# firgr

Parks-McClellan FIR filter

## Syntax

```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) ```

## Description

`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

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

`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)`

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`.

`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`.

## Examples

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.

```fvtool(b1,1,b2,1,b3,1) legend('Filter b1','Filter b2','Filter b3');``` ## References

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.