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## Dividing by zero smoothly

divide by zero zero divide Think of any real numbers x, y, and f and any program containing x=y/f. How can we change the program so that it never divides by zero? A popular answer is to change x=y/f to , where is any tiny value. When ,then x is approximately y/f as expected. But when the divisor f 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 divisionzero divide, but it is a good way, and it permeates the subject of signal analysis.

To apply this method in the Fourier domain, suppose that X, Y, and F are complex numbers. What do we do then with X=Y/F? We multiply the top and bottom by the complex conjugate ,and again add to the denominator. Thus,
 (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 filter ! inverse X=Y/F, to filtering with ,which is called matched filtering.'' filter ! matched Notice that for any complex number F, the phase of 1/F equals the phase of ,so the filters have the same phase.

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Stanford Exploration Project
4/27/2004