The Fourier series is a powerful mathematical tool, and it applies to multiple branches of engineering and mathematics. The design of a frequency tripler is a good example of the Fourier series in action. For computer and electrical engineers, the Fourier series provides a way to represent any periodic signal as a sum of complex sinusoids via Euler’s formulas.
The purpose of the frequency tripler is to output a sinusoidal signal with a frequency that’s exactly three times the frequency of the input sinusoid.
![[Credit: Illustration by Mark Wickert, PhD]](https://cdn.prod.website-files.com/6634a8f8dd9b2a63c9e6be83/669d58e766a45ab4108acd1a_378853.image0.jpeg)
The circuit design for the tripler uses radio frequency circuit design principles, but the theory of operation is firmly rooted in signals and systems theory — the Fourier series modeling, to be exact.
The input to the tripler is
The output is
The action of the limiter circuit, which is a nonlinear system, is to clip the sinusoidal input and convert it to a square wave. The function sign() in Python acts as an ideal limiter because it outputs 1 when the input is greater than 0 and –1 when the input is less than 0.
A square wave contains only the odd harmonics due to the odd half-wave symmetry property. Here, 3f0 is of specific interest. The band-pass filter, centered on 3f0, allows you to keep the 3f0 term of the Fourier series and reject harmonics at other frequencies. Mathematically speaking, the frequency response of the band-pass filter will pass signals only in the vicinity of the center frequency, which is 3f0in this case.
The limiter needs to clip symmetrically to ensure odd half-wave symmetry. If odd half-wave symmetry is destroyed, even harmonics appear, and even harmonics are difficult to remove with a practical band-pass filter design because the second and fourth harmonics lie closer to the third harmonic signal you want to retain.
Therefore, this is a band-pass filter design issue you want to avoid. A small deviation from odd half-wave symmetry is acceptable because the second and fourth harmonics remain small.
