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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.


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 (although linear-phase is generally considered a good thing) i.e. 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 (we should point out 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';

If we zoom in, we can see that the filter has a passband peak ripple of about 0.008 and 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);
h = fvtool(b,1,bgr,1);
h.MagnitudeDisplay = 'Magnitude';
h.Color = 'White';
legend(h,'FIRLPNORM design','FIRGR design');
Order: 30

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');

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);
h = fvtool(b,1,bm,1,'Color','White');
legend(h,'FIRLPNORM design','FIRGR minimum-phase design');
Order: 32

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');

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');

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')

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');

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');

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.