DUALITY AND CLASSIFICATION OF BILINEAR TIME-FREQUENCY SIGNAL REPRESENTATIONS

Bilinear time-frequency representations (BTFR's) of signals, like Wigner distribution, ambiguity function, and spectrogram, are both an important signal theoretic concept and a powerful tool for signal analysis and processing. This paper discusses a fundamental duality principle of BTFR's and presents a systematic classification of BTFR's which is consistent with BTFR duality. BTFR's 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 property of the C-domain, and vice versa. With this duality principle as a background, a classification of BTFR's is given. This classification is based on two dual shift-invariance properties and a self-dual scale-invariance property of BTFR's. The mathematical description of BTFR's by means of kernel functions is simplified inside the respective BTFR classes. It is shown that BTFR's which are both shift invariant and scale invariant can be represented as superpositions of generalized Wigner distributions (E-domain) or generalized ambiguity functions (C-domain).