title: Nonlinear distortion
text: # Nonlinear distortion
  
  Nonlinear distortion is a term used (in fields such as electronics, audio and telecommunications) to describe the phenomenon of a non-linear relationship between the "input" and "output" signals of - for example - an electronic device.
  
  ## Model
  
  For many devices, a linear model is accurate only for small signal levels. For example, at 2 volts input, a typical audio amplifier might put out 20 V, meaning the linear gain is 10 V/V. For 3 V input, it might then output 30 V. However, the model implies that at 50 V input it would produce 500 V, which is not possible with most amplifiers.
  
  Mathematically, the input-output relationship of many devices should be described by a polynomial or Taylor series, as shown below.
  
  $v=\sum _{k=1}^{\infty }a_{k}u^{k}$
  
  For larger values of u, the higher order coefficients such as $a_{2}$ and $a_{3}$ come into play.
  
  ## Effects of nonlinearity
  
  Nonlinearity can have several effects, which are unwanted in typical situations. The $a_{3}$ term for example would, when the input is a sine wave with frequency $\omega$, result in an extra sine wave at $3\omega$, as shown below.
  
  $v=(a_{1}+{\frac {3}{4}}a_{3})sin(\omega t)-{\frac {1}{4}}a_{3}sin(3\omega t)$
  
  In certain situations, this spurious signal can be filtered away because the "harmonic" $3\omega$ lies far outside the frequency range used, but in cable television, for example, third order distortion could cause a 200 MHz signal to interfere with the regular channel at 600 MHz.
  
  Nonlinear distortion applied to a superposition of two signals at different frequencies causes the circuit to act as a frequency mixer, creating intermodulation distortion.
  
  