# firgr

Parks-McClellan FIR filter

## Description

example

b = firgr(n,f,a,w) uses a minimax filter design algorithm to design a real linear phase FIR filter b of length n+1. The filter has the best approximation to the desired frequency response described by f and a in the minimax sense. w is a vector of weights, one per band. When you omit w, all bands are weighted equally. For more information on the input arguments, see firpm.

b = firgr(n,f,a,ftype) design a filter of type specified by ftype

b = firgr(m,f,a,r), designs filters repeatedly until the minimum order filter as specified in m and that meets the specifications f,a is found. m can be one of 'minorder', 'mineven' or 'minodd'. r is a vector containing the peak ripple per frequency band. You must specify r. When you specify 'mineven' or 'minodd', the minimum even or odd order filter is found.

b = firgr({m,ni},f,a,r) where m is one of 'minorder', 'mineven' or 'minodd', uses ni as the initial estimate of the filter order. ni is optional for common filter designs, but it must be specified for designs in which firpmord cannot be used, such as while designing differentiators or Hilbert transformers.

b = firgr(n,f,a,w,e) specifies independent approximation errors for different bands. Use this syntax to design extra ripple or maximal ripple filters. These filters have interesting properties such as having the minimum transition width. e is a cell array of character vectors specifying the approximation errors to use. Its length must equal the number of bands.

b = firgr(n,f,a,s) is used to design filters with special properties s at certain frequency points.

b = firgr(n,f,a,s,w,e) specifies weights and independent approximation errors for filters with special properties. The weights and properties are included in vectors w and e. Sometimes, you may need to use independent approximation errors to get designs with forced values to converge. For example,

b = firgr(82,[0 0.055 0.06 0.1 0.15 1], [0 0 0 0 1 1],...
{'n' 'i' 'f' 'n' 'n' 'n'}, [10 1 1] ,{'e1' 'e2' 'e3'});

b = firgr(___,'1') designs a type 1 filter (even-order symmetric). You can specify type 2 (odd-order symmetric), type 3 (even-order antisymmetric), and type 4 (odd-order antisymmetric) filters as well. Note that restrictions apply to a at f = 0 or f = 1 for FIR filter types 2, 3, and 4.

b = firgr(___,'minphase') designs a minimum-phase FIR filter. You can use the argument 'maxphase' to design a maximum phase FIR filter.

b = firgr(___,'check') returns a warning when there are potential transition-region anomalies.

b = firgr(___,{lgrid}), where {lgrid} is a scalar cell array. The value of the scalar controls the density of the frequency grid by setting the number of samples used along the frequency axis.

[b,err] = firgr(___) returns the unweighted approximation error magnitudes. err contains one element for each independent approximation error returned by the function.

[b,err,res] = firgr(___) returns the structure res comprising optional results computed by firgr.

b = firgr(n,f,fresp,w) returns a length n + 1 FIR filter which has the best approximation to the desired frequency response as returned by the user-defined function fresp.

firgr includes a predefined frequency response function named 'firpmfrf2'. You can write your own based on the simpler 'firpmfrf'. See the help for private/firpmfrf for more information.

b = firgr(n,f,{fresp,p1,p2,...},w) specifies optional arguments p1, p2,…, pn to be passed to the response function fresp.

b = firgr(n,f,a,w) is a synonym for b = firgr(n,f,{'firpmfrf2',a},w), where a is a vector containing your specified response amplitudes at each band edge in f. By default, firgr designs symmetric (even) FIR filters. 'firpmfrf2' is the predefined frequency response function. If you do not specify your own frequency response function (the fresp variable), firgr uses 'firpmfrf2'.

b = firgr(___,'h') and b = firgr(__,'d') design antisymmetric (odd) filters. When you omit the 'h' or 'd' arguments from the firgr command syntax, each frequency response function fresp can tell firgr to design either an even or odd filter. Use the command syntax sym = fresp('defaults',{n,f,[],w,p1,p2,...}).

firgr expects fresp to return sym = 'even' or sym = 'odd'. If fresp does not support this call, firgr assumes even symmetry.

## Examples

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Design an FIR filter with two single-band notches at 0.25 and 0.55.

b1 = firgr(42,[0 0.2 0.25 0.3 0.5 0.55 0.6 1],[1 1 0 1 1 0 1 1],...
{'n' 'n' 's' 'n' 'n' 's' 'n' 'n'});

Design a highpass filter whose gain at 0.06 is forced to be zero. The gain at 0.055 is indeterminate since it should abut the band.

b2 = firgr(82,[0 0.055 0.06 0.1 0.15 1],[0 0 0 0 1 1],...
{'n' 'i' 'f' 'n' 'n' 'n'});

Design a second highpass filter with forced values and independent approximation errors.

b3 = firgr(82,[0 0.055 0.06 0.1 0.15 1], [0 0 0 0 1 1], ...
{'n' 'i' 'f' 'n' 'n' 'n'}, [10 1 1] ,{'e1' 'e2' 'e3'});

View the frequency response of the filters.

filterAnalyzer(b1,1,b2,1,b3,1,FilterNames=["Filter_b1","Filter_b2","Filter_b3"]);

## Input Arguments

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Filter order, specified as an integer greater than or equal to 3.

Normalized frequency points, specified as a real-valued vector. The argument must be in the range [0, 1] , where 1 corresponds to the Nyquist frequency. The length of f must be even. The frequencies must be in increasing order.

Desired amplitudes at the points specified in f, specified as a vector. f and a must be of the same length. The length of a must be even.

Weights used to adjust the fit in each frequency band, specified as a real-valued vector. The length of w is half the length of f and a, so there is exactly one weight per band.

