Editing Ring modulation (section)

== Simplified operation ==
Denoting the carrier signal by <math>c(t)</math>, the modulator signal by <math>x(t)</math> and the output signal by <math>y(t)</math> (where <math>t</math> denotes time), ring modulation approximates [[multiplication]]:

:<math>y(t)=x(t) \; c(t).</math>

[[File:Frequenzy Mixer Mini-Circuits SBL-1.png|thumb|Double balanced high level [[frequency mixer]] Mini-Circuits SBL-1 with four Schottky diodes. [[Local oscillator|LO]] level +7 dBm (1.41 V<sub>p-p</sub>) and RF 1–500 MHz (ADE-1: 0.5–500 MHz)]]
[[File:Frequency mixer Mini Circuits ADE-1 macro.png|thumb|Macro of the ADE-1]]
[[File:Ring modulation two forms Diode-clipping or 'chopper' RM.svg|thumb|upright=1.1|An example of ring modulation on a sine wave of frequency <math>f</math> and a square wave of frequency <math>12f</math>, resulting in a complex sound using analog FM known as ''diode-clipping'' or ''chopper'' RM, producing a variation in amplitude of the square wave-like frequency on <math>12f</math><ref name="Roads"/>]]

If <math>c(t)</math> and <math>x(t)</math> are sine waves with frequencies <math>f_c</math> and <math>f_x</math>, respectively, then <math>y(t)</math> is the sum of two ([[Phase shift|phase-shifted]]) sine waves, one of frequency <math>f_c+f_x</math> and the other of frequency <math>f_c-f_x</math>. This is a consequence of the [[trigonometric identity]]:

:<math>\sin(u) \, \sin(v)=\frac{1}{2}\left(\cos(u-v)-\cos(u+v)\right).</math>

Alternatively, one can use the fact that multiplication in the [[time domain]] is the same as [[convolution]] in the [[frequency domain]].

Ring modulators thus output the [[combination tone|sum and difference]] of the frequencies present in each waveform. This process of ring modulation produces a signal rich in [[Partial (music)|partials]]. Neither the carrier nor the incoming signal are prominent in the output, and ideally, not present at all.

Two oscillators, whose frequencies were harmonically related and ring modulated against each other, produce sounds that still adhere to the harmonic partials of the notes but contain a very different spectral makeup. When the oscillators' frequencies are ''not'' harmonically related, ring modulation creates [[inharmonic]]s, often producing bell-like or otherwise metallic sounds.

If the carrier signal is a [[Square wave (waveform)|square wave]] of frequency <math>f_c</math>, whose [[Fourier expansion]] contains the [[Fundamental frequency|fundamental]] and a series of reducing-amplitude odd [[harmonic]]s:

:<math>c(t) = \sin f_ct + \frac 1 3 \sin 3f_ct + \frac 1 5 \sin 5f_ct + \frac 1 7 \sin 7f_ct + \ldots</math>

and the carrier frequency <math>f_c</math> is at least twice the maximum frequency of the modulating signal ''<math>x(t)</math>'', then the resulting output is a series of duplicates of ''<math>x(t)</math>'' at increasing regions of the frequency spectrum.<ref name="analog.com" /> For example, let ''<math>x(t)</math>'' represent a sine wave at 100&nbsp;Hz, and the carrier ''<math>c(t)</math>'' be an ideal square wave at 300&nbsp;Hz. The output then includes sine waves at 100±300&nbsp;Hz, 100±900&nbsp;Hz, 100±1500&nbsp;Hz, 100±2100&nbsp;Hz, etc., at decreasing amplitudes according to the Fourier expansion of the carrier square wave. If the carrier frequency is less than twice the upper frequency of the signal then the resulting output signal contains spectral components from both the signal and the carrier that combine in the time domain. 

Because the output contains neither the individual modulator or carrier components, the ring modulator is said to be a ''double-balanced'' mixer,<ref>{{cite web|url=https://www.electronics-notes.com/articles/radio/rf-mixer/double-balanced-mixer.php|title=Double Balanced Mixer – Theory; Circuit; Operation – Tutorial – Electronics Notes}}</ref> where both input signals are suppressed (not present in the output)—the output is composed entirely of the sum of the products of the frequency components of the two inputs.