# Minimum Phase IIR Filter from Attenuation Data 1.0

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Created : Sep 8, 2007

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## MINPHASEFITIIR Minimum Phase Transfer Function Fit ...

MINPHASEFITIIR minimum_phase Transfer Function Fit from Attenuation Data.

N, D, K]=MINPHASEFITIIR(Fk, Ak, NP) finds the iir_filter transfer function K*N(z)/D(z) of minimum order that fits the attenuation data in Ak and Fk. Ak is a vector of attenuation in dB and Fk is a vector of normalized frequencies associated with Ak. That is, the n-th values in Ak and Fk, Ak(n) and Fk(n) specify an attenuation and associated frequency to be met by the filter. The algorithm assumes that the input data is a piecewise linear description of the attenuation specifications to be met.

k(1)>0 is required and it is assumed that the attenuation between 0 and Fk(1) has zero slope. Fk(end) must be less than 1. 0, which is the Nyquist frequency (i. e. , one half the sampling frequency) as used in the Signal Processing Toolbox. It is also assumed that the attenuation slope is zero beyond Fk(end).

NP is the number of poles (and zeros) to add to the minimum number determined by the algorithm. If not given, NP=2 is chosen. The optimum value for NP cannot be predetermined. Simple attenuation characteristics may work well with NP=0. Attenuation characteristics having sharper transitions or a greater number of passbands often require a greater NP. Increasing NP does not always lead to greater fit accuracy.

MINPHASEFITIIR(Fk, Ak, NP, OPTIONS) includes the structure variable OPTIONS which sets options for the function FMINSEARCH that is used to find the optimum transfer function. See the help text for FMINSEARCH for information on setting options.

Notes: (1) No phase information is needed since the fit is to a minimum phase transfer function for which there is a unique relationship between the gain (or attenuation) and phase. (2) Attenuation is inverse of gain, so in dB it is the negative of the filter gain in dB. (3) The bilinear transformation is used to map the discrete time requirements into continuous time requirements. The design is then implemented using the Irons-Gilbert algorithm, and the final results are transformed back into the discrete domain.

Example: A bandpass filter with arithmetic passband symmetry.

Fk=linspace(0. 25, 0. 75, 51);

Ak=25*(1-cos(3*pi*(Fk-0. 5)));

NP=2;

Result is 6th order and has peak error of about 0. 4 dB in the passband.

N, D, K]=MINPHASEFITIIR(Fk, Ak, NP) finds the iir_filter transfer function K*N(z)/D(z) of minimum order that fits the attenuation data in Ak and Fk. Ak is a vector of attenuation in dB and Fk is a vector of normalized frequencies associated with Ak. That is, the n-th values in Ak and Fk, Ak(n) and Fk(n) specify an attenuation and associated frequency to be met by the filter. The algorithm assumes that the input data is a piecewise linear description of the attenuation specifications to be met.

k(1)>0 is required and it is assumed that the attenuation between 0 and Fk(1) has zero slope. Fk(end) must be less than 1. 0, which is the Nyquist frequency (i. e. , one half the sampling frequency) as used in the Signal Processing Toolbox. It is also assumed that the attenuation slope is zero beyond Fk(end).

NP is the number of poles (and zeros) to add to the minimum number determined by the algorithm. If not given, NP=2 is chosen. The optimum value for NP cannot be predetermined. Simple attenuation characteristics may work well with NP=0. Attenuation characteristics having sharper transitions or a greater number of passbands often require a greater NP. Increasing NP does not always lead to greater fit accuracy.

MINPHASEFITIIR(Fk, Ak, NP, OPTIONS) includes the structure variable OPTIONS which sets options for the function FMINSEARCH that is used to find the optimum transfer function. See the help text for FMINSEARCH for information on setting options.

Notes: (1) No phase information is needed since the fit is to a minimum phase transfer function for which there is a unique relationship between the gain (or attenuation) and phase. (2) Attenuation is inverse of gain, so in dB it is the negative of the filter gain in dB. (3) The bilinear transformation is used to map the discrete time requirements into continuous time requirements. The design is then implemented using the Irons-Gilbert algorithm, and the final results are transformed back into the discrete domain.

Example: A bandpass filter with arithmetic passband symmetry.

Fk=linspace(0. 25, 0. 75, 51);

Ak=25*(1-cos(3*pi*(Fk-0. 5)));

NP=2;

Result is 6th order and has peak error of about 0. 4 dB in the passband.

**• MATLAB Release: R14SP1**

**Demands:****Minimum Phase IIR Filter from Attenuation Data 1.0 scripting tags:**attenuation, iir filter, minimum phase, attenuation data, function.

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