Nontrivial harmonic waves with positive instantaneous frequency

The concept of time-varying frequency is fundamental in communications and nature. One description of a time-varying frequency is the instantaneous frequency of a signal. Our concern is the design of analytic signals with nonlinear phase and positive instantaneous frequency. In this paper, a kind of nontrivial orthonormal harmonic waves are derived by applying the Gram–Schmidt procedure to the Blaschke products. Furthermore, we show the orthonormal harmonic waves satisfy the analytic condition and have positive instantaneous frequencies which are tightly associated with the average frequencies of the two mono-component analytic signals at each time. As a consequence, non-stationary signals can be decomposed into the superposition of nontrivial harmonic waves. Related conclusions are established for the orthonormal analytic signals on the whole time range.