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Convolution is the operation we do on polynomial coefficients
when we multiply polynomials.
Deconvolution is likewise for polynomial division.
Often these ideas are described
as polynomials in the variable *Z*.
Take *X*(*Z*) to denote the polynomial
whose coefficients are samples of input data,
and let *A*(*Z*) likewise denote the filter.
The convention I adopt here is that the first coefficient
of the filter has the value +1, so the filter's polynomial
is .To see how to convolve, we now identify the coefficient
of *Z*^{k} in the product *Y*(*Z*)=*A*(*Z*)*X*(*Z*).
The usual case (*k* larger than the number *N*_{a} of filter coefficients) is
| |
(1) |

Convolution computes *y*_{k} from *x*_{k} whereas deconvolution
(also called back substitution) does the reverse.
Rearranging (1) we get
| |
(2) |

where now we are finding the output *x*_{k} from
its past outputs *x*_{k-i} and from the present input *y*_{k}.
We see that the deconvolution process is essentially
the same as the convolution process,
except that the filter coefficients
are used with opposite polarity;
and they are applied to the past *outputs*
instead of the past *inputs*.
That is why deconvolution must be done sequentially
while convolution can be done in parallel.

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Stanford Exploration Project

4/27/2004