An Apodization Function
![\begin{displaymath}
f(x)=1,
\end{displaymath}](u_134.gif) |
(1) |
having Instrument Function
The peak (in units of
) is 2. The extrema are given by letting
and solving
![\begin{displaymath}
{d\over d\beta} (\beta\sin\beta)={\sin\beta-\beta\cos\beta\over\beta^2}=0
\end{displaymath}](u_140.gif) |
(3) |
![\begin{displaymath}
\sin\beta-\beta\cos\beta=0
\end{displaymath}](u_141.gif) |
(4) |
![\begin{displaymath}
\tan\beta=\beta.
\end{displaymath}](u_142.gif) |
(5) |
Solving this numerically gives
,
,
, ...for the first few solutions. The
second of these is the peak Positive sidelobe, and the third is the peak Negative sidelobe. As a fraction of the peak,
they are 0.128375 and
. The Full Width at Half Maximum is found by setting
![\begin{displaymath}
\mathop{\rm sinc}\nolimits (x)={\textstyle{1\over 2}},
\end{displaymath}](u_148.gif) |
(6) |
and solving for
, yielding
![\begin{displaymath}
x_{1/2}=2\pi k_{1/2} a=1.89549.
\end{displaymath}](u_150.gif) |
(7) |
Therefore, with
,
![\begin{displaymath}
{\rm FWHM}=2k_{1/2} = {0.603353\over a} ={1.20671\over L}.
\end{displaymath}](u_152.gif) |
(8) |
See also Apodization Function
© 1996-9 Eric W. Weisstein
1999-05-26