Parks-McClellan optimal FIR filter order estimation


[n,fo,ao,w] = firpmord(f,a,dev)
[n,fo,ao,w] = firpmord(f,a,dev,fs)
c = firpmord(f,a,dev,fs,'cell')


[n,fo,ao,w] = firpmord(f,a,dev) finds the approximate order, normalized frequency band edges, frequency band amplitudes, and weights that meet input specifications f, a, and dev.

  • f is a vector of frequency band edges (between 0 and Fs/2, where Fs is the sampling frequency), and a is a vector specifying the desired amplitude on the bands defined by f. The length of f is two less than twice the length of a. The desired function is piecewise constant.

  • dev is a vector the same size as a that specifies the maximum allowable deviation or ripples between the frequency response and the desired amplitude of the output filter for each band.

Use firpm with the resulting order n, frequency vector fo, amplitude response vector ao, and weights w to design the filter b which approximately meets the specifications given by firpmord input parameters f, a, and dev.

b = firpm(n,fo,ao,w)

[n,fo,ao,w] = firpmord(f,a,dev,fs) specifies a sampling frequency fs. fs defaults to 2 Hz, implying a Nyquist frequency of 1 Hz. You can therefore specify band edges scaled to a particular application's sampling frequency.

c = firpmord(f,a,dev,fs,'cell') generates a cell-array whose elements are the parameters to firpm.


In some cases, firpmord underestimates or overestimates the order n. If the filter does not meet the specifications, try a higher order such as n+1 or n+2.


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Design a minimum-order lowpass filter with a 500 Hz passband cutoff frequency and 600 Hz stopband cutoff frequency. Specify a sampling frequency of 2000 Hz. Require at least 40 dB of attenuation in the stopband and less than 3 dB of ripple in the passband.

rp = 3;           % Passband ripple
rs = 40;          % Stopband ripple
fs = 2000;        % Sampling frequency
f = [500 600];    % Cutoff frequencies
a = [1 0];        % Desired amplitudes

Convert the deviations to linear units. Design the filter and visualize its magnitude and phase responses.

dev = [(10^(rp/20)-1)/(10^(rp/20)+1)  10^(-rs/20)]; 
[n,fo,ao,w] = firpmord(f,a,dev,fs);
b = firpm(n,fo,ao,w);
title('Lowpass Filter Designed to Specifications')

Note that the filter falls slightly short of meeting the stopband attenuation and passband ripple specifications. Using n+1 in the call to firpm instead of n achieves the desired amplitude characteristics.

Design a lowpass filter with a 1500 Hz passband cutoff frequency and 2000 Hz stopband cutoff frequency. Specify a sampling frequency of 8000 Hz. Require a maximum stopband amplitude of 0.1 and a maximum passband error (ripple) of 0.01.

[n,fo,ao,w] = firpmord([1500 2000],[1 0],[0.01 0.1],8000);
b = firpm(n,fo,ao,w);

Obtain an equivalent result by having firpmord generate a cell array. Visualize the frequency response of the filter.

c = firpmord([1500 2000],[1 0],[0.01 0.1],8000,'cell');
B = firpm(c{:});


firpmord uses the algorithm suggested in [1]. This method is inaccurate for band edges close to either 0 or the Nyquist frequency, fs/2.


[1] Rabiner, Lawrence R., and Otto Herrmann. “The Predictability of Certain Optimum Finite-Impulse-Response Digital Filters.” IEEE® Transactions on Circuit Theory. Vol.  20, Number 4, 1973, pp. 401–408.

[2] Rabiner, Lawrence R., and Bernard Gold. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975, pp. 156–157.

Extended Capabilities

Introduced before R2006a