Misconceptions about instantaneous frequency, and complex signal representations through pole-zero manipulations

摘要

Instantaneous frequency is an important characteristic of time-varying or non-stationary signals. The definition and interpretation of instantaneous frequency have been the subject of discussion and debate for decades. The most common approach is due to Gabor, whereby a specific complex signal, called the analytic signal, is associated with a given real signal by inverting the spectrum of the real signal over only the positive frequency axis; the instantaneous frequency is then taken to be the derivative of the phase. Other approaches for associating a particular complex signal to a given real signal, and hence obtaining different instantaneous frequencies, have also been proposed. One way to define the associated complex signal / instantaneous frequency is by imposing physical constraints, which we discuss. We also discuss the common interpretation of instantaneous frequency as the average frequency at each time, and point out when this interpretation holds, which is not usually the case. This leads to the question of what is the "average frequency at each time?" The answer, coupled with physical constraints on the complex signal representation, leads to a quadrature-AM / FM signal model. Finally, we consider methods that manipulate the poles and zeros of the signal to obtain a complex representation.