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** Up:** FAMILIAR OPERATORS
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convolution
convolution ! transient
The next operator we examine is convolution.
It arises in many applications; and it could be derived in many ways.
A basic derivation is from the multiplication of two polynomials, say
*X*(*Z*) = *x*_{1} + *x*_{2} *Z* + *x*_{3} *Z*^{2} + *x*_{4} *Z*^{3} + *x*_{5} *Z*^{4} + *x*_{6} *Z*^{5} times
*B*(*Z*) = *b*_{1} + *b*_{2} *Z* + *b*_{3} *Z*^{2} + *b*_{4} *Z*^{3}.^{}
Identifying the *k*-th power of *Z* in the product
*Y*(*Z*)=*B*(*Z*)*X*(*Z*) gives the *k*-th row of the convolution transformation
(4).

| |
(4) |

Notice that columns of
equation (4)
all contain the same signal,
but with different shifts.
This signal is called the filter's impulse response.
Equation (4) could be rewritten as

| |
(5) |

In applications we can choose between
and
.In one case
the output is dual to the filter ,and in the other case
the output is dual to the input .Sometimes we must solve
for and sometimes for ;so sometimes we use equation (5) and
sometimes (4).
Such solutions begin from the adjoints.
The adjoint of (4) is
| |
(6) |

The adjoint *crosscorrelates* with the filter
instead of convolving with it (because the filter is backwards).
Notice that each row in
equation (6) contains all the filter coefficients
and there are no rows where the filter somehow uses zero values
off the ends of the data as we saw earlier.
In some applications it is important not to assume zero values
beyond the interval where inputs are given.
The adjoint of (5) crosscorrelates a fixed portion
of filter input across a variable portion of filter output.

| |
(7) |

Module `tcai1`
is used for
and module `tcaf1`
is used for
.tcai1transient convolution
tcaf1transient convolution

The polynomials
*X*(*Z*),
*B*(*Z*), and
*Y*(*Z*)
are called *Z* transforms.
An important fact in real life
(but not important here)
is that the *Z* transforms are
Fourier transforms in disguise.
Each polynomial is a sum of terms
and the sum
amounts to a Fourier sum when we take .The very expression
*Y*(*Z*)=*B*(*Z*)*X*(*Z*) says that a product in the frequency domain
(*Z* has a numerical value) is a convolution in the time domain
(that's how we multipy polynomials, convolve their coefficients).

** Next:** Internal convolution
** Up:** FAMILIAR OPERATORS
** Previous:** Adjoint derivative
Stanford Exploration Project

4/27/2004