Editing Frequency multiplier (section)

==Theory==
A pure [[sine wave]] has a single frequency ''f'' 
:<math>x(t) = A\sin(2 \pi ft)\,</math>
If the sine wave is applied to a [[linear circuit]], such as a  non–distortion [[amplifier]], the output is still a sine wave (but may acquire a phase shift).
However, if the sine wave is applied to a [[nonlinear circuit]], the resulting distortion creates [[harmonic]]s; frequency components at integer multiples ''nf'' of the fundamental frequency ''f''. The distorted signal can be described by a [[Fourier series]] in ''f''.

:<math>x(t) = \sum_{k=-\infty}^{\infty} c_k e^{j 2 \pi k f t}.</math>

The nonzero ''c<sub>k</sub>'' represent the generated harmonics.  The Fourier coefficients are given by integrating over the fundamental period ''T'':

:<math>c_k = \frac{1}{2\pi}\int_{0}^{T} x(t) \, e^{-j 2 \pi k t / T}\, dt</math>

So a frequency multiplier can be built from a nonlinear electronic component which generates a series of harmonics, followed by a [[bandpass filter]] which passes one of the harmonics to the output and blocks the others.

From a conversion efficiency standpoint, the nonlinear circuit should maximize the coefficient for the desired harmonic and minimize the others. Consequently, the transcribing function is often specially chosen. Easy choices are to use an even function to generate even harmonics or an odd function for odd harmonics. See [[Even and odd functions#Harmonics]].  A full wave rectifier, for example, is good for making a doubler. To produce a times-3 multiplier, the original signal may be input to an amplifier that is over driven to produce nearly a square wave. This signal is high in 3rd order harmonics and can be filtered to produce the desired
x3 outcome.

YIG multipliers often want to select an arbitrary harmonic, so they use a stateful distortion circuit that converts the input sine wave into an approximate [[Dirac comb|impulse train]]. The ideal (but impractical) impulse train generates an infinite number of (weak) harmonics.  In practice, an impulse train generated by a monostable circuit will have many usable harmonics.  YIG multipliers using step recovery diodes may, for example, take an input frequency of 1 to 2&nbsp;GHz and produce outputs up to 18&nbsp;GHz.<ref>For example, the old Hewlett Packard 83590A.</ref> Sometimes the frequency multiplier circuit will adjust the width of the impulses to improve conversion efficiency for a specific harmonic.