polystab

Stabilize polynomial

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    Syntax

    b = polystab(a)

    Description

    b = polystab(a) stabilizes a polynomial of coefficient vector a with respect to the unit circle, reflects roots with magnitudes greater than 1 inside the unit circle, and returns a stabilized polynomial with a coefficient vector b.

    example

    Examples

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    Open Live Script

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

    hmin0 = polystab(h);
hmin = hmin0/norm(hmin0)*norm(h);

hmin_linphase = islinphase(hmin)
    hmin_linphase = logical
   0


    hmin_minphase = isminphase(hmin)
    hmin_minphase = logical
   1

    Plot the phase responses of the filters.

    phasez(h,1)
hold on
phasez(hmin,1)
hold off
legend("h","hmin")

    Figure 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.

    [hMag,hW] = freqz(h,1);
[hminMag,hminW] = freqz(hmin,1);

plot(hW,mag2db(abs(hMag)),".-")
hold on
plot(hminW,mag2db(abs(hminMag)))
hold off
grid on
xlabel('Normalized Frequency (\times\pi rad/sample)')
ylabel('Magnitude (dB)')
legend("h","hmin")

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

    Input Arguments

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    Input polynomial coefficients, specified as a vector. Each element (a1, a2, …, am+1) represents the polynomial coefficients, typically in the z-domain:

    A(z)=a1+a2z−1⋯+amz−(m−1)+am+1z−m

    Data Types: single | double
    Complex Number Support: Yes

    Output Arguments

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    Stabilized polynomial coefficients, returned as a row vector.

    Algorithms

    The polystab function finds the roots of the polynomial, maps the roots found outside the unit circle, and relocates them 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

    See Also