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 | Model fitting by least squares |  |
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Another way to say
is to say
is small, or
is small.
This does not solve the problem of
going to zero,
so we need the idea that
does not get too big.
To find
, we minimize the quadratic function in
.
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(2) |
The second term is called a ``damping factor,''
because it prevents
from going to
when
.
Set
, which gives:
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(3) |
Equation (3) yields our earlier common-sense guess
.
It also leads us to wider areas of application in which the elements are complex
vectors and matrices.
With Fourier transforms,
the signal
is a complex number at each frequency
.
Therefore we generalize equation (2) to:
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(4) |
To minimize
, we could use a real-values approach,
where we express
in terms of two real values
and
,
and then set
and
.
The approach we take, however,
is to use complex values,
where we set
and
.
Let us examine
:
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(5) |
The derivative
is
the complex conjugate of
.
Therefore, if either is zero, the other is also zero.
Thus, we do not need to specify both
and
.
I usually set
equal to zero.
Solving equation (5) for
gives equation (1).
Equation (1) solves
for
,
giving the solution for what is called
``the deconvolution problem with a known wavelet
.''
Analogously, we can use
when the filter
is unknown,
but the input
and output
are given.
Simply interchange
and
in the derivation and result.
 |
 |
 |
 | Model fitting by least squares |  |
![[pdf]](icons/pdf.png) |
Next: Smoothing the denominator spectrum
Up: UNIVARIATE LEAST SQUARES
Previous: Dividing by zero smoothly
2014-12-01