Constrained-least-squares FIR multiband filter design
n+1 linear phase FIR filter
The frequency-magnitude characteristics of this filter match those
given by vectors
f is a vector of transition frequencies
in the range from 0 to 1, where 1 corresponds to the Nyquist frequency.
The first point of
f must be
the last point
1. The frequency
points must be in increasing order.
amp is a vector describing the
piecewise-constant desired amplitude of the frequency response. The
amp is equal to the number of bands in
the response and should be equal to
vectors with the same length as
amp. They define
the upper and lower bounds for the frequency response in each band.
fircls always uses an even filter order for
configurations with a passband at the Nyquist frequency (that is,
highpass and bandstop filters). This is because for odd orders, the
frequency response at the Nyquist frequency is necessarily 0.
If you specify an odd-valued
it by 1.
you to monitor the filter design, where
'trace', for a textual display
of the design error at each iteration step.
'plots', for a collection of plots
showing the filter's full-band magnitude response and a zoomed view
of the magnitude response in each sub-band. All plots are updated
at each iteration step. The O's on the plot are the estimated extremals
of the new iteration and the X's are the estimated extremals of the
previous iteration, where the extremals are the peaks (maximum and
minimum) of the filter ripples. Only ripples that have a corresponding
O and X are made equal.
'both', for both the textual display
Normally, the lower value in the stopband will be specified
as negative. By setting
Design a 150th-order lowpass filter with a normalized cutoff frequency of rad/sample. Specify a maximum absolute error of 0.02 in the passband and 0.01 in the stopband. Display plots of the bands.
n = 150; f = [0 0.4 1]; a = [1 0]; up = [1.02 0.01]; lo = [0.98 -0.01]; b = fircls(n,f,a,up,lo,'both');
Bound Violation = 0.0788344298966 Bound Violation = 0.0096137744998 Bound Violation = 0.0005681345753 Bound Violation = 0.0000051519942 Bound Violation = 0.0000000348656 Bound Violation = 0.0000000006231
The Bound Violations denote the iterations of the procedure as the design converges. Display the magnitude response of the filter.
fircls uses an iterative least-squares algorithm
to obtain an equiripple response. The algorithm is a multiple exchange
algorithm that uses Lagrange multipliers and Kuhn-Tucker conditions
on each iteration.
 Selesnick, I. W., M. Lang, and C. S. Burrus. "Constrained Least Square Design of FIR Filters without Specified Transition Bands." Proceedings of the 1995 International Conference on Acoustics, Speech, and Signal Processing. Vol. 2, 1995, pp. 1260–1263.
 Selesnick, I. W., M. Lang, and C. S. Burrus. "Constrained Least Square Design of FIR Filters without Specified Transition Bands." IEEE® Transactions on Signal Processing. Vol. 44, Number 8, 1996, pp. 1879–1892.