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Let .The coefficients y_{t} can be found from the coefficients
x_{t} and b_{t}
by convolution in the time domain or
by multiplication in the frequency domain.
For the latter, we would
evaluate both X(Z) and B(Z) at uniform locations around the unit circle,
i.e., compute Fourier sums X_{k} and B_{k} from x_{t} and b_{t}.
Then we would form
C_{k}=X_{k} B_{k} for all k, and inverse Fourier transform to y_{t}.
The values y_{t} come out the same as by the timedomain convolution method,
roughly that of our calculation precision
(typically fourbyte arithmetic or about one part in 10^{6}).
The only way in which you need to be cautious is to use
zero padding greater
than the combined lengths of x_{t} and b_{t}.
An example is shown in Figure 8.
It is the result of a Fourierdomain computation
which shows that the convolution of a rectangle function
with itself gives a triangle.
Notice that the triangle is cleanthere are no unexpected end effects.
box2triangle
Figure 7
Top shows a rectangle transformed to a sinc.
Bottom shows the sinc squared, back transformed to a triangle.

 
Because of the fast method of Fourier transform described next,
the frequencydomain calculation is quicker when both X(Z) and B(Z)
have more than roughly 20 coefficients.
If either X(Z) or B(Z) has less than roughly 20 coefficients,
then the timedomain calculation is quicker.
Next: SETTING UP THE FAST
Up: SYMMETRIES
Previous: Plot interpretation
Stanford Exploration Project
10/21/1998