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
*Z*-transform notation, Wilson's method implies solving the equation

(79) |

(80) |

Burg (1998, personal communication) recognized that dividing both sides of equation () 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:

(81) |

Equation () 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:

- 1.
- Compute the left side of equation () using forward and adjoint polynomial division.
- 2.
- Abandon negative lags, to keep only the causal part of the
signal, and also keep half of the zero lag. This gives us
*A*_{t+1}(*Z*)/*A*_{t}(*Z*). - 3.
- Multiply out (convolve) the denominator
*A*_{t}(*Z*). Now we have the desired result*A*_{t+1}(*Z*). - 4.
- Iterate until convergence.

iter | a_{0} |
a_{1} |
a_{2} |
a_{3} |

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 on a simple 1-D signal. The autocorrelation
*S*(*Z*) in this case is , and the
corresponding minimum-phase signal is *A*(*Z*) = (2+*Z*)(3+*Z*)(4+*Z*) = 24 +
26 *Z* + 9 *Z ^{2}* +

12/28/2000