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# Computing the Aliased Ambiguity Surface

IP.com Disclosure Number: IPCOM000103651D
Original Publication Date: 1993-Jan-01
Included in the Prior Art Database: 2005-Mar-18
Document File: 2 page(s) / 82K

IBM

Feig, E: AUTHOR

## Abstract

A major drawback of computing the cross ambiguity surface by first computing the actual function and then taking its magnitude squared (it is quite standard to call this the surface, or as in [1,2], the intensity function) is that aliasing in the computation of the function does not relate well to the radar problem. Radar theory implies that the aliased cross ambiguity surface is well approximated as a weighted superposition of aliased auto ambiguity surfaces of the transmitted signal. It is not true that the aliased cross ambiguity function is well approximated as a weighted sum of auto-ambiguity functions. Therefore, it will be advantageous to have a very efficient method for computing the aliased surface.

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Computing the Aliased Ambiguity Surface

A major drawback of computing the cross ambiguity surface
by first computing the actual function and then taking its magnitude
squared (it is quite standard to call this the surface, or as in
[1,2], the intensity function) is that aliasing in the computation of
the function does not relate well to the radar problem.  Radar theory
implies that the aliased cross ambiguity surface is well approximated
as a weighted superposition of aliased auto ambiguity surfaces of the
transmitted signal.  It is not true that the aliased cross ambiguity
function is well approximated as a weighted sum of auto-ambiguity
functions.  Therefore, it will be advantageous to have a very
efficient method for computing the aliased surface.

We can do so because of the following identity:

(Image Omitted)

The first term in the equation is essentially the two-dimensional
Fourier transform of the cross ambiguity surface.  We say essentially
because the signs in the exponential factor are reversed for the two
dimensions.  Computationally, however, evaluating this integral
operator is just like doing a two-dimensional FFT.  Note that by
taking f=g the equation says that the auto-ambiguity surface is
essentially self dual.

From a computational point of view, the equation suggests a
novel way of evaluating the aliased cross ambiguity surface.  It can
be done by first evaluating its Fourier transform on a sparse grid,
and then doing a two-dimensional FFT.  But its Fourier transform is
the product of two auto-ambiguity functions.  As was shown in ,
these can be very easily evaluated on spar... 