# Documentation

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# 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:

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

## Examples

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

h = fir1(25,0.4); h_linphase = islinphase(h) h_minphase = isminphase(h) 
h_linphase = logical 1 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_minphase = isminphase(hmin) hfvt = fvtool(h,1,hmin,1,'Analysis','phase'); legend(hfvt,'h','hmin') 
hmin_linphase = logical 0 hmin_minphase = logical 1 

Verify that the two filters have identical magnitude responses.

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

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### 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);