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iirlpnormc

Constrained least Pth-norm optimal IIR filter

Syntax

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

Description

[num,den] = iirlpnormc(n,d,f,edges,a) returns a filter having 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. A constrained Newton-type algorithm is employed. n and d should be chosen so that the zeros and poles are used effectively. See the Hints section. Always check the resulting filter using fvtool.

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

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

[num,den] = iirlpnormc(n,d,f,edges,a,w,radius) returns a filter having a maximum pole radius of radius where 0<radius<1. radius defaults to 0.999999. Filters that have a reduced pole radius may retain better transfer function accuracy after you quantize them.

[num,den] = iirlpnormc(n,d,f,edges,a,w,radius,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 the string 'inspect', no optimization will occur. This can be used to inspect the initial pole/zero placement.

[num,den] = iirlpnormc(n,d,f,edges,a,w,radius,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] = iirlpnormc(n,d,f,edges,a,w,radius,p,dens,initnum,initden) allows for the specification of the initial estimate of the filter numerator and denominator coefficients in vectors initnum and initden. 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] = iirlpnormc(...) returns the least-Pth approximation error err.

[num,den,err,sos,g] = iirlpnormc(...) returns the second-order section representation in the matrix SOS and gain G. For numerical reasons you may find SOS and G beneficial in some cases.

Hints

  • This is a weighted least-pth optimization.

  • Check the radii and location of the resulting poles and zeros.

  • If the zeros are all on the unit circle and the poles are well inside of 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 radius and the zeros are well inside of the unit circle, try increasing the order of the denominator or reducing the error weight in the passband.

  • If you reduce the pole radius, you might need to increase the order of the denominator.

The message

Poorly conditioned matrix. See the "help" file.

indicates that iirlpnormc cannot accurately compute the optimization because either:

  1. The approximation error is extremely small (try reducing the number of poles or zeros — refer to the hints above).

  2. The filter specifications have huge variation, such as a=[1 1e9 0 0].

Examples

This example returns a lowpass filter whose pole radius is constrained to 0.8

[b,a,err,s,g] = iirlpnormc(6,6,[0 .4 .5 1],[0 .4 .5 1],... 
[1 1 0 0],[1 1 1 1],.8);
hd = dfilt.df1sos(s,g); % Construct second-order sections filter.
fvtool(hd); % View filter's magnitude response

From the magnitude response shown here you see the lowpass nature of the filter. The pole/zero plot following shows that the poles are constrained to 0.8 as specified in the command.

References

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

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