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Equation (1) is the solution to an optimization problem
that arises in many applications.
Now that we know the solution, let us formally define the problem.
First, we solve a simpler problem with real values.
We choose to minimize the quadratic function of *x*
| |
(2) |

The second term is called a *damping factor*
because it prevents *x* from going to when *f*=0.
Set *dQ*/*dx*=0 getting
| |
(3) |

which yields the earlier answer .
With Fourier transforms,
the signal *X* is a complex number at each frequency .So we generalize equation (2) to

| |
(4) |

To minimize *Q* we could use a real-values approach where we express
*X*=*u*+*iv* in terms of two real values *u* and *v*
and then set and .Alternately,
we can use a complex-values approach where we set
and .Let us examine .
| |
(5) |

The derivative is
the complex conjugate of .So if one is zero, the other is too.
Thus we don't need to specify both
and because either one is enough.
I usually take
.Solving (5) for *X* gives the answer (1).

** Next:** Unknown filter
** Up:** HOW TO DIVIDE NOISY
** Previous:** Dividing by zero smoothly
Stanford Exploration Project

1/13/1998