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A technique is described whereby two-dimensional windows are designed for use with algorithms, as used in Range-Doppler radar applications, for computing the ambiguity function around the origin.
English (United States)
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Designing Windows for Filtering the Ambiguity Function in Range-Doppler
A technique is described whereby two-dimensional windows are designed
for use with algorithms, as used in Range-Doppler radar applications, for
computing the ambiguity function around the origin.
Basic Range-Doppler radar relies on the computation of a portion of the
ambiguity function around the origin. This computation is so costly that the
function is sampled on a small set of range coordinates. Since windowed
ambiguity functions are required to estimate sampled values (*), the concept
described herein provides a method of generating good windows for this
The question of how to find two-dimensional windows K(x,p), whose inverse
Fourier transforms have the following form is addressed. It is straightforward to
see that this requires W to be of the form
First, a generalization allows windows to be viewed as images of linear
operators acting on rank-one tensors. Two linear operators on L2(R) are defined
L1(K)(x,y) = K(ax+by,cx+dy) where a, b, c, d e R and ad-bc * 0, and
The first operator is an invertible change of basis and the second is the
Fourier transform with respect to the first variable, followed by a change of
variables. Operator A is defined as the composition Observe that for functions of
the form K(t,x) = f(t)g*(x) and b=c=d=1, a=0, the image of the linear operator A is
precisely a window of the type desired. Therefore, the tensor-product notation
(fRg*)(x,y) = f(x)g* (y) is used. In the following, the inner products will be
evaluated of the form < A(fRg*), K > + < fRg*, A*(K) > where, as usual, A*
denotes the adjoint of A.
Every K e L2(R) acts on L2(R2) as a Hilbert-Schmidt...