Np -k
A(z) = 1 - SUM p(k) z .
k=1
The corresponding all-pole filter is 1/A(z). The filter with the radially
scaled poles is 1/A(b*z). The bandwith expanded polynomial A(b*z) has
coefficients,
p'(k) = p(k) * b^k for 1 <= k <= Np.
The effective bandwidth expansion of a sharp resonance can be determined from that for an isolated pole pair. Consider a simple filter with poles at z1 = r exp (j w0) and z2 = r exp (-j w0),
z^2 - 1
H(z) = ------------------------- .
z^2 - 2r cos(w0) z + r^2
The frequency response of this filter has a peak at w = w0 with a 3 dB
bandwidth
BW = pi/2 - 2 atan (r^2) normalized radians.
If the pole positions are scaled radially, r' = r * b, with b < 1, the bandwidth of the resonance increases. The increase in bandwidth can be computed to be
dBW = 2 atan (r^2 (1-b^2) / (1 + b^2 r^4))For a 8 kHz sampling rate, the bandwidths expansions for different values of b are shown below (calculated for r=1).
b BW exp
1.000 0
0.996 10.2 Hz
0.995 12.8 Hz
0.994 15.3 Hz
0.990 25.6 Hz
0.980 51.4 Hz
For speech analysis, bwexp is often chosen to give a bandwidth expansion of
10 to 25 Hz.
Predictor coefficients are usually expressed algebraically as vectors with 1-offset indexing. The correspondence to the 0-offset C-arrays is as follows.
p(1) <==> pc[0] predictor coefficient corresponding to lag 1 p(k) <==> pc[k-1] 1 <= k < Np p'(1) <==> pcb[0] predictor coefficient corresponding to lag 1 p'(k) <==> pcb[k-1] 1 <= k < Np