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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-p^{th} 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

[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*
0.0028 0.0036 0.0018 0.0005 0.0001 0.0000

`den = `*1×13*
1.0000 -6.6498 22.6915 -51.5423 85.9098 -109.9888 110.4802 -87.5412 54.2750 -25.7000 8.8451 -1.9963 0.2255

Display the magnitude response in fvtool

fvtool(num,den)

This is a weighted least-p

^{th}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.

`filter`

| `freqz`

| `iirgrpdelay`

| `iirlpnormc`

| `zplane`