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Here we examine an example of the general idea that
adjoints of products are reverse-ordered products of adjoints.
For this example we use the Fourier transformation.
No details of Fourier transformation are given here
and we merely use it as an example of a square matrix .We denote the complex-conjugate transpose (or adjoint) matrix
with a prime,
i.e., .The adjoint arises naturally whenever we consider energy.
The statement that Fourier transforms conserve energy is
where .Substituting gives , which shows that
the inverse matrix to Fourier transform
happens to be the complex conjugate of the transpose of .
With Fourier transforms,
zero padding and truncation are especially prevalent.
Most modules transform a dataset of length of 2^{n},
whereas dataset lengths are often of length .The practical approach is therefore to pad given data with zeros.
Padding followed by Fourier transformation can be expressed in matrix algebra as

| |
(13) |

According to matrix algebra, the transpose of a product,
say ,is the product in reverse order.
So the adjoint routine is given by
| |
(14) |

Thus the adjoint routine
*truncates* the data *after* the inverse Fourier transform.
This concrete example illustrates that common sense often represents
the mathematical abstraction
that adjoints of products are reverse-ordered products of adjoints.
It is also nice to see a formal mathematical notation
for a practical necessity.
Making an approximation need not lead to collapse of all precise analysis.

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Stanford Exploration Project

4/27/2004