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Revision difference : convolution |
| Version 5 |
Version 4 |
| Let $\Sigma$ be an alphabet, $\#$ a symbol not in $\Sigma$. |
Let $\Sigma$ be an alphabet, $\#$ a symbol not in $\Sigma$. |
| Let $x_1x_2\ldots x_{|x|},y_1y_2\ldots y_{|y|},z_1z_2\ldots z_{|z|},\ldots$ be $n$ words over $\Sigma^*$. Let $l$ denote the maximum length. |
Let $x_1x_2\ldots x_{|x|},y_1y_2\ldots y_{|y|},z_1z_2\ldots z_{|z|},\ldots$ be $n$ words over $\Sigma^*$. Let $l$ denote the maximum length. |
| The \emph{convolution} of these words is |
The \emph{convolution} of these words is |
| $$(x_1,y_1,\ldots)(x_2,y_2,\ldots)\ldots(x_l,y_l,\ldots)$$where for any index greater than the length of the word, the symbol is $\#$. This is a new word in $((\Sigma\cup\{\#\})^n)^*$. |
$$(x_1,y_1,\ldots)(x_2,y_2,\ldots)\ldots(x_l,y_l,\ldots)$$where for any index greater than the length of the word, the symbol is $\#$. This is a new word in $((\Sigma\cup\{\#\})^n)^*$. |
| The convolution of $x,y,z,\ldots$ is sometimes denoted conv($x,y,z,\ldots$), or $x\star y\star z\star\ldots$ |
The convolution of $x,y,z,\ldots$ is sometimes denoted conv($x,y,z,\ldots$), or $x\star y\star z\star\ldots$ |
| \subsection*{Example} |
\subsection*{Example} |
| The convolution of $and, fish, be$ is |
The convolution of $and, fish, be$ is |
| $$(a,f,b)(n,i,e)(d,s,\#)(\#,h,\#)$$ |
$$(a,f,b)(n,i,e)(d,s,\#)(\#,h,\#)$$ |
| \subsection*{Notes} |
\subsection*{Notes} |
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This definition bears no \PMlinkescapetext{mathematical relation} to the notion of \PMlinkname{convolution}{Convolution} of functions.
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This definition bears no real relation to the notion of convolution of functions.
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