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# firpm

Parks-McClellan optimal FIR filter design

## Syntax

b = firpm(n,f,a)
b = firpm(n,f,a,w)
b = firpm(n,f,a, 'ftype')
b = firpm(n,f,a,w, 'ftype')
b = firpm(...,{lgrid})
[b,err] = firpm(...)
[b,err,res] = firpm(...)
b = firpm(n,f,@fresp,w)
b = firpm(n,f,@fresp,w,'ftype')

## Description

firpm designs a linear-phase FIR filter using the Parks-McClellan algorithm [1]. The Parks-McClellan algorithm uses the Remez exchange algorithm and Chebyshev approximation theory to design filters with an optimal fit between the desired and actual frequency responses. The filters are optimal in the sense that the maximum error between the desired frequency response and the actual frequency response is minimized. Filters designed this way exhibit an equiripple behavior in their frequency responses and are sometimes called equiripple filters. firpm exhibits discontinuities at the head and tail of its impulse response due to this equiripple nature.

b = firpm(n,f,a) returns row vector b containing the n+1 coefficients of the order n FIR filter whose frequency-amplitude characteristics match those given by vectors f and a.

The output filter coefficients (taps) in b obey the symmetry relation:

Vectors f and a specify the frequency-magnitude characteristics of the filter:

• f is a vector of pairs of normalized frequency points, specified in the range between 0 and 1, where 1 corresponds to the Nyquist frequency. The frequencies must be in increasing order.

• a is a vector containing the desired amplitudes at the points specified in f.

The desired amplitude 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 at frequencies between pairs of points (f(k), f(k+1)) for k even is unspecified. The areas between such points are transition or "don't care" regions.

• f and a must be the same length. The length must be an even number.

The relationship between the f and a vectors in defining a desired frequency response is shown in the illustration below.

firpm always uses an even filter order for configurations with even symmetry and a nonzero passband at the Nyquist frequency. This is because for impulse responses exhibiting even symmetry and odd orders, the frequency response at the Nyquist frequency is necessarily 0. If you specify an odd-valued n, firpm increments it by 1.

b = firpm(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.

 Note   b = firpm(n,f,a,w) is a synonym for b = firpm(n,f,{@firpmfrf,a},w), where, @firpmfrf is the predefined frequency response function handle for firpm. If desired, you can write your own response function. Use help private/firpmfrf for information.

b = firpm(n,f,a, 'ftype') and

b = firpm(n,f,a,w, 'ftype') specify a filter type, where 'ftype' is

• 'hilbert', for linear-phase filters with odd symmetry (type III and type IV)

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,

h = firpm(30,[0.1 0.9],[1 1],'hilbert');

designs an approximate FIR Hilbert transformer of length 31.

• 'differentiator', for type III and type IV filters, using a special weighting technique

For nonzero amplitude bands, it weights 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).

b = firpm(...,{lgrid}) uses the integer lgrid to 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. Note that the {lgrid} argument must be a 1-by-1 cell array.

[b,err] = firpm(...) returns the maximum ripple height in err.

[b,err,res] = firpm(...) returns a structure res with the following fields.

 res.fgrid Frequency grid vector used for the filter design optimization res.des Desired frequency response for each point in res.fgrid res.wt Weighting for each point in opt.fgrid res.H Actual frequency response for each point in res.fgrid res.error Error at each point in res.fgrid (res.des-res.H) res.iextr Vector of indices into res.fgrid for extremal frequencies res.fextr Vector of extremal frequencies

You can also use firpm to write a function that defines the desired frequency response. The predefined frequency response function handle for firpm is @firpmfrf, which designs a linear-phase FIR filter.

b = firpm(n,f,@fresp,w) returns row vector b containing the n+1 coefficients of the order n FIR filter whose frequency-amplitude characteristics best approximate the response returned by function handle @fresp. The function is called from within firpm with the following syntax.

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

The arguments are similar to those for firpm:

• n is the filter order.

• f is the vector of normalized frequency band edges that appear monotonically between 0 and 1, where 1 is the Nyquist frequency.

• gf is a vector of grid points that have been linearly interpolated over each specified frequency band by firpm. gf determines the frequency grid at which the response function must be evaluated, and contains the same data returned by cfirpm in the fgrid field of the opt structure.

• w is a vector of real, positive weights, one per band, used during optimization. w is optional in the call to firpm; if not specified, it is set to unity weighting before being passed to fresp.

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

b = firpm(n,f,@fresp,w,'ftype') designs antisymmetric (odd) filters, where 'ftype' is either 'd' for a differentiator or 'h' for a Hilbert transformer. If you do not specify an ftype, a call is made to fresp to determine the default symmetry property sym. This call is made using the syntax.

sym = fresp('defaults',{n,f,[],w,p1,p2,...})

The arguments n, f, w, etc., may be used as necessary in determining an appropriate value for sym, which firpm expects to be either 'even' or 'odd'. If fresp does not support this calling syntax, firpm defaults to even symmetry.

## Examples

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### Parks-McClellan Bandpass Filter

Use the Parks-McClellan algorithm to design an FIR bandpass filter of order 17. Specify normalized stopband frequencies of and rad/sample and normalized passband frequencies of and rad/sample. Plot the ideal and actual magnitude responses.

f = [0 0.3 0.4 0.6 0.7 1];
a = [0 0.0 1.0 1.0 0.0 0];
b = firpm(17,f,a);

[h,w] = freqz(b,1,512);
plot(f,a,w/pi,abs(h))
legend('Ideal','firpm Design')
xlabel 'Radian Frequency (\omega/\pi)', ylabel 'Magnitude'

## Tips

If your filter design fails to converge, it is possible that the filter design is correct. Verify the design by checking the frequency response.

If your filter design fails to converge and the resulting filter design is not correct, attempt one or more of the following:

• Increase the filter order

• Relax the filter design by reducing the attenuation in the stopbands and/or broadening the transition regions

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### Algorithms

firpm is a MEX-file version of the original Fortran code from [1], altered to design arbitrarily long filters with arbitrarily many linear bands.

firpm designs type I, II, III, and IV linear-phase filters. Type I and type II are the defaults for n even and n odd, respectively, while type III (n even) and type IV (n odd) are obtained with the 'hilbert' and 'differentiator' flags. The different types of filters have different symmetries and certain constraints on their frequency responses (see [5] for more details).

Linear Phase Filter TypeFilter OrderSymmetry of CoefficientsResponse H(f), f = 0Response H(f), f = 1 (Nyquist)

Type I

Even

even:

No restriction

No restriction

Type II

Odd

even:

No restriction

H(1)=0

firpm increments the filter order by 1 if you attempt to construct a type II filter with a nonzero passband at the Nyquist frequency.

Type III

Even

odd:

H(0) = 0

H(1) = 0

Type IVOdd

odd:

H(0) = 0

No restriction

## References

[1] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Programs for Digital Signal Processing. New York: IEEE Press, 1979, algorithm 5.1.

[2] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Selected Papers in Digital Signal Processing. Vol. II. New York: IEEE Press, 1976.

[3] Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, p. 83.

[4] Rabiner, Lawrence R., James H. McClellan, and Thomas W. Parks. "FIR Digital Filter Design Techniques Using Weighted Chebyshev Approximation." Proceedings of the IEEE®. Vol. 63, Number 4, 1975, pp. 595–610.

[5] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999, p. 486.