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Hamming Function

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

An Apodization Function chosen to minimize the height of the highest sidelobe. The Hamming function is given by

A(x)=0.54+0.46\cos\left({\pi x\over a}\right),
\end{displaymath} (1)

Its Full Width at Half Maximum is $1.05543a$. The corresponding Instrument Function is
I(k)={a(1.08-0.64a^2k^2)\mathop{\rm sinc}\nolimits (2\pi ak)\over 1-4a^2k^2}.
\end{displaymath} (2)

This Apodization Function is close to the one produced by the requirement that the Apparatus Function goes to 0 at $ka=5/4$. From Apodization Function, a general symmetric apodization function $A(x)$ can be written as a Fourier Series
A(x)=a_0+2\sum_{n=1}^\infty a_n\cos\left({n\pi x\over b}\right),
\end{displaymath} (3)

where the Coefficients satisfy
a_0+2\sum_{n=1}^\infty a_n=1.
\end{displaymath} (4)

The corresponding apparatus function is

I(t)=2b\{a_0\mathop{\rm sinc}\nolimits (2\pi kb)+\sum_{n=1}^...
...s (2\pi kb+n\pi)+\mathop{\rm sinc}\nolimits (2\pi kb-n\pi)]\}.
\end{displaymath} (5)

To obtain an Apodization Function with zero at $ka=3/4$, use
\end{displaymath} (6)

a_0\mathop{\rm sinc}\nolimits ({\textstyle{5\over 2}}\pi)+a_...
...}\pi)+\mathop{\rm sinc}\nolimits ({\textstyle{3\over 2}}\pi)=0
\end{displaymath} (7)

(1-2a_1){2\over 5\pi}-a_1\left({{2\over 7\pi}+{2\over 3\pi}}...
...\over 5}}-a_1({\textstyle{1\over 7}}+{\textstyle{1\over 3}})=0
\end{displaymath} (8)

a_1({\textstyle{1\over 7}}+{\textstyle{1\over 3}}+{\textstyle{2\over 5}})={\textstyle{1\over 5}}
\end{displaymath} (9)

$\displaystyle a_1$ $\textstyle =$ $\displaystyle {{\textstyle{1\over 5}}\over{\textstyle{2\over 5}}+{\textstyle{1\...
...7}}+{\textstyle{1\over 3}}} = {7\cdot 3\over 2\cdot 3\cdot 7+3\cdot 5+5\cdot 7}$  
  $\textstyle =$ $\displaystyle {\textstyle{21\over 92}}\approx 0.2283$ (10)
$\displaystyle a_0$ $\textstyle =$ $\displaystyle 1-2a_1={92-2\cdot 21\over 92} = {92-42\over 92}$  
  $\textstyle =$ $\displaystyle {\textstyle{50\over 92}} = {\textstyle{25\over 46}}\approx 0.5435.$ (11)

The FWHM is 1.81522, the peak is 1.08, the peak Negative and Positive sidelobes (in units of the peak) are $-0.00689132$ and 0.00734934, respectively.

See also Apodization Function, Hanning Function, Instrument Function


Blackman, R. B. and Tukey, J. W. ``Particular Pairs of Windows.'' In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 98-99, 1959.

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© 1996-9 Eric W. Weisstein