Parks-McClellan FIR filter

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## Syntax

`b = firgr(n,f,a,w)`

`b = firgr(n,f,a,ftype)`

`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)`

`b = firgr(___,'h')`

## 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)`

isa synonym for `b = firgr(n,f,{'firpmfrf2',a},w)`

,where `a`

is a vector containing your specified responseamplitudes at each band edge in `f`

. By default, `firgr`

designssymmetric (even) FIR filters. `'firpmfrf2'`

is thepredefined frequency response function. If you do not specify yourown 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.

## Examples

collapse all

### Design a Filter Using firgr

Open Live Script

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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`n`

— Filter order

integer greater than or equal to 3

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

`f`

— Normalized frequency points

real-valued vector

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.

`a`

— Desired amplitude

vector

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.

`w`

— Weights

real-valued vector

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.

`ftype`

— Filter type

`'hilbert'`

| `'differentiator'`

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).

`m`

— Type of minimum order filter

`'minorder'`

| `'mineven'`

| `'minodd'`

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.

`r`

— Peak ripple per frequency band

real-valued vector

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.

`ni`

— Initial estimate of filter order

integer greater than or equal to 3

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.

`e`

— Approximation errors

cell array of character vectors | array of strings

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`

`s`

— Special properties at certain frequency points

cell array of character vectors | array of strings

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`

`lgrid`

— Density of frequency grid

`16`

(default) | 1-by-1 cell array with integer value

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`

.

`fresp`

— Frequency response

function handle

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

collapse all

`b`

— Filter coefficients

row vector

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

`err`

— Unweighted approximation error magnitude

scalar | vector

Unweighted approximation error magnitudes, returned as a scalar or a vector. `err`

contains one element for each independent approximation error returned by the function.

`res`

— Frequency response characteristics

structure

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 |

`res.wt` | Weights on |

`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 |

`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 |

`res.iterations` | Number of |

`res.evals` | Number of function evaluations for the optimization |

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

All inputs must beconstant. Expressions or variables are allowed if their values donot change.

Does not support syntaxes that have cell array input.

## Version History

**Introduced in R2011a**

expand all

### R2022b: Approximation errors and special properties support array of strings

Starting in R2022b, you can specify the approximation errors (`e`

) and the special properties (`s`

) as an array of strings.

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"]);

## See Also

butter | cheby1 | cheby2 | ellip | freqz | filter | firls | fircls | firpm

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