Scale roots of polynomial
b = polyscale(a,alpha)
b = polyscale(a,alpha) scales
the roots of a polynomial in the z-plane, where
a is a vector containing the polynomial
alpha is the scaling factor.
alpha is a real value in the range
[0 1], then the roots of
radially scaled toward the origin in the z-plane.
Complex values for
alpha allow arbitrary changes
to the root locations.
Roots of Unity
Express the solutions to the equation as the roots of a polynomial. Plot the roots in the complex plane.
pp = [1 0 0 0 0 0 0 -1]; zplane(pp,1)
Scale the roots of
p in and out of the unit circle. Plot the results.
hold on for sc = [1:-0.2:0.2 1.2 1.4]; b = polyscale(pp,sc); plot(roots(b),'o') end axis([-1 1 -1 1]*1.5) hold off
Bandwidth Expansion of LPC Speech Spectrum
Load a speech signal sampled at . The file contains a recording of a female voice saying the word "MATLAB®."
Model a 100-sample section of the signal using a 12th-order autoregressive polynomial. Perform bandwidth expansion of the signal by scaling the roots of the autoregressive polynomial by 0.85.
Ao = lpc(mtlb(1000:1100),12); Ax = polyscale(Ao,0.85);
Plot the zeros, poles, and frequency responses of the models.
subplot(2,2,1) zplane(1,Ao) title('Original') subplot(2,2,3) zplane(1,Ax) title('Flattened') subplot(1,2,2) [ho,w] = freqz(1,Ao); [hx,w] = freqz(1,Ax); plot(w/pi,abs([ho hx])) legend('Original','Flattened')
By reducing the radius of the roots in an autoregressive polynomial, the bandwidth of the spectral peaks in the frequency response is expanded (flattened). This operation is often referred to as bandwidth expansion.