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Constrained-least-squares FIR multiband filter design

`b = fircls(n,f,amp,up,lo)`

fircls(n,f,amp,up,lo,'* design_flag*')

`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`

:

`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`0`

and 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 length of`amp`

is equal to the number of bands in the response and should be equal to`length(f)-1`

.`up`

and`lo`

are 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 `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.

`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.

[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.