fircls
Constrained-least-squares FIR multiband filter design
Syntax
b = fircls(n,f,amp,up,lo)
fircls(n,f,amp,up,lo,'design_flag')
Description
b = fircls(n,f,amp,up,lo) generates
a length n+1 linear phase FIR filter b.
The frequency-magnitude characteristics of this filter match those
given by vectors f and amp:
fis a vector of transition frequencies in the range from 0 to 1, where 1 corresponds to the Nyquist frequency. The first point offmust be0and the last point1. The frequency points must be in increasing order.ampis a vector describing the piecewise-constant desired amplitude of the frequency response. The length ofampis equal to the number of bands in the response and should be equal tolength(f)-1.upandloare vectors with the same length asamp. 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 n, fircls increments
it by 1.
fircls(n,f,amp,up,lo,' enables
you to monitor the filter design, where design_flag')'design_flag' can
be
'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 and plots.
Note
Normally, the lower value in the stopband will be specified
as negative. By setting lo equal to 0 in
the stopbands, a nonnegative frequency response amplitude can be obtained.
Such filters can be spectrally factored to obtain minimum phase filters.
Examples
Constrained Least-Squares Lowpass Filter
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.
fvtool(b)
Algorithms
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.
References
[1] 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.
[2] 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.