Least P-norm optimal FIR filter
b = firlpnorm(n,f,edges,a)
b = firlpnorm(n,f,edges,a,w)
b = firlpnorm(n,f,edges,a,w,p)
b = firlpnorm(n,f,edges,a,w,p,dens)
b = firlpnorm(n,f,edges,a,w,p,dens,initnum)
b = firlpnorm(...,'minphase')
[b,err] = firlpnorm(...)
b = firlpnorm(n,f,edges,a) returns
a filter of numerator order
n which represents
the best approximation to the frequency response described by
the least-Pth norm sense. P is set to 128 by default, which essentially
equivalent to the infinity norm. Vector
the band-edge frequencies for multiband designs.
an unconstrained quasi-Newton algorithm to design the specified filter.
a must have the
same number of elements, which can exceed the number of elements in
This lets you specify filters with any gain contour within each band.
However, the frequencies in
edges must also be
f. Always use
check the resulting filter.
b = firlpnorm(n,f,edges,a,w) uses
the weights in
w to weight the error.
w has one entry per frequency point (the same length as
firlpnorm how much emphasis to put
on minimizing the error in the vicinity of each frequency point relative
to the other points. For example,
b = firlpnorm(20,[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, and with emphasis placed on minimizing the error in the stopband.
b = firlpnorm(n,f,edges,a,w,p) where
a two-element vector [
pmin pmax] lets you specify
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.
be even numbers. The design algorithm starts optimizing the filter
pmin and moves toward an optimal filter in
pmax sense. When
p is the
not optimize the resulting filter. You might use this feature to inspect
the initial zero placement.
b = firlpnorm(n,f,edges,a,w,p,dens) specifies
the grid density
dens used in the optimization.
The number of grid points is [
default is 20. You can specify
dens as a single-element
cell array. The grid is equally spaced.
b = firlpnorm(n,f,edges,a,w,p,dens,initnum) lets
you determine the initial estimate of the filter numerator coefficients
initnum. This can prove helpful for difficult
optimization problems. The pole-zero editor in Signal Processing Toolbox™ software
can be used for generating
b = firlpnorm(...,'minphase') where
minphase' is the last argument in the argument
list generates a minimum-phase FIR filter. By default,
mixed-phase filters. Specifying input option '
firlpnorm to use a different optimization
method to design the minimum-phase filter. As a result of the different
optimization used, the minimum-phase filter can yield slightly different
[b,err] = firlpnorm(...) returns
the least-pth approximation error
firlpnorm, here are two
examples — the first designs a lowpass filter and the second
a highpass, minimum-phase filter.
% Lowpass filter with a peak of 1.4 in the passband. b = firlpnorm(22,[0 .15 .4 .5 1],[0 .4 .5 1],[1 1.4 1 0 0],... [1 1 1 2 2]); fvtool(b)
From the figure you see the resulting filter is lowpass, with the desired 1.4 peak in the passband (notice the 1.4 specified in vector a).
Now for the minimum-phase filter.
% Highpass minimum-phase filter optimized for the 4-norm. b = firlpnorm(44,[0 .4 .45 1],[0 .4 .45 1],[0 0 1 1],[5 1 1 1],... [2 4],'minphase'); fvtool(b)
As shown in the next figure, this is a minimum-phase, highpass filter.
The next zero-pole plot shows the minimum phase nature more clearly.
Saramaki, T, Finite Impulse Response Filter Design, Handbook for Digital Signal ProcessingMitra, S.K. and J.F. Kaiser Eds. Wiley-Interscience, N.Y., 1993, Chapter 4.