Instantaneous frequency and the Wigner-Gabor signal

Determining the amplitude and phase of a signal is important in many areas of science and engineering. The derivative of the phase is typically called the "instantaneous frequency," which in principle mathematically describes (and ideally coincides with) the common physical experiences of variable-frequency phenomena, such as a siren. However, there is an infinite number of different amplitude-phase pairs that will all generate the same real signal, and hence there is an unlimited number of "instantaneous frequencies" for a given real signal. Gabor gave a procedure for associating a specific complex signal to a given real signal, from which a unique definition of the amplitude and phase, and consequently the instantaneous frequency, of the real signal is obtained. This complex signal, called the analytic signal, is obtained by inverting the Fourier spectrum of the real signal over the positive frequency range only. We introduce a new complex signal representation by applying Gabor's idea to the Wigner time-frequency distribution. The resulting complex signal, which we call the Wigner-Gabor signal, has a number of interesting properties that we discuss and compare with the analytic signal. In general the Wigner-Gabor signal is not the analytic signal, although for a pure tone A cos(omega(0)t) the Wigner-Gabor and analytic signals both equal A exp(j omega(0)t). Also, for a time-limited signal s(t) = 0, vertical bar t vertical bar > T, the analytic signal is not time-limited, but the Wigner-Gabor signal is time-limited.