Padded convolution of words ($u\otimes v$)

The padded convolution of words reads two finite words over an alphabet in parallel, extending the shorter with a fresh padding symbol $\#$ until the longer ends. Its length is $\max (|u|,|v|)$. Padding may only occur as a suffix, and a $(\#,\#)$ column is excluded. This encodes synchronous two-tape relations as formal languages over a finite paired alphabet.