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)