We describe a filter parameter optimization technique carried out in the frequency domain that is representative of frequency domain design methods.

The design is most easily carried out with the system function for the IIR filter expressed in the cascade form as

The error in magnitude at the frequency ωk is G.A(ωk) – Ad(ωk) for 0 ≤ |ω| ≤ Ï€, where Ad(ωk) is the desired magnitude response at ωk.

As a performance index for determining the filter parameters, one can choose any arbitrary function of the errors in magnitude and delay.

In the error function ‘p’ denotes the 4K dimension vector of the filter coefficients.

We are led to define the error in delay as Tg(ωk) – Tg(ω0) – Td(ωk), where Tg(ω0) is the filter delay at some nominal centre frequency in the pass band of the filter.

Instead of dealing with the phase response Ï´(ω), it is more convenient to deal with the envelope delay as a function of frequency.

We know that the error in delay is defined as Tg(ωk) – Td(ωk). However, the choice of Td(ωk) for error in delay is complicated by the difficulty in assigning a nominal delay of the filter.

Similarly as in the previous question, the error in delay at ωk is defined as Tg(ωk)-Td(ωk), where Td(ωk) is the desired delay response.

Instead of dealing with the phase response Ï´(ω), it is more convenient to deal with the envelope delay as a function of frequency, which is