Abstract
In this paper, instantaneous frequency expressions are mathematically derived, where each of the expressions is a function of sine-wave samples in time-differential domain space. At first, we introduce differential operator, divided finite difference operator, shifting operator, and mean operator. Then, we derive a strict expression of instantaneous frequency by applying the operators to the governing equation, which generates sine-wave. It is shown that the derived expression is identical to the Prony's method. Next, the other five expressions of instantaneous frequency are derived by the operator-applying method. Numerical results show that precise instantaneous frequencies are calculated by using the expressions. All the expressions are not under constraint of the uncertainty principle, and one of them is not under constraint of the sampling theorem.
