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Ramp Function

\begin{figure}\begin{center}\BoxedEPSF{RampFunction.epsf scaled 800}\end{center}\end{figure}

$\displaystyle R(x)$ $\textstyle \equiv$ $\displaystyle xH(x)$ (1)
  $\textstyle =$ $\displaystyle \int^x_{-\infty} H(x')\,dx'$ (2)
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty H(x')H(x-x')\,dx'$ (3)
  $\textstyle =$ $\displaystyle H(x)*H(x),$ (4)

where $H(x)$ is the Heaviside Step Function and $*$ is the Convolution. The Derivative is
R'(x) = -H(x).
\end{displaymath} (5)

The Fourier Transform of the ramp function is given by
{\mathcal F}[R(x)]=\int_{-\infty}^\infty e^{-2\pi ikx}R(x)\,dx = \pi i\delta'(2\pi k)-{1\over 4\pi^2 k^2},
\end{displaymath} (6)

where $\delta(x)$ is the Delta Function and $\delta'(x)$ its Derivative.

See also Fourier Transform--Ramp Function, Heaviside Step Function, Rectangle Function, Sgn, Square Wave

© 1996-9 Eric W. Weisstein