Nonlinear Filters

Smoothing windows, IIR filters, and FIR filters are linear because they satisfy the superposition and proportionality principles

L {ax(t) + by(t)} = aL {x(t)} + bL{y(t)},

where a and b are constants, x(t) and y(t) are signals, L{•} is a linear filtering operation, and their inputs and outputs are related through the convolution operation.

A nonlinear filter does not meet the preceding conditions, and you cannot obtain its output signals through the convolution operation, because a set of coefficients cannot characterize the impulse response of the filter. Nonlinear filters provide specific filtering characteristics that are difficult to obtain using linear techniques. The median filter is a nonlinear filter that combines lowpass filter characteristics to remove high-frequency noise and high-frequency characteristics to detect edges.