* Corresponding author. Tel.: 351-2184-18286; fax: 351-2184-18291. E-mail address: vab@isr.ist.utl.pt (V. Barroso). Journal of the Franklin Institute 337 (2000) 303}316 De"nitions of Instantaneous Frequency under physical constraints Paulo M. Oliveira, Victor Barroso* Escola Naval, Base Naval de Lisboa, Portugal Instituto Superior Te & cnico, Instituto de Sistemas e Robo H tica, Torre Norte, Piso. 7, Av Rovisco Pais, 1049-001 Lisbon, Portugal Accepted 22 March 2000 Abstract Ville's de"nition of Instantaneous Frequency (IF), even though widely used and accepted, fails in most cases of practical interest. This failure has been often reported, namely in the multicomponent case. In this paper, we will analyze the advantages and shortcomings of that de"nition, and determine when and why it fails, integrating all previously reported cases of failure in a simple uni"ed theory. We will also be able to extrapolate and predict the behavior of the traditional de"nition for any type of multicomponent signal. Alternative de"nitions of IF are discussed.  2000 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Time}frequency signal analysis; Instantaneous frequency; Mean frequency; Multicomponent signals; Time-varying signals 1. Introduction The analysis of signals with time varying spectral contents have always presented considerable di$culties. These di$culties are not only of an operational nature (the loss of ergodicity being one of these), but also of a deeper and more pervasive nature: the very essence of the concepts involved, which severely hampers our ability to understand and, ultimately, reason in a physically sound way. The concept of Instantaneous Frequency (IF) is a paradigm of this. Even though physically intuitive at "rst sight (e.g. in the context of FM theory), it has evaded proper 0016-0032/00/$20.00  2000 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 0 ) 0 0 0 2 9 - 6