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Least square linear-phase FIR filter design

`b = firls(n,f,a)b = firls(n,f,a,w)b = firls(n,f,a,'ftype')b = firls(n,f,a,w,'ftype')`

`firls` designs a linear-phase FIR filter
that minimizes the weighted, integrated squared error between an ideal
piecewise linear function and the magnitude response of the filter
over a set of desired frequency bands.

`b = firls(n,f,a)` returns
row vector

These are type I (`n` odd) and type II (`n` even)
linear-phase filters. Vectors `f` and `a` specify
the frequency-amplitude characteristics of the filter:

`f`is a vector of pairs of frequency points, specified in the range between 0 and 1, where 1 corresponds to the Nyquist frequency. The frequencies must be in increasing order. Duplicate frequency points are allowed and, in fact, can be used to design a filter exactly the same as those returned by the`fir1`and`fir2`functions with a rectangular (`rectwin`) window.`a`is a vector containing the desired amplitude at the points specified in`f`.The desired amplitude function at frequencies between pairs of points (

*f*(*k*),*f*(*k*+1)) for*k*odd is the line segment connecting the points (*f*(*k*),*a*(*k*)) and (*f*(*k*+1),*a*(*k*+1)).The desired amplitude function at frequencies between pairs of points (

*f*(*k*),*f*(*k*+1)) for*k*even is unspecified. These are transition or "don't care" regions.`f`and`a`are the same length. This length must be an even number.

`firls` always uses an even filter order for
configurations with a passband at the Nyquist frequency. This is because
for odd orders, the frequency response at the Nyquist frequency is
necessarily 0. If you specify an odd-valued `n`, `firls` increments
it by 1.

The figure below illustrates the relationship between the `f` and `a` vectors
in defining a desired amplitude response.

`b = firls(n,f,a,w)` uses the weights
in vector `w` to weight the fit in each frequency
band. The length of `w` is half the length of `f` and `a`,
so there is exactly one weight per band.

`b = firls(n,f,a,'ftype')` and

`b = firls(n,f,a,w,'ftype')` specify
a filter type, where

`'hilbert'`for linear-phase filters with odd symmetry (type III and type IV). The output coefficients in`b`obey the relationThis class of filters includes the Hilbert transformer, which has a desired amplitude of 1 across the entire band.

`'differentiator'`for type III and type IV filters, using a special weighting technique. For nonzero amplitude bands, the integrated squared error has a weight of (1/*f*)^{2}so that the error at low frequencies is much smaller than at high frequencies. For FIR differentiators, which have an amplitude characteristic proportional to frequency, the filters minimize the relative integrated squared error (the integral of the square of the ratio of the error to the desired amplitude).

One of the following diagnostic messages is displayed when an incorrect argument is used:

F must be even length. F and A must be equal lengths. Requires symmetry to be 'hilbert' or 'differentiator'. Requires one weight per band. Frequencies in F must be nondecreasing. Frequencies in F must be in range [0,1].

A more serious warning message is

Warning: Matrix is close to singular or badly scaled.

This tends to happen when the product of the filter length and transition width grows large. In this case, the filter coefficients b might not represent the desired filter. You can check the filter by looking at its frequency response.

[1] Parks, Thomas W., and C. Sidney Burrus. *Digital
Filter Design*. New York: John Wiley & Sons, 1987,
pp. 54–83.

[2] Oppenheim, Alan V., Ronald W. Schafer,
and John R. Buck. *Discrete-Time Signal Processing*.
Upper Saddle River, NJ: Prentice Hall, 1999.

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