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MMA Memo 218:
Level of Negative Side Lobes in an Array Beam.

L. Kogantex2html_wrap_inline220
(1) - National Radio Astronomy Observatory, Socorro, New Mexico, USA

Abstract:

It is shown that the level of negative side lobes in any array beam (PSF - Point Spread Function) is equal tex2html_wrap_inline222, where N is number of elements at the array. For the central symmetry arrays all negative side lobes are tangent to the horizontal line at the level tex2html_wrap_inline226. For large N positive side lobes are bigger than negative ones (in absolute values). For example, for N = 36 (accepted value for MMA now) the value of negative side lobes is 0.0286. The level of negative side lobes in the natural weighting case does not depend on the array configuration and is determined completely by the number of the elements at the array

The approach to the problem

Let's suppose the vector tex2html_wrap_inline252 determines the position of the array element at the aperture of the array measured at wavelengths. Then the beam pattern of the array is determined by all nonzero baselines and can be verified by the following equation in the natural weighting case.


 equation39

 where   is the vector of the direction on the sky


       		 N is the number of elements in the array.
Now let's calculate the similar double sum including zero baselines.


 equation55

 where    is a voltage beam pattern.
What is the difference between tex2html_wrap_inline256 and tex2html_wrap_inline258 ? The number of summands corresponding to the zero baselines at (2) is equal N and all of them are equal 1.
Therefore
 equation93
It is seen from equation (2) that tex2html_wrap_inline262 0 elsewhere. All minimums of the tex2html_wrap_inline258 are equal zero. Thus, as it follows from equation (3), all minimums of the tex2html_wrap_inline256 are equal tex2html_wrap_inline226. For a central symmetry array, function tex2html_wrap_inline270 is real and continuous. So it intersects zero line going from its positive lobes to negative ones. Therefore, the minimums of the function tex2html_wrap_inline272 are equal zero and all negative side lobes of tex2html_wrap_inline256 are tangent to the horizontal line at the level tex2html_wrap_inline226. Let's illustrate this statement by the one dimension homogeneous array of N=10 and N=36 elements with spacing d.
The functions tex2html_wrap_inline284 and tex2html_wrap_inline286 are shown at figures (1) and (2) respectively.
The confirmation of the statement has been found in the [1], where the negative side lobes for N=36 and N=63 were found very close to tex2html_wrap_inline222

Conclusion

The level of negative side lobes in the natural weighting case does not depend on the array configuration and is determined completely by the number of the elements at the array

References

1
J. Conway, MMA memo  216, 1998

  figure131
Figure 1: Point spread function PSFtex2html_wrap_inline294 and Ptex2html_wrap_inline294 for one dimension homogeneous array of N=10 elemments with spacing d

  figure136
Figure 2: Point spread function PSFtex2html_wrap_inline294 and Ptex2html_wrap_inline294 for one dimension homogeneous array of N=36 elements with spacing d