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# IIR filters

Module by: Ricardo Vargas. E-mail the author

An IIR filter describes a system with input x(n)x(n) and output y(n)y(n), related by the following expression

y ( n ) = k = 0 M b ( k ) x ( n - k ) - k = 1 N a ( k ) y ( n - k ) y ( n ) = k = 0 M b ( k ) x ( n - k ) - k = 1 N a ( k ) y ( n - k )
(1)

Since the current output y(n)y(n) depends on the input as well as on NN previous output values, the output of an IIR filter might not be zero well after x(n)x(n) becomes zero (hence the name “Infinite”). Typically IIR filters are described by a rational transfer function of the form

H ( z ) = B ( z ) A ( z ) = b 0 + b 1 z - 1 + + b M z - M 1 + a 1 z - 1 + + a N z - N H ( z ) = B ( z ) A ( z ) = b 0 + b 1 z - 1 + + b M z - M 1 + a 1 z - 1 + + a N z - N
(2)

where

H ( z ) = n = 0 h ( n ) z - n H ( z ) = n = 0 h ( n ) z - n
(3)

and h(n)h(n) is the infinite impulse response of the filter. Its frequency response is given by

H ( ω ) = H ( z ) | z = e j ω H ( ω ) = H ( z ) | z = e j ω
(4)

Substituting Equation 2 into Equation 4 we obtain

H ( ω ) = B ( ω ) A ( ω ) = n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n H ( ω ) = B ( ω ) A ( ω ) = n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n
(5)

Given a desired frequency response D(ω)D(ω), the l2l2 IIR design problem consists of solving the following problem

min a n , b n B ( ω ) A ( ω ) - D ( ω ) 2 2 min a n , b n B ( ω ) A ( ω ) - D ( ω ) 2 2
(6)

for the M+N+1M+N+1 real filter coefficients an,bnan,bn with ωΩωΩ (where ΩΩ is the set of frequencies for which the approximation is done). A discrete version of Equation 6 is given by

min a n , b n ω k n = 0 M b n e - j ω k n 1 + n = 1 N a n e - j ω k n - D ( ω k ) 2 min a n , b n ω k n = 0 M b n e - j ω k n 1 + n = 1 N a n e - j ω k n - D ( ω k ) 2
(7)

where ωkωk are the LL frequency samples over which the approximation is made. Clearly, Equation 7 is a nonlinear least squares optimization problem with respect to the filter coefficients.

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