maxflat

Generalized digital Butterworth filter design

Syntax

[b,a] = maxflat(n,m,Wn)

b = maxflat(n,'sym',Wn)

[b,a,b1,b2] = maxflat(n,m,Wn)

[b,a,b1,b2,sos,g] = maxflat(n,m,Wn)

[___] = maxflat(n,m,Wn,designflag)

Description

[b,a] = maxflat(n,m,Wn) returns the nth-order numerator coefficients b and mth-order denominator coefficients a of a lowpass Butterworth filter with normalized cutoff frequency Wn.

b = maxflat(n,'sym',Wn) returns the coefficients b of a symmetric FIR Butterworth filter. n must be even.

[b,a,b1,b2] = maxflat(n,m,Wn) returns two polynomials b1 and b2 whose product is equal to the numerator polynomial b (that is, b = conv(b1,b2)).

[b,a,b1,b2,sos,g] = maxflat(n,m,Wn) returns the second-order sections representation of the filter as the filter matrix sos and the gain g.

[___] = maxflat(n,m,Wn,designflag) specifies the option to display the filter design as a table, plot, or both using designflag. You can use any of the output combinations from previous syntaxes.

Examples

Generalized Butterworth Filter

Design a generalized Butterworth filter with normalized cutoff frequency $$0.2\pi$$ rad/s. Specify a numerator order of 10 and a denominator order of 2. Visualize the frequency response of the filter.

n = 10;
m = 2;
Wn = 0.2;

[b,a] = maxflat(n,m,Wn);
freqz(b,a)

Monitor Filter Design with Display Option

Design a generalized Butterworth filter with normalized cutoff frequency 0.5$\pi$ rad/s. Specify a numerator order of 8 and a denominator order of 2. Display the design table and the plots of the filter characteristics.

n = 8;
m = 2;
Wn = 0.5;
b = maxflat(n,m,Wn,'both');

 Table:
 
    L         M         N         wo_min/pi wo_max/pi
 
    8.0000         0    2.0000         0    0.2707
    7.0000    1.0000    2.0000    0.2707    0.3710
    6.0000    2.0000    2.0000    0.3710    0.4581
    5.0000    3.0000    2.0000    0.4581    0.5419
    4.0000    4.0000    2.0000    0.5419    0.6290
    3.0000    5.0000    2.0000    0.6290    0.7293
    2.0000    6.0000    2.0000    0.7293    1.0000

Input Arguments

n — Numerator coefficient order

real positive scalar

Numerator coefficient order, specified as a real positive scalar.

Data Types: single | double

m — Denominator coefficient order

real positive scalar

Denominator coefficient order, specified as a real positive scalar.

Data Types: single | double

Wn — Normalized cutoff frequency

scalar in the range [0, 1]

Normalized cutoff frequency at which the magnitude response of the filter is equal to $$1/\sqrt{2}$$, specified as a scalar in the range [0, 1], where 1 corresponds to the Nyquist frequency.

Data Types: single | double

designflag — Filter design display

'trace' | 'plots' | 'both'

Filter design display, specified as one of these values:

• 'trace' for a textual display of the design table used in the design

• 'plots' for plots of the filter magnitude, group delay, and zeros and poles

• 'both' for both the textual display and plots

Output Arguments

b — Numerator coefficients

vector

Numerator coefficients, returned as a vector.

a — Denominator coefficients

vector

Denominator coefficients, returned as a vector.

b1, b2 — Polynomials

vectors

Polynomials, returned as vectors. The product of b1 and b2 is equal to the numerator polynomial b. b1 contains all of the zeros at z = -1, and b2 contains all of the other zeros.

sos — Second-order section coefficients

matrix

Second-order section coefficients, returned as a matrix.

g — Gain

real-valued scalar

Gain of the filter, returned as a real-valued scalar.

References

[1] Selesnick, Ivan W., and C. Sidney Burrus. “Generalized Digital Butterworth Filter Design.” IEEE^® Transactions on Signal Processing 46, no. 6, (June 1998): 1688–94.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

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

Version History

Introduced before R2006a