On amplitude and frequency demodulation using energy operators

It is shown that the nonlinear energy-tracking signal operator Ψ(x)=(dx/dt)²-xd ²x/dt² and its discrete-time counterpart can estimate the AM and FM modulating signals. Specifically, Ψ can approximately estimate the amplitude envelope of AM signals and the instantaneous frequency of FM signals. Bounds are derived for the approximation errors, which are negligible under general realistic conditions. These results, coupled with the simplicity of Ψ, establish the usefulness of the energy operator for AM and FM signal demodulation. These ideas are then extended to a more general class of signals that are sine waves with a time-varying amplitude and frequency and thus contain both an AM and an FM component; for such signals it is shown that Ψ can approximately track the product of their amplitude envelope and their instantaneous frequency. The theoretical analysis is done for both continuous- and discrete-time signals