Nonlinear Audio Processing

Nonlinear Systems

Digital signal processing is mainly based on linear time-invariant systems. The assumption of linearity and time invariance is undoubtedly valid for various technical systems, especially for scenarios where input and output signals are bounded to a specific amplitude range. To be precise, several analog audio processing devices have nonlinearities. These are valve amplifiers, analog effect devices, analog tape recorders, loudspeakers, and the human hearing mechanism at the end of the chain; compensation and simulation of these nonlinearities need nonlinear signal processing and, of course, a theory of nonlinear systems.

Nonlinear Device

An example of a nonlinear device could be a compressor with any ratio other than 1:1. When a signal passes through a nonlinear system, different kinds of distortion occur—the less linear the system, the more profound the distortion. One type of distortion is harmonic distortion, which essentially means adding harmonics to the existing frequency content of the audio signal. Analog components cannot be entirely linear. A total harmonic distortion (THD) specification measures the harmonic distortion content produced by an analog device under standard test conditions.

Analog Distortion

There are different flavors to an analog distortion. The ratio between low-order harmonics produced by a tube differs from that produced by a transistor. This significantly contributes to the different sounds that the tune and solid-state equipment produce. Although technically speaking, the lower the distortion, the better; harmonic distortion is an intimate part of the analog sound in general and the characteristics of the analog gear in particular. Digital systems can be perfectly linear and thus might not produce harmonic distortion. Although technically superior, many people find the digital sound lifeless and pale compared to the analog sound.

Inter-Modulation (IMD)

Another type of distortion is intermodulation. Like total harmonic distortion, it can be measured, and the specification given is called intermodulation (IMD). Like harmonic distortion, intermodulation distortion involves additional frequencies, but unlike harmonic distortion, these are not necessarily harmonically related to the sound. Therefore, intermodulation is usually harsh and unwanted. It is an integral part of any nonlinear system.

Nonlinear Mixing

One more example of nonlinearity in audio processing is nonlinear mixing. It accounts for producing so-called “phantom partials” in piano tones. The “phantom partials” appear at frequencies exactly harmonic to normal inharmonic string partials and at frequencies equal to the sums of the frequencies of normal inharmonic partials. The reason why nonlinear mixing can occur is that the tension varies during transverse vibration. This produces longitudinal string forces of phantom-partial frequency. They appear at the soundboard bridge and are coupled to the soundboard. “Phantom partials” were found in piano tones, in the motion of a piano bridge, in the longitudinal vibrations of a monochord string, in the acoustic output of a sound board coupled to a monochord string, and in the acoustic work of a guitar. Any plucked-string or struck-string instrument having an appreciable acoustical response to longitudinal string forces could be expected to produce phantom partials. The relation in frequency between phantom and normal partials, which varies with inharmonicity, may play a part in differentiating the timbre of tones at the same fundamental frequency in pianos of different sizes and designs.