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Parks-McClellan FIR filter


b = firgr(n,f,a,w)
b = firgr(n,f,a,'hilbert')
b = firgr(m,f,a,r),
b = firgr({m,ni},f,a,r)
b = firgr(n,f,a,w,e)
b = firgr(n,f,a,s)
b = firgr(n,f,a,s,w,e)
b = firgr(...,'1')
b = firgr(...,'minphase')
b = firgr(..., 'check')
b = firgr(...,{lgrid}),
[b,err] = firgr(...)
[b,err,res] = firgr(...)
b = firgr(n,f,fresp,w)
b = firgr(n,f,{fresp,p1,p2,...},w)
b = firgr(n,f,a,w)


firgr is a minimax filter design algorithm you use 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

b = firgr(n,f,a,w) returns a length n+1 linear phase FIR filter which 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, refer to firpm in Signal Processing Toolbox™ User's Guide.

b = firgr(n,f,a,'hilbert') and b = firgr(n,f,a,'differentiator') design FIR Hilbert transformers and differentiators. For more information on designing these filters, refer to firpm in Signal Processing Toolbox User's Guide.

b = firgr(m,f,a,r), where m is one of 'minorder', 'mineven' or 'minodd', designs filters repeatedly until the minimum order filter, as specified in m, that meets the specifications is found. 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. Entries of e must be in the form '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'. Note that 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).

b = firgr(n,f,a,s) is used to design filters with special properties at certain frequency points. s is a cell array of character vectors and must be the same length as f and a. Entries of s must be one of:

  • 'n' — normal frequency point.

  • 's' — single-point band. The frequency “band” is given by a single point. The corresponding gain at this frequency point must be specified in a.

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

  • '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 stop bands (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.

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. res contains the following fields.

Structure Field



Vector containing the frequency grid used in the filter design optimization


Desired response on fgrid


Weights on fgrid


Actual frequency response on the frequency grid


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


Vector of indices into fgrid of external frequencies


Vector of external frequencies


Filter order


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.


Number of s iterations for the optimization


Number of function evaluations for the optimization

firgr is also a “function function,” allowing you to write a function that defines the desired frequency response.

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. Use the following firgr syntax to call fresp:

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


  • 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.

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.

For more information about the input arguments to firgr, refer to firpm.


collapse all

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'});

Use the filter visualization tool to view the results of the filters.

hvft = fvtool(b1,1,b2,1,b3,1);
legend(hvft,'Filter b1','Filter b2','Filter b3')

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 3 objects of type line. These objects represent Filter b1, Filter b2, Filter b3.


Shpak, D.J. and A. Antoniou, “A generalized Remez method for the design of FIR digital filters,” IEEE® Trans. Circuits and Systems, pp. 161-174, Feb. 1990.

Extended Capabilities

Version History

Introduced in R2011a