Even and Odd Functions - Harmonics

Harmonics

In signal processing, harmonic distortion occurs when a sine wave signal is sent through a memoryless nonlinear system, that is, a system whose output at time only depends on the input at time and does not depend on the input at any previous times. Such a system is described by a response function . The type of harmonics produced depend on the response function :

• When the response function is even, the resulting signal will consist of only even harmonics of the input sine wave;
  • The fundamental is also an odd harmonic, so will not be present.
  • A simple example is a full-wave rectifier.
  • The component represents the DC offset, due to the one-sided nature of even-symmetric transfer functions.
• When it is odd, the resulting signal will consist of only odd harmonics of the input sine wave;
  • The output signal will be half-wave symmetric.
  • A simple example is clipping in a symmetric push-pull amplifier.
• When it is asymmetric, the resulting signal may contain either even or odd harmonics;
  • Simple examples are a half-wave rectifier, and clipping in an asymmetrical class A amplifier.

Note that this does not hold true for more complex waveforms. A sawtooth wave contains both even and odd harmonics, for instance. After even-symmetric full-wave rectification, it becomes a triangle wave, which, other than the DC offset, contains only odd harmonics.