Sound Waves

Sum and difference tones

Non-linearity and distortion

When we drive a loudspeaker with a fluctuating voltage output by an amplifier, if the voltage is a purely sinusoidal fluctuation at a single frequency, we hope that the loudspeaker will produce a sound at the same single frequency. However, most real loudspeakers are slightly non-linear in their response; the sound output is not always perfectly directly proportional to the voltage input. This means that some additional frequency components get produced in the sound output even when only a single frequency is applied to the input. This is sometimes called non-linear distortion.

Non-linearity in a hi-fi system produces unwanted distortion, but for the ear, non-linear behaviour is a critical feature that enables the large dynamic range of human hearing. The non-linearity of the ear has been known for over a century, but it was relatively recently that the primary causes of non-linearity have been identified. The middle ear is quite linear at normal listening levels. It is the inner ear non-linearity that produces distortion, which can be heard, and measured in the ear canal. In fact, the measurement of distortion products in the ear canal is used as a hearing test for newborn infants and children, since the distortion products are absent for certain forms of hearing impairment! But what is a distortion product ?

Distortion (or intermodulation) products

If two frequencies, f₁ and f₂, are present in a sound, in principle, the ear non-linearity would produce intermodulation products equal to the sum and difference of the two frequencies, that is m f₁ = n f₂, for any whole numbers m and n. In particular, it turns out that a difference tone of 2f₁ – f₂ is quite audible, for certain choices of frequencies. The demonstration below illustrates this phenomenon. It is simply a constant sine wave and a downward swept sine wave on two channels. Nonlinearities in the ear cause the listener to hear an intermodulation tone when both channels are played at the same time. As a result, a rising tone is heard. If you don’t notice it at first try increasing the volume of your loudspeakers.

This technique allows psychoacousticians to probe the nonlinearities of the ear and to test the healthiness of the auditory system.