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Least P-norm optimal IIR filter


[num,den] = iirlpnorm(n,d,f,edges,a)
[num,den] = iirlpnorm(n,d,f,edges,a,w)
[num,den] = iirlpnorm(n,d,f,edges,a,w,p)
[num,den] = iirlpnorm(n,d,f,edges,a,w,p,dens)
[num,den] = iirlpnorm(n,d,f,edges,a,w,p,dens,initnum,initden)
[num,den,err] = iirlpnorm(...)
[num,den,err,sos,g] = iirlpnorm(...)


[num,den] = iirlpnorm(n,d,f,edges,a) returns a filter having a numerator order n and denominator order d which is the best approximation to the desired frequency response described by f and a in the least-pth sense. The vector edges specifies the band-edge frequencies for multi-band designs. An unconstrained quasi-Newton algorithm is employed and any poles or zeros that lie outside of the unit circle are reflected back inside. n and d should be chosen so that the zeros and poles are used effectively. See the Hints section. Always use freqz to check the resulting filter.

[num,den] = iirlpnorm(n,d,f,edges,a,w) uses the weights in w to weight the error. w has one entry per frequency point (the same length as f and a) which tells iirlpnorm how much emphasis to put on minimizing the error in the vicinity of each frequency point relative to the other points. f and a must have the same number of elements, which may exceed the number of elements in edges. This allows for the specification of filters having any gain contour within each band. The frequencies specified in edges must also appear in the vector f. For example,

[num,den] = iirlpnorm(5,12,[0 .15 .4 .5 1],[0 .4 .5 1],... 
[1 1.6 1 0 0],[1 1 1 10 10])

is a lowpass filter with a peak of 1.6 within the passband.

[num,den] = iirlpnorm(n,d,f,edges,a,w,p) where p is a two-element vector [pmin pmax] allows for the specification of the minimum and maximum values of p used in the least-pth algorithm. Default is [2 128] which essentially yields the L-infinity, or Chebyshev, norm. Pmin and pmax should be even. If p is 'inspect', no optimization will occur. This can be used to inspect the initial pole/zero placement.

[num,den] = iirlpnorm(n,d,f,edges,a,w,p,dens) specifies the grid density dens used in the optimization. The number of grid points is (dens*(n+d+1)). The default is 20. dens can be specified as a single-element cell array. The grid is not equally spaced.

[num,den] = iirlpnorm(n,d,f,edges,a,w,p,dens,initnum,initden) allows for the specification of the initial estimate of the filter numerator and denominator coefficients in vectors initnum and initden. initnum should be of length n+1, and initden should be of length d+1. This may be useful for difficult optimization problems. The pole-zero editor in Signal Processing Toolbox™ software can be used for generating initnum and initden.

[num,den,err] = iirlpnorm(...) returns the least-pth approximation error, err.

[num,den,err,sos,g] = iirlpnorm(...) returns the second-order section representation in the matrix sos and gain g. For numerical reasons it may be beneficial to use sos and g in some cases.


Least Pth-Norm Optimal IIR Filter Design

Design a Lowpass Filter with a Peak of 4.015 dB in Passband

[num,den] = iirlpnorm(5,12,[0 .15 .4 .5 1],[0 .4 .5 1],... 
[1 1.6 1 0 0],[1 1 1 10 10])
num = 1×6

    0.0025    0.0037    0.0019    0.0005    0.0001    0.0000

den = 1×13

    1.0000   -6.6504   22.6937  -51.5460   85.9115 -109.9830  110.4643  -87.5191   54.2543  -25.6864    8.8388   -1.9944    0.2252

Display the magnitude response.



  • This is a weighted least-pth optimization.

  • Check the radii and locations of the poles and zeros for your filter. If the zeros are on the unit circle and the poles are well inside the unit circle, try increasing the order of the numerator or reducing the error weighting in the stopband.

  • Similarly, if several poles have a large radii and the zeros are well inside of the unit circle, try increasing the order of the denominator or reducing the error weighting in the passband.


Antoniou, A., Digital Filters: Analysis, Design, and Applications, Second Edition, McGraw-Hill, Inc. 1993.

Extended Capabilities

Version History

Introduced in R2011a