Some remarks on instantaneous amplitude and frequency of signals

Abstract

The most common way to define instantaneous amplitude, phase and frequency of a real signal is to use its analytic signal. One of the main advantages of this procedure is that it establishes a one-to-one relationship between any real signal and the pair of functions defining its instantaneous amplitude and phase. However, there are some problems in its physical interpretation but it is shown that for narrow-band signals the analytical signal is well related to some physical procedures allowing the measurements of amplitude and frequency. As any analytical signal is a very specific function, any arbitrary pair of functions cannot be considered as the amplitude and phase of a real signal. This point is especially discussed in the case of phase signals, which means signals with constant instantaneous amplitude. The phase must satisfy very specific conditions related to the theory of Blaschke functions and analyzed in the regular as well as in the singular cases. These results are applied to hyperbolic and parabolic chirp signals. Hyperbolic signals are true phase signals corresponding to the singular case. On the other hand parabolic signals, often called signals with instantaneous frequency varying linearly in time, are not phase signals and their amplitude is not constant. This point is analyzed in detail and the structure of their amplitude and phase is explicitly calculated. Various calculations allow the evaluation of the errors appearing when assuming that these signals are true phase signals. Some extensions to other cases of chirp signals are discussed.