info prev up next book cdrom email home

Impulse Pair

\begin{figure}\begin{center}\BoxedEPSF{ImpulsePairEven.epsf}\end{center}\end{figure}

The even impulse pair is the Fourier Transform of $\cos(\pi k)$,

\begin{displaymath}
\mathop{\rm II}\nolimits (x)\equiv {\textstyle{1\over 2}}\de...
...r 2}})+{\textstyle{1\over 2}}\delta(x-{\textstyle{1\over 2}}).
\end{displaymath} (1)

It satisfies
\begin{displaymath}
\mathop{\rm II}\nolimits (x)*f(x)={\textstyle{1\over 2}}f(x+...
...1\over 2}})+{\textstyle{1\over 2}}f(x-{\textstyle{1\over 2}}),
\end{displaymath} (2)

where $*$ denotes Convolution, and
\begin{displaymath}
\int_{-\infty}^\infty\mathop{\rm II}\nolimits (x)\,dx = 1.
\end{displaymath} (3)


\begin{figure}\begin{center}\BoxedEPSF{ImpulsePairOdd.epsf}\end{center}\end{figure}

The odd impulse pair is the Fourier Transform of $i\sin (\pi s)$,

\begin{displaymath}
\mathop{{\rm I}\lower3pt\hbox{{\rm I}}}\nolimits (x)\equiv {...
...r 2}})-{\textstyle{1\over 2}}\delta(x-{\textstyle{1\over 2}}).
\end{displaymath} (4)




© 1996-9 Eric W. Weisstein
1999-05-26