Integral bounds for radar ambiguity functions and Wigner distributions
An upper bound is proved for the L p norm of Woodward’s ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p >2. A lower bound is proved for 1 ≤p < 2. In addition, a lower bound is proved for the ent
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Integral bounds for radar ambiguity functions and Wigner distributions Elliott H. Ueb Departments of Mathematics and Physics, Princeton University, P. O. Box 708, Princeton, New Jersey 08544
(Received 10 November 1989; accepted for publication 22 November 1989) An upper bound is proved for the LP norm of Woodward's ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p > 2. A lower bound is proved for I 2 (Theorem I) and universally bounded below when I 2 in their footnote 10. The Price-Hofstetter bounds have found application in the work of Janssen' for example. The next theorem gives reversed inequalities for p < 2. Theorem 2: Assume that t >->.f(t-~T)g·(t+~T) is in L 1for every T, so that the definition (1.1) of AJC'(T,O)) makes sense. (This L 1 condition can be satisfied, for example, by assuming thatfEL a and gELP for some 12 and that the reverse inequality holds for all 0
I. Equality is achieved iff and g are a malched Gaussian pair. Remarks: (5) It is possible to show that equality is achieved in Theorem 3 only whenfand g are matched Gaussians. The proof is complicated and dll not be given; the reader is invited to find a simple proof. The method of proof of these three theorems follows closely the methods used in Ref. 7 to prove U bounds of coherent state transforms. The coherent state transform off is Af,g( - 7, - 0) exp(hT0)7) with g being the fixed Gaussian g(t) = 7~\/4 exp( - 1'12). From the mathematical point of view there is, however, a genuinely new development in the present paper, namely the proof that Gaussians uniquely saturate the bounds. This uses Ref. 4.
II. PRELIMINARY LEMMAS
The following convention for the Fourier transfornV' of a functionfwill be employed: 1(0) 596
=
ff(t)e~'·iw'dl,
J. Math. Phys., Vol. 31, No.3, March 1990
(2.1)
The equality (1.6) follows from (2.3). Some other important facts about Aig which follow easily from (2.3), the Cauchy-Schwarz inequality and a change of integration variables are Aig (7,0)
=A~g(
A Jg(7,0) =A gJ (
-0),7), -
7, -
(2.4 )
0),
(2.5)
(2.6)
IAig (7,0) I I,b> I, as in Theorems 1 and 2, Holder's inequality yields the pointwise bound (2.7) Inequality (2.6) is important because it implies that InlAf,g (7,0) 1'
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