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An IIR filter describes a system with input and output , related by the following expression
Since the current output depends on the input as well as on previous output values, the output of an IIR filter might not be zero well after becomes zero (hence the name “Infinite”). Typically IIR filters are described by a rational transfer function of the form
where
and is the infinite impulse response of the filter. Its frequency response is given by
Substituting [link] into [link] we obtain
Given a desired frequency response , the IIR design problem consists of solving the following problem
for the real filter coefficients with (where is the set of frequencies for which the approximation is done). A discrete version of [link] is given by
where are the frequency samples over which the approximation is made. Clearly, [link] is a nonlinear least squares optimization problem with respect to the filter coefficients.
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