Filter type for linear-phase filters with odd symmetry (type III and type IV), specified as either 'hilbert' or 'differentiator':

• 'hilbert' — The output coefficients in b obey the relation b(k) = –b(n + 2 – k), k = 1, ..., n + 1. This class of filters includes the Hilbert transformer, which has a desired amplitude of 1 across the entire band.

For example,

b = firgr(30,[0.1 0.9],[1 1],'hilbert');

designs an approximate FIR Hilbert transformer of length 31.

• 'differentiator' — For nonzero amplitude bands, the filter weighs the error by a factor of 1/f 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, these filters minimize the maximum relative error (the maximum of the ratio of the error to the desired amplitude).

Type of minimum order filter, specified as 'minorder', 'mineven', or 'minodd'. When you specify 'mineven' or 'minodd', the function designs a minimum even order or minimum odd order filter.

Peak ripple per frequency band, specified as a real-valued vector. The length of r is half the length of f and a, so there is exactly one weight per band.

Initial estimate of the filter order, specified as an integer greater than or equal to 3. The function accepts this value in the b = firgr({m,ni},f,a,r} syntax. ni is optional for common filter designs, but it must be specified for designs in which firpmord cannot be used, such as while designing differentiators or Hilbert transformers.

Independent approximation errors to use for different bands, specified as a cell array of character vectors or an array of strings. The length of e must equal the number of bands. Entries of e must be in the form 'e#' or "e#" where # indicates which approximation error to use for the corresponding band. For example, when e = ["e1","e2","e1"], the first and third bands use the same approximation error "e1" and the second band uses a different one "e2". When all bands use the same approximation error, such as ["e1","e1","e1",...], it is equivalent to omitting e, as in b = firgr(n,f,a,w).

Data Types: char | string

Special properties at certain frequency points, specified as a cell array of character vectors or an array of strings. s must have the same length as f and a. Entries of s must be one of:

• 'n' or "n" — Normal frequency point.

• 's' or "s" — Single-point band. The frequency “band” is given by a single point. Specify the corresponding gain at this frequency point in a.

• 'f' or "f" — Forced frequency point. Forces the gain at the specified frequency band to be the value specified.

• 'i' or "i" — Indeterminate frequency point. Use this argument when adjacent bands abut one another (no transition region).

For example, the following command designs a bandstop filter with zero-valued single-point stopbands (notches) at 0.25 and 0.55.

b = firgr(42,[0 0.2 0.25 0.3 0.5 0.55 0.6 1],...
[1 1 0 1 1 0 1 1],["n" "n" "s" "n" "n" "s" "n" "n"])

b = firgr(82,[0 0.055 0.06 0.1 0.15 1],[0 0 0 0 1 1],["n" "i" "f" "n" "n" "n"]) designs a highpass filter with the gain at 0.06 forced to be zero. The band edge at 0.055 is indeterminate since the first two bands actually touch. The other band edges are normal.

Data Types: char | string

Control the density of the frequency grid, which has roughly (lgrid*n)/(2*bw) frequency points, where bw is the fraction of the total frequency band interval [0,1] covered by f. Increasing lgrid often results in filters that more exactly match an equiripple filter, but that take longer to compute. The default value of 16 is the minimum value that should be specified for lgrid.

Frequency response, specified as a function handle. The function is called from within firgr with this syntax:

[dh,dw] = fresp(n,f,gf,w)

where:

• fresp identifies the function that you use to define your desired filter frequency response.

• n is the filter order.

• f is the vector of frequency band edges which must appear monotonically between 0 and 1, where 1 is one-half of the sampling frequency. The frequency bands span f(k) to f(k+1) for k odd. The intervals f(k+1) to f(k+2) for k odd are “transition bands” or “don't care” regions during optimization.

• gf is a vector of grid points that have been chosen over each specified frequency band by firgr, and determines the frequencies at which firgr evaluates the response function.

• w is a vector of real, positive weights, one per band, for use during optimization. w is optional in the call to firgr. If you do not specify w, it is set to unity weighting before being passed to fresp.

• dh and dw are the desired frequency response and optimization weight vectors, evaluated at each frequency in grid gf.

## Output Arguments

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Filter coefficients, returned as a row vector of length n + 1.

The firgr uses the minimax filter design algorithm to design the following types of real FIR filters:

• Types 1-4 linear phase:

• Type 1 is even order, symmetric

• Type 2 is odd order, symmetric

• Type 3 is even order, antisymmetric

• Type 4 is odd order, antisymmetric

• Minimum phase

• Maximum phase

• Minimum order (even or odd)

• Extra ripple

• Maximal ripple

• Constrained ripple

• Single-point band (notching and peaking)

• Forced gain

• Arbitrary shape frequency response curve filters

Unweighted approximation error magnitudes, returned as a scalar or a vector. err contains one element for each independent approximation error returned by the function.

Frequency response characteristics, returned as a structure. The structure res has the following fields:

Structure Field

Contents

res.fgrid

Vector containing the frequency grid used in the filter design optimization

res.des

Desired response on fgrid

res.wt

Weights on fgrid

res.h

Actual frequency response on the frequency grid

res.error

Error at each point (desired response - actual response) on the frequency grid

res.iextr

Vector of indices into fgrid of external frequencies

res.fextr

Vector of external frequencies

res.order

Filter order

res.edgecheck

Transition-region anomaly check. One element per band edge. Element values have the following meanings: 1 = OK, 0 = probable transition-region anomaly, -1 = edge not checked. Computed when you specify the 'check' input option in the function syntax.

res.iterations

Number of s iterations for the optimization

res.evals

Number of function evaluations for the optimization

## Version History

Introduced in R2011a

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