polystab

Stabilize polynomial

Syntax

b = polystab(a)

Description

polystab stabilizes a polynomial with respect to the unit circle; it reflects roots with magnitudes greater than 1 inside the unit circle.

b = polystab(a) returns a row vector b containing the stabilized polynomial. a is a vector of polynomial coefficients, normally in the z-domain:

$A (z) = a (1) + a (2) z^{- 1} + \dots + a (m + 1) z^{- m} .$

Examples

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Convert Linear-Phase Filter to Minimum-Phase

Open Live Script

Use the window method to design a 25th-oder FIR filter with normalized cutoff frequency $0.4 π$ rad/sample. Verify that it has linear phase but not minimum phase.

h = fir1(25,0.4);

h_linphase = islinphase(h)

h_linphase = logical
   1

h_minphase = isminphase(h)

h_minphase = logical
   0

Use polystab to convert the linear-phase filter into a minimum-phase filter. Plot the phase responses of the filters.

hmin = polystab(h)*norm(h)/norm(polystab(h));

hmin_linphase = islinphase(hmin)

hmin_linphase = logical
   0

hmin_minphase = isminphase(hmin)

hmin_minphase = logical
   1

hfvt = fvtool(h,1,hmin,1,'Analysis','phase');
legend(hfvt,'h','hmin')

Figure Figure 1: Phase Response contains an axes object. The axes object with title Phase Response, xlabel Normalized Frequency ( times pi blank rad/sample), ylabel Phase (radians) contains 2 objects of type line. These objects represent h, hmin.

Verify that the two filters have identical magnitude responses.

hfvt = fvtool(h,1,hmin,1);
legend(hfvt,'h','hmin')

Figure Figure 2: Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB), xlabel Normalized Frequency ( times pi blank rad/sample), ylabel Magnitude (dB) contains 2 objects of type line. These objects represent h, hmin.

Algorithms

polystab finds the roots of the polynomial and maps those roots found outside the unit circle to the inside of the unit circle:

v = roots(a);
vs = 0.5*(sign(abs(v)-1)+1);
v = (1-vs).*v + vs./conj(v);
b = a(1)*poly(v);

Version History

Introduced before R2006a