The Wilson-Burg method of spectral factorization with application to helical filtering |
Spectral factorization constructs a minimum-phase signal from its spectrum. The algorithm, suggested by Wilson (1969), approaches this problem directly with Newton's iterative method. In a -transform notation, Wilson's method implies solving the equation
Burg (1998, personal communication) recognized that dividing both sides of equation (2) by leads to a particularly convenient form, where the terms on the left are symmetric, and the two terms on the right are correspondingly strictly causal and anticausal:
Equation (3) leads to the Wilson-Burg algorithm, which accomplishes spectral factorization by a recursive application of convolution (polynomial multiplication) and deconvolution (polynomial division). The algorithm proceeds as follows:
iter | ||||
0 | 1.000000 | 0.000000 | 0.000000 | 0.000000 |
1 | 36.523964 | 23.737839 | 6.625787 | 0.657103 |
2 | 26.243151 | 25.726116 | 8.471050 | 0.914951 |
3 | 24.162354 | 25.991493 | 8.962727 | 0.990802 |
4 | 24.001223 | 25.999662 | 9.000164 | 0.999200 |
5 | 24.000015 | 25.999977 | 9.000029 | 0.999944 |
6 | 23.999998 | 26.000002 | 9.000003 | 0.999996 |
7 | 23.999998 | 26.000004 | 9.000001 | 1.000000 |
8 | 23.999998 | 25.999998 | 9.000000 | 1.000000 |
9 | 24.000000 | 26.000000 | 9.000000 | 1.000000 |
An example of the Wilson-Burg convergence is shown in Table 1 on a simple 1-D signal. The autocorrelation in this case is , and the corresponding minimum-phase signal is . A quadratic rate of convergence is visible from the table. The convergence slows down for signals whose polynomial roots are close to the unit circle (Wilson, 1969).
The clear advantage of the Wilson-Burg algorithm in comparison with the original Wilson algorithm is in the elimination of the expensive matrix inversion step. Only convolution and deconvolution operations are used at each iteration step.
The Wilson-Burg method of spectral factorization with application to helical filtering |