In electronics, a frequency multiplier is an electronic circuit that generates an output signal and that output frequency is a harmonic (multiple) of its input frequency. Frequency multipliers consist of a nonlinear circuit that distorts the input signal and consequently generates harmonics of the input signal. A subsequent bandpass filter selects the desired harmonic frequency and removes the unwanted fundamental and other harmonics from the output. Frequency multipliers are often used in frequency synthesizers and communications circuits. It can be more economical to develop a lower frequency signal with lower power and less expensive devices, and then use a frequency multiplier chain to generate an output frequency in the microwave or millimeter wave range. Some modulation schemes, such as frequency modulation, survive the nonlinear distortion without ill effect (but schemes such as amplitude modulation do not). Frequency multiplication is also used in nonlinear optics. The nonlinear distortion in crystals can be used to generate harmonics of laser light. Theory A pure sine wave has a single frequency f 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 harmonics; frequency components at integer multiples nf of the fundamental frequency f. The distorted signal can be described by a Fourier series in f. The nonzero ck represent the generated harmonics. The Fourier coefficients are given by integrating over the fundamental period T: 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