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.


Applets by Michael Ross