  Natural numbers satisfying a certain unusual property are defined by the
author in a previous note. Later, the author called such numbers
$v$-palindromic numbers and proved a periodic phenomenon pertaining to such
numbers and repeated concatenations of the digits of a number. It was left as a
problem of further investigation to find the smallest period. In this paper, we
provide a method to find the smallest period. Some theorems from signal
processing are used, but we also supply our own proofs.
