(1) |

Now we introduce another wavelet
**d** which will have the same number of
components as **c**.
We call **d** the desired output of the filter.
We saw that **c** is the actual output.
The actual output **c** was seen to be a
function of the input **b** and the filter **f**.
The problem now is to determine **f**
so that **c** and **d** are very much alike.
Specifically we will choose
**f** so that the difference vector
has minimum length squared
(in *n* + *m* + 1 dimensional space).
In other words,
we use the method of least squares to
solve the overdetermined equations

(2) |

(3) |

The formulas of this section may also be used
to attempt to predict a time series from its past.
For example is a
prediction filter of *x*_{t+10} from
if we solve by least squares the equations

(4) |

If the matrix of (4) is very much higher than it is wide, it may be desirable to treat the end effects differently. If one uses instead

(5) |

Of special interest is the filter which is designed from the equations

(6) |

Such a filter is called the
*prediction error filter for unit span*
because the *a*_{k} operate on
attempting to cancel *x*_{t}.
Thus, the *a*_{k} on the gives the negative of a best prediction of *x*_{t},
based on .The normal equations implied by (6)
are the square set

(7) |

Solutions to Toeplitz equations when the right-hand side takes the more arbitrary form (3) are not generally minimum-phase, but the Levinson recursion may be generalized to make the calculation speedy. This is done in Section 7.5 on the multichannel Levinson recursion.

- Find a three-term zero delay inverse
to the wavelet (1,2). Compare the error to the error of
(2,1). Compare the waveform.
An extensive discussion of the error in least-squares
inverse filters is given in Reference 26.
One conclusion is that the sum of the squared errors
goes to zero as the filter length becomes infinite in
two situations:
(

*a*) Zero delay inverse if and only if the wavelet being inverted is minimum-phase.(

*b*) If the wavelet being inverted in not minimum-phase, the error goes to zero only if the output is delayed, that it, .Calculate a three-term delayed inverse to (1,2), that is, try*d*= (0, 1, 0, 0) or*d*= (0, 0, 1, 0). - A pressure sensor in a deep well records upgoing seismic
waves and, at some time
*t*later, identical downgoing waves of opposite sign. Determine delayed and non-delayed least-squares filters of length_{0}*m*to eliminate the double pulse. (You should be able to guess the solution to large matrices of this type. Try filters of the form where and are scalars.) What is the error as a function of the filter length? - Let .Find by least squares the best one-term filter which
predicts
*b*_{t}, using only*b*_{t-1}. Find the best two-term filter using*b*_{t-1}and*b*_{t-2}. Likewise find the best three-term filter. What is the error as a function of time in each case?

10/30/1997