frequency components removed). Of course you can also experiment with filtering by clicking on the amplitude spectrum to add or remove harmonics.
While using the applet, examine your two modes of listening. Normally we listen to a tone holistically, however we listen analytically to detect the presence or absence of an individual harmonic.
Filtering White Noise
White noise is a special signal that contains equal-amplitude components at all frequencies, not just at integer multiples of a fundamental frequency. In the applet below you can play white noise, low-pass filtered white noise (frequencies above 10,000 Hz removed), high-pass filtered white noise (frequencies below 1000 Hz removed), and band-pass filtered white noise (frequencies below 1000 Hz and above 10,000 Hz removed). We do not include frequency components above 20,000 Hz because they cannot be heard by most humans.