Regularity and unitarity of bilinear time-frequency signal representations

Abstract

Two structural properties of bilinear time-frequency representations (BTFRs) of signals are introduced and studied. The definition of these properties is based on a linear-operator description of BTFRs. The first property, termed regularity, has important implications with respect to the recovery of signals from the BTFR outcome, the derivation of other bilinear signal representations from a BTFR, the BTFRs reaction to linear signal transformations, and the construction of bases of induced BTFR-domain spaces. The second property, called unitarity, is equivalent to validity of Moyal's formula (1949). Unitarity is thus necessary and sufficient for a closed-form solution of optimal signal synthesis and for a BTFR formulation or optimal detection/estimation methods. Unitarity also allows the systematic construction of BTFR product relations like the Wigner distribution's interference formula and the ambiguity function's self-transformation property. Unitarity permits the construction of induced orthogonal projection operators and guarantees the orthonormality of induced basis functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>