On analytic signals with nonnegative instantaneous frequencies

In this paper, we characterize all analytic signals with band-limited amplitudes and polynomial phases. We show that a signal with band-limited amplitude and polynomial phase is analytic if and only if it has nonneg-ative constant instantaneous frequency, i.e., the derivative of the phase is a nonnegative constant, and the constant is greater than or equal to the minimum bandwidth of the amplitude. 1 Introduction Both concepts of analytic signals and instantaneous frequencies play important roles in many areas including communication systems , physics, and joint time-frequency analysis in signal processing. They have been extensively studied, see for example 1-13]. A signal is analytic if its Fourier spectrum vanishes at negative frequencies. This implies that a nonzero analytic signal must be complex-valued. The instantaneous frequency (IF) of a complex-valued signal is commonly deened as the derivative of the phase of the signal. The IF is used as a generalization of the conventional frequency from the global sense to the local sense. With these two deenitions, it is not unusual to expect that the IF of an analytic signal are nonnegative. Trivial examples of analytic signals with nonnegative IF are single tone signals exp(j! 0 t) for nonnega-tive constants ! 0. It is, however, not always true for an analytic signal to have nonnegative IF. In this paper, we consider the class of signals with band-limited amplitudes and polynomial phases. This class is quite broad in our real applications, such as in communication systems. We characterize all analytic signals in the class as follows. Let f(t) be a signal with polynomial phase and band-limited amplitude of minimal bandwidth B. It is shown that the signal f(t) is analytic if and only if it has a constant nonnegative IF ! 0 with ! 0 B for all t except possibly isolated points of t. Although this result looks simple, it turns out that the proof is not trivial. 2 Main Results Before going to the main results, let us recall a characterization for a general analytic signal. In what follows, the letter z always stands for a complex value with its real part x and its imaginary part y, i.e., z = x + jy. Let be a region of the complex plane. A