The time function (2, 1) has the same spectrum as the time function (1, 2).
The autocorrelation is (2, 5, 2).
We may utilize this observation to explore the multiplicity
of all time functions with the same autocorrelation and spectrum.
It would seem that the time reverse of any function would have the
same autocorrelation as the function.
Actually, certain applications will involve complex time series;
therefore we should make the more precise statement
that any wavelet and its complex-conjugate time-reverse share the
same autocorrelation and spectrum.
Let us verify this for simple two-point time functions.
The spectrum of (*b _{0}*,

(1) |

(2) |

(3) |

(4) |

Now, what can we do to change the wavelet (3)
which will leave its spectrum (4) unchanged?
Clearly, *b _{2}* may be multiplied by any complex number of unit magnitude.
What is left of (4) can be broken
up into a product of factors of form .But such a factor is just like (3).
The time function of (

(5) |

Now let us discuss the calculation of *B*(*Z*) from a given *R*(*Z*).
First, the roots of *R*(*Z*) are by definition the solutions to *R*(*Z*) = 0.
If we multiply *R*(*Z*) by *Z*^{N} (where *R*(*Z*) has been given up to degree *N*),
then
*Z*^{N} *R*(*Z*) is a polynomial and the solutions *Z*_{i} to *Z*^{N} *R*(*Z*) = 0 will
be the same as the solutions of *R*(*Z*) = 0.
Finding all roots of a polynomial is a standard though difficult task.
Assuming this to have been done we may
then check to see if the roots come in the pairs *Z*_{i} and .If they do not,
the *R*(*Z*) was not really a spectrum.
If they do, then for every zero inside the unit circle,
we must have one outside.
Refer to Figure 1.

3-1
Roots of .Figure 1 |

Thus, if we decide to make *B*(*Z*) be a minimum-phase wavelet
with the spectrum *R*(*Z*),
we collect all of the roots outside the unit circle.
Then we create *B*(*Z*) with

(6) |

This then summarizes the calculation of a minimum-phase wavelet from a given
spectrum.
When *N* is large,
it is computationally very awkward compared to methods yet to be discussed.
The value of the root method is that it shows certain basic principles.

- 1.
- Every spectrum has a minimum-phase wavelet which is unique within a complex scale factor of unit magnitude.
- 2.
- There are infinitely many time functions with any given spectrum.
- 3.
- Not all functions are possible autocorrelation functions.

The root method of spectral factorization was apparently developed by
economists in the 1920s and 1930s.
A number of early references may be found
in Wold's book, *Stationary Time Series*.

- How can you find the scale factor
*b*_{N}in (6)? - Compute the autocorrelation of each of the four wavelets (4, 0, -1), (2, 3, -2), (-2, 3, 2), (1, 0, -4).
- A power spectrum is observed to fit the form .What are some wavelets with this spectrum? Which is minimum phase? (HINT: ;use quadratic formula.)
- Show that if a wavelet is real,
the roots of the spectrum
*R*come in the quadruplets*Z*, 1/_{0}*Z*, , and .Look into the case of roots exactly on the unit circle and on the real axis. What is the minimum multiplicity of such roots?_{0}

10/30/1997