


 Basic operators and adjoints  

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Here, we examine an example of the general idea that
adjoints of products are reverseordered 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 complexconjugate 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 ;
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.
Therefore, 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 reverseordered products of adjoints.
It is also nice to see a formal mathematical notation
for a practical necessity.
Making an approximation need not lead to the collapse of all precise analysis.



 Basic operators and adjoints  

Next: Nearestneighbor coordinates
Up: FAMILIAR OPERATORS
Previous: Zero padding is the
20140927