Nonlinear Systems


Digital signal processing is mainly based on linear time-invariant systems. The assumption of linearity and time invariance is for sure valid for a great variety of technical systems. Especially for the systems 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 at the end of the chain the human hearing mechanism. Compensation and simulation of these nonlinearities need nonlinear signal processing and of course a theory of nonlinear systems.


Nonlinear device


An example of the nonlinear device could be a compressor with any ratio rather than 1:1. When a signal passes through a nonlinear system, different kind of distortion is occurs. 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 are incapable of being entirely linear. A specification called total harmonic distortion (THD) 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 ration between low-order harmonics produced by a tube is different from that produced by a transistor. This is a significant contributor 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 are capable of being perfectly linear and thus might not produce harmonic distortion. Although it is technically superior, many different many find the digital sound lifeless and pale compared to the analog sound.


Inter-Modulation (IMD)


Another type of distortion is inter-modulation. Like total harmonic distortion, it can be measured and the specification given is called inter-modulation (IMD). Just like harmonic distortion, inter-modulation distortion involves additional frequencies, but unlike harmonic distortion, these are not necessarily harmonically related to the sound. Therefore, inter-modulation 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 appears to account for the production of so-called “phantom partials” in piano tones. The “Phantom partials” are those that 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 acoustical output of a sound board coupled to a monochord string, and in the acoustical output 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 partials 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 size and design.


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