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Hamming Function

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Hamming

An apodization function chosen to minimize the height of the highest sidelobe (Hamming and Tukey 1949, Blackman and Tukey 1959). The Hamming function is given by

A(x)=0.54+0.46cos((pix)/a),
(1)

and its full width at half maximum is 1.05543a.

The corresponding instrument function is

I(k)=(a(1.08-0.64a^2k^2)sinc(2piak))/(1-4a^2k^2).
(2)

This apodization function is close to the one produced by the requirement that the instrument function goes to 0 at ka=5/4. The FWHM is 1.81522/L, the peak is 1.08, and the peak negative and positive sidelobes (in units of the peak) are -0.00689132 and 0.00734934, respectively.

From the apodization function, a general symmetric apodization function A(x) can be written as a Fourier series

A(x)=a_0+2sum_(n=1)^inftya_ncos((npix)/b),
(3)

where the coefficients satisfy

a_0+2sum_(n=1)^inftya_n=1.
(4)

The corresponding instrument function is

I(t)=2b{a_0sinc(2pikb)+sum_(n=1)^infty[sinc(2pikb+npi)+sinc(2pikb-npi)]}.
(5)

To obtain an apodization function with zero at ka=3/4, use

a_0+2a_1=1,
(6)

so

a_0sinc(5/2pi)+a_1[sinc(7/2pi)+sinc(3/2pi)=0
(7)
(1-2a_1)2/(5pi)-a_1(2/(7pi)+2/(3pi))=(1-2a_1)1/5-a_1(1/7+1/3)=0
(8)
a_1(1/7+1/3+2/5)=1/5
(9)
a_1=(21)/(92) approx 0.2283
(10)
a_0=(25)/(46) approx 0.5435.
(11)

See also

Apodization Function, Hanning Function, Instrument Function

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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.Hamming, R. W. and Tukey, J. W. "Measuring Noise Color." Unpublished memorandum, 1949.

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Hamming Function

Cite this as:

Weisstein, Eric W. "Hamming Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HammingFunction.html

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