On an ambiguity in the definition of the amplitude and phase of a signal

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We point out that the conventional definition of instantaneous amplitude and frequency, namely as the magnitude and derivative of the phase, respectively, of a complex representation of the signal, sometimes contains an ambiguity, even for a unique complex representation (e.g., the analytic signal). There are at least two choices for resolving this ambiguity when it arises. One choice yields a nonnegative amplitude but an instantaneous frequency with infinite spikes, and one yields a bounded instantaneous frequency but an instantaneous amplitude with positive and negative values. Historically, both solutions (i.e., both amplitudes) have been important in radio engineering, and both can be measured with real devices. The former choice is more commonly used for defining the instantaneous amplitude and frequency of signals, but the latter choice is equally acceptable and may be preferred in some situations.