The Apodization Function
![\begin{displaymath}
f(x)=1-{\vert x\vert\over a}
\end{displaymath}](b_175.gif) |
(1) |
which is a generalization of the one-argument Triangle Function. Its Full Width at Half Maximum is
.
It has Instrument Function
Letting
in the first part therefore gives
Rewriting (2) using (3) gives
Integrating the first part and using the integral
![\begin{displaymath}
\int x\cos(bx)\,dx={1\over b^2}\cos(bx)+{x\over b}\sin(bx)
\end{displaymath}](b_185.gif) |
(5) |
for the second part gives
where
is the Sinc Function. The peak (in units of
) is 1. The function
is always positive, so
there are no Negative sidelobes. The extrema are given by letting
and solving
![\begin{displaymath}
{d\over d\beta} \left({\sin\beta\over\beta}\right)^2 = 2{\sin\beta\over\beta}{\sin\beta-\beta\cos\beta\over\beta^2}=0
\end{displaymath}](b_193.gif) |
(7) |
![\begin{displaymath}
\sin\beta(\sin\beta-\beta\cos\beta)=0
\end{displaymath}](b_194.gif) |
(8) |
![\begin{displaymath}
\sin\beta-\beta\cos\beta=0
\end{displaymath}](b_195.gif) |
(9) |
![\begin{displaymath}
\tan\beta=\beta.
\end{displaymath}](b_196.gif) |
(10) |
Solving this numerically gives
for the first maximum, and the peak Positive sidelobe is 0.047190. The
full width at half maximum is given by setting
and solving
![\begin{displaymath}
\mathop{\rm sinc}\nolimits ^2 x={\textstyle{1\over 2}}
\end{displaymath}](b_199.gif) |
(11) |
for
, yielding
![\begin{displaymath}
x_{1/2}=\pi k_{1/2} a=1.39156.
\end{displaymath}](b_201.gif) |
(12) |
Therefore, with
,
![\begin{displaymath}
{\rm FWHM}=2k_{1/2} ={0.885895\over a} ={1.77179\over L}.
\end{displaymath}](b_203.gif) |
(13) |
See also Apodization Function, Parzen Apodization Function, Triangle Function
References
Bartlett, M. S. ``Periodogram Analysis and Continuous Spectra.'' Biometrika 37, 1-16, 1950.
© 1996-9 Eric W. Weisstein
1999-05-26