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Least Pth-Norm Optimal FIR Filter Design

This example shows how to design least Pth-norm FIR filters with the firlpnorm function. This function uses a least-Pth unconstrained optimization algorithm to design FIR filters with arbitrary magnitude response.

FIRLPNORM Syntax

The syntax for firlpnorm is similar to that of iirlpnorm (see the Least Pth-Norm Optimal IIR Filter Design example for details) except that the denominator order is not specified.

The function designs optimal FIR filters in the least-Pth sense. However the filter is not constrained to have linear-phase, that is, the impulse response has no special symmetry properties.

However, the linear-phase constraint also results in filters with larger order than the more general nonlinear-phase designs. Note that in some hardware implementations, one can reduce the number of multipliers in half when implementing linear-phase filters because of the symmetry in the coefficients. For example, consider the following firlpnorm design

N = 30;
F = [0 0.3 0.45 1];
E = F;
A = [1 1 0 0];
W = [1 1 10 10];
b = firlpnorm(N,F,E,A,W);
h = fvtool(b);
h.MagnitudeDisplay = 'Magnitude';
h.Color = 'White';

Figure Figure 1: Magnitude Response contains an axes object. The axes object with title Magnitude Response contains an object of type line.

If we zoom in, we can see that the filter has a passband peak ripple of about 0.008 and a stopband peak ripple of about 0.000832. A firpm or firgr design with comparable specs will require a 37th order filter. This is especially significant considering that firgr will provide the lowest order linear-phase FIR filter that meets the specifications.

dev = [0.008 0.000832];
bgr = firgr('minorder',F,A,dev);
orderfirgr = length(b)-1;
fprintf('Order: %d\n',orderfirgr);
Order: 30
h = fvtool(b,1,bgr,1);
h.MagnitudeDisplay = 'Magnitude';
h.Color = 'White';
legend(h,'FIRLPNORM design','FIRGR design');

Figure Figure 2: Magnitude Response contains an axes object. The axes object with title Magnitude Response contains 2 objects of type line. These objects represent FIRLPNORM design, FIRGR design.

Another way to look at this is by using the firceqrip function which also designs linear-phase equiripple filters, but whose specifications are given in a different way to firgr (see the constrained equiripple FIR filter design example for details). If we want a linear-phase filter of 30th order that meets the passband and stopband ripple that the design from firlpnorm achieves we need to live with a larger transition width.

bceq = firceqrip(30,(F(2)+F(3))/2,dev);
h = fvtool(b,1,bceq,1,'Color','White');
legend(h,'FIRLPNORM design','FIRCEQRIP design');

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 2 objects of type line. These objects represent FIRLPNORM design, FIRCEQRIP design.

Minimum-Phase Designs

Of course it is also possible to design nonlinear-phase filters with firgr by specifying the minphase option. Doing so allows us to obtain an FIR filter of lower order than in the linear-phase case and still meet the required specs. However, even in this case, the result is a non-optimal nonlinear-phase filter because the filter order is larger than the minimum required for a nonlinear-phase equiripple filter to meet the specs as is evident from the following example.

bm = firgr('minorder',F,A,dev,'minphase');
orderfirgrmin = length(bm)-1;
fprintf('Order: %d\n',orderfirgrmin);
Order: 32
h = fvtool(b,1,bm,1,'Color','White');
legend(h,'FIRLPNORM design','FIRGR minimum-phase design');

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 2 objects of type line. These objects represent FIRLPNORM design, FIRGR minimum-phase design.

Minimum-Phase Designs with FIRLPNORM

firlpnorm does allow for the option to constrain the zeros to lie on or inside the unit circle, resulting in a minimum-phase design. The constraint, however, results in larger passband ripple and less stopband attenuation than the unconstrained design.

bmlp = firlpnorm(30,F,E,A,W,'minphase');
h = fvtool(b,1,bmlp,1,'Color','White');
legend(h,'FIRLPNORM design','FIRLPNORM minimum-phase design');

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 2 objects of type line. These objects represent FIRLPNORM design, FIRLPNORM minimum-phase design.

If we increase the order to that of the minimum-phase filter designed with firgr we can see that we meet the specs met by both the 30th order firlpnorm (nonminimum-phase) design and the 32nd order firgr minimum-phase design.

bmlp = firlpnorm(orderfirgrmin,F,E,A,W,'minphase');
h = fvtool(b,1,bm,1,bmlp,1,'Color','White');
legend(h,'FIRLPNORM design',...
    'FIRGR minimum-phase design',...
    'FIRLPNORM minimum-phase design');

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 3 objects of type line. These objects represent FIRLPNORM design, FIRGR minimum-phase design, FIRLPNORM minimum-phase design.

Changing the Pth-Norm

Like iirlpnorm and iirlpnormc, firlpnorm allows for the specification of the Pth-norm used to optimize the filter. The Pth-norm is specified in the exact same way as in iirlpnorm, i.e. a two element vector with Pinit and Pfinal as its elements. Pinit specifies the initial Pth-norm used by the algorithm (this aids in the convergence) and Pfinal specifies the final Pth-norm with which the filter is optimized.

For example, a least-squares design for the above specs can be obtained as follows:

N = 40;
F = [0 0.4 0.45 1];
E = F;
A = [0 0 1 1];
W = [1 1 10 10];
P = [2 2];
bl2 = firlpnorm(N,F,E,A,W,P);
h = fvtool(bl2,1,'Color','White');
legend(h,'FIRLPNORM design')

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains an object of type line. This object represents FIRLPNORM design.

Comparing to FIRLS

In comparison, we design a linear-phase least-squares filter using firls. Once again, for the same filter order, the linear-phase constraint results in less stopband attenuation and a larger passband ripple.

W = [1 20];
bls = firls(N,F,A,W);
h = fvtool(bl2,1,bls,1,'Color','White');
legend(h,'FIRLPNORM design','FIRLS design');

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 2 objects of type line. These objects represent FIRLPNORM design, FIRLS design.

Other Norms

Equiripple designs are useful when one requires the smallest possible order to meet a set of design specifications. To meet the same specs with a least-squares design requires a higher order filter. However, the higher order does provide extra attenuation (less ripple) for a large portion of the stopband (passband). In fact least-squares design minimize the energy of the stopband. Compromises between equiripple design and least-squares design can be reached by using intermediate norms. For example we show the design of a filter with the same specs, but optimized for the following norms: 2, 4, 12, 256 (approx. infinity norm).

W = [1 1 10 10];
P4 = [2 4];
bl4 = firlpnorm(N,F,E,A,W,P4);
P12 = [2 12];
bl12 = firlpnorm(N,F,E,A,W,P12);
Pinf = [2 256];
blinf = firlpnorm(N,F,E,A,W,Pinf);
h = fvtool(bl2,1,bl4,1,bl12,1,blinf,1,'Color','White');
legend(h,'P = 2','P = 4','P = 12','P = 256');

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 4 objects of type line. These objects represent P = 2, P = 4, P = 12, P = 256.

In order to meet the minimum stopband attenuation of the equiripple (256-norm) case, it is necessary to increase the order of the other designs.