# polystab

Stabilize polynomial

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:

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

## Examples

collapse all

Use the window method to design a 25th-oder FIR filter with normalized cutoff frequency $0.4\pi$ 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')

Verify that the two filters have identical magnitude responses.

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