![\begin{displaymath}
\amalg\mkern-10.5mu\amalg (x)\equiv \sum_{n=-\infty}^\infty \delta(x-n)
\end{displaymath}](s1_836.gif) |
(1) |
where
is the Delta Function, so
for
(i.e.,
not an Integer). The
shah function obeys the identities
for
(i.e.,
a half-integer).
It is normalized so that
![\begin{displaymath}
\int_{n-1/2}^{n+1/2} \amalg\mkern-10.5mu\amalg (x)\,dx=1.
\end{displaymath}](s1_847.gif) |
(5) |
The ``sampling property'' is
![\begin{displaymath}
\amalg\mkern-10.5mu\amalg (x)f(x)=\sum_{n=-\infty}^\infty f(n)\delta(x-n)
\end{displaymath}](s1_848.gif) |
(6) |
and the ``replicating property'' is
![\begin{displaymath}
\amalg\mkern-10.5mu\amalg (x)*f(x)=\sum_{n=-\infty}^\infty f(x-n),
\end{displaymath}](s1_849.gif) |
(7) |
where
denotes Convolution.
See also Convolution, Delta Function, Impulse Pair
© 1996-9 Eric W. Weisstein
1999-05-26