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An Analog-Style Frequency Shifter

In this post, I will give an overview of a well-known method for designing frequency shifter effects in analog electronics, and some notes on implementing a digital version.

Frequency shifting is a special case of single sideband (SSB) modulation, a device often used in radio electronics. SSB in telecommunications involves shifting audio frequencies by e.g. 10 MHz so they can be transmitted as radio waves, then shifting down 10 MHz at the receiver. For this reason, SSB is well-studied, and there are a few different ways to implement it. This post describes a design that has found use in electronic music applications, where the shift is small (up to ±20 kHz, and often in tiny shifts like 1 Hz). Wikipedia calls it a "Hartley modulator," but I can't find any source to corroborate that terminology.

A fantastic reference to look at for the design of an analog frequency shifter is [Haible1996], which provides a full set of schematics of a high-quality real-world analog unit. I'll only provide details up to an abstract block diagram that can be easily digitized.

This post starts off pretty theoretical, with an attempt to explain how the frequency shifter works. Admittedly, this is an explanation more for myself than it is for others, but I hope it helps. If not, feel free to skip the first two sections and check out the following links which might explain it better: [Smith2007] [Boschen2016] [Boschen2020]

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Exploring Modal Synthesis

Modal synthesis is simple in theory: a method of synthesizing sound where an excitation signal is fed into a parallel bank of resonators [Bilbao2006]. Each resonator, known as a "mode," has the impulse response of an exponentially decaying sine wave and has three parameters: frequency, amplitude, and decay time. It's a powerful method that excels in particular for pitched percussion like bells and mallet instruments.

Despite its conceptual simplicity, modal synthesis has a daunting number of parameters to tune -- three times the number of modes, to be exact -- and setting them by hand isn't practical or musically expressive. How are we supposed to get great sound design when we have so many degrees of freedom? In interface design parlance, what we want is a divergent mapping [Rovan1997] that takes a small number of controls, maybe four knobs, and maps them to the several dozen frequencies, amplitudes, and decay times that control the modal synthesizer.

While it is possible to reverse engineer modes from a recorded sample (see [Ren2012] for a particularly impressive application), I'm most interested in "parametric" modal synthesis where each model derives from mathematical formulas that allow the user to navigate around a multidimensional space of timbres. I found a number of such algorithms while shopping around the literature from both computer music and mechanical engineering, so I decided to summarize them in a concise reference for sound designers and composers.

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