• In an ideal, perfectly linear system, a monochromatic signal, that is, one made of a single sinusoidal component at frequency ${\displaystyle f_0}$, can only be changed into another monochromatic signal at the same frequency.
• If instead the system shows non-linear effects, which always happens for large enough amplitudes of the input signal, it can distort the signal, which simply means introducing new harmonics that do not belong to the spectrum of the original signal.
• An easy example of distortion occurs when a piece of music is played too loud (or at normal volume, but on low-quality equipment). The signal is "clipped" or flattened at its extremes, so the output signal is no longer a faithful reproduction of the input signal. Essentially, the main aim of high fidelity is to ensure a response that is as linear as possible over the widest range of frequencies and intensities of the input signal.
• Intermodulation, however, can also be exploited usefully to "enrich" with harmonics a musical sample whose sound would otherwise be "flat".
• You can experience the effect of clipping distortion directly with the "Aumenta e taglia" (amplify and clip) function of the Fourier app. For a guide, see experiments on distortion.

Intermodulation in radio and in the human ear

In the simplest case, intermodulation distortion arises from the non-linear combination of two sinusoids. One operation that produces this combination is multiplying the signals together. Recall, in fact, that the only operations allowed by a linear transformation are the algebraic sum of signals and the multiplication of signals by a constant. Multiplying signals together, instead, produces components with frequencies equal to the sum and difference of the initial frequencies, as can be seen immediately from the following trigonometric identity:

${\displaystyle \sin ({\omega }_{1}t)\cos ({\omega }_{2}t)=\frac{1}{2}\sin [({\omega }_1+{\omega }_2)t]+\frac{1}{2}\sin [({\omega }_1-{\omega }_2)t]}$.

A non-linear system produces these products even without an actual multiplier. If, for example, the output y contains, besides a term proportional to the input x, a small quadratic term,

${\displaystyle y=ax+b{x}^2}$,

and the input is the sum of two sinusoids, ${\displaystyle x=\sin ({\omega }_{1}t)+\sin ({\omega }_{2}t)}$, the term ${\displaystyle x^2}$ contains the product

${\displaystyle 2\sin ({\omega }_{1}t)\sin ({\omega }_{2}t)=\cos [({\omega }_1-{\omega }_2)t]-\cos [({\omega }_1+{\omega }_2)t]}$,

that is, the difference and sum frequencies. The other two terms of ${\displaystyle x^2}$, ${\displaystyle {\sin }^2({\omega }_{1}t)}$ and ${\displaystyle {\sin }^2({\omega }_{2}t)}$, produce the doubled frequencies ${\displaystyle 2f_1}$ and ${\displaystyle 2f_2}$ and a constant component.

In radio this operation is performed in order to transmit (modulate) an audio signal, that is, one at acoustic frequencies, on an electromagnetic carrier at radio frequency. Several annotated examples can be found on the page on radio.

In the human ear, a multiplication of this kind seems to take place when intense signals are received. The ear therefore does not simply record and faithfully reproduce what it receives from outside, but adds new harmonic components to the sound, which in some cases can be perceived as extraneous sounds. On this subject see the pages on acoustic effects and illusions and on beats.

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