Radar Ambiguity Functions and Group Theory

Abstract

P. M. Woodward in the early 1950’s introduced a mapping from a radar signal f to a function of two variables $W(f)$, called the ambiguity function, that plays a central role in the radar design problem. We may think of W as a nonlinear operator from $L^2 (\mathbb{R})$ into $L^2 (\mathbb{R}^2 )$. The description of the range of W has been an open problem. This paper provides, in terms of special functions in $L^2 (\mathbb{R})$ and $L^2 (\mathbb{R}^2 )$ a fairly complete description of $W(L^2 (\mathbb{R}))$. We show also that $W(L^2 (\mathbb{R}))$ is a closed subset of $L^2 (\mathbb{R}^2 )$ and if $W(f) + W(g) = W(h),f,g,h \in L^2 (\mathbb{R})$ then $f = \lambda g,\lambda $ a constant.