Duality and classification of bilinear time-frequency signal representations

The author discusses a fundamental duality principle of BTFRs (bilinear time-frequency representations) and presents a systematic classification of BTFRs which is consistent with BTFR duality. BTFRs are first grouped into two basic domains, namely, the energy density domain (E-domain) with energetic interpretation, and the correlation domain (C-domain) with correlative interpretation. It is shown that these domains are related by a Fourier transform duality: to any BTFR, BTFR relation, or BTFR property of the E-domain, there corresponds a dual BTFR, BTFR relation, or BTFR properties of the C-domain, and vice versa. With this duality principle as a background, a classification of BTFRs is given. This classification is based on two dual shift-invariance properties and a self-dual scale-invariance property of BTFRs. The mathematical description of BTFRs by means of kernel functions is simplified, inside the respective BTFR classes. It is shown that BTFRs which are both shift and scale invariant can be represented as superpositions of generalized Wigner distributions (E-domain) or generalized ambiguity function (C-domain)