Hamming Function

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

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

Its Full Width at Half Maximum is $1.05543a$. The corresponding Instrument Function is

$$I(k)={a(1.08-0.64a^2k^2)\mathop{\rm sinc}\nolimits (2\pi ak)\over 1-4a^2k^2}.$$ (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

$$A(x)=a_0+2\sum_{n=1}^\infty a_n\cos\left({n\pi x\over b}\right),$$ (3)

where the Coefficients satisfy

$$a_0+2\sum_{n=1}^\infty a_n=1.$$ (4)

The corresponding apparatus function is

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

To obtain an Apodization Function with zero at $ka=3/4$, use

$$a_0+2a_1=1,$$ (6)

so

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

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

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

$$a_1 = {{\textstyle{1\over 5}}\over{\textstyle{2\over 5}}+{\textstyle{1\... ...7}}+{\textstyle{1\over 3}}} = {7\cdot 3\over 2\cdot 3\cdot 7+3\cdot 5+5\cdot 7}$$

$$= {\textstyle{21\over 92}}\approx 0.2283$$ (10)

$$a_0 = 1-2a_1={92-2\cdot 21\over 92} = {92-42\over 92}$$

$$= {\textstyle{50\over 92}} = {\textstyle{25\over 46}}\approx 0.5435.$$ (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

References

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.