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Think of any real numbers , , and and any
program containing .
How can we change the program so that it never divides by zero?
A popular answer is to change
to
, where is any tiny value.
When
,
then is approximately as expected.
But when the divisor vanishes,
the result is safely zero instead of infinity.
The transition is smooth,
but some criterion is needed to choose the value of .
This method may not be the only way or the best way
to cope with
zero division,
but it is a good method,
and permeates the subject of signal analysis.
To apply this method in the Fourier domain,
suppose that , , and are complex numbers.
What do we do then with ?
We multiply the
top and bottom by the complex conjugate ,
and again add to the denominator.
Thus,
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(1) |
Now,
the denominator must always be a positive number greater than zero,
so division is always safe.
Equation (1) ranges continuously from
inverse filtering, with
, to filtering with
,
which is called ``matched filtering.''
Notice that for any complex number ,
the phase of equals the phase of ,
so the filters
have the same phase.
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| Model fitting by least squares | |
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Next: Damped solution
Up: UNIVARIATE LEAST SQUARES
Previous: UNIVARIATE LEAST SQUARES
2014-12-01