Complex Envelope Properties, Interpretation, Filtering, and Evaluation

摘要

The complex envelope of a narrowband waveform y(t) typically has logarithmicsingularities, due to discontinuities in y(t) or its derivatives, which have little physical significance. The complex envelope also has a very slow decay in time, due to the discontinuous spectrum associated with its very definition; this