firpmord

Parks-McClellan optimal FIR filter order estimation

Syntax

[n,fo,ao,w] = firpmord(f,a,dev)

[___] = firpmord(___,fs)

c = firpmord(___,'cell')

Description

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

[___] = firpmord(___,fs) specifies a sampling frequency fs. fs defaults to 2 Hz, implying a Nyquist frequency of 1 Hz. You can specify band edges scaled to a particular application's sample rate. You can use this with any of the previous input syntaxes.

c = firpmord(___,'cell') returns a cell array c whose elements are the parameters to firpm.

Examples

Minimum-Order Lowpass Filter

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 in dB 
rs = 40;          % Stopband ripple in dB
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);
freqz(b,1,1024,fs)
title('Lowpass Filter Designed to Specifications')

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

Parks-McClellan Order of Lowpass Filter

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{:});
freqz(B,1,1024,8000)

Input Arguments

f — Frequency band edges

real-valued vector

Frequency band edges, specified as a real-valued vector. The argument must be in the range [0, F_s/2], where F_s is the Nyquist frequency. The number of elements in the vector is always a multiple of 2. The frequencies must be in increasing order.

a — Desired amplitude

vector

Desired amplitudes at the points contained in f, specified as a vector. f and a must satisfy the condition length(f) = 2length(a)–2. The desired function is piecewise constant.

dev — Maximum allowable deviation

vector

Maximum allowable deviation, specified as a vector. dev has the same size as a. It specifies the maximum allowable deviation or ripples between the frequency response and the desired amplitude of the output filter for each band.

fs — Sample rate

2 Hz (default) | real scalar

Sample rate, specified as a real scalar.

Output Arguments

n — Filter order

positive integer

Filter order, returned as a positive integer.

fo — 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 number of elements in the vector is always a multiple of 2. The frequencies must be in increasing order.

ao — Amplitude response

real-valued vector

Amplitude response, returned as a real-valued vector.

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.

c — FIR filter parameters

cell array

FIR filter parameters, returned as a cell array.

Algorithms

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

Note

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.

References

[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

C/C++ Code Generation

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

Usage notes and limitations:

All inputs must be constants. Expressions or variables are allowed if their values do not change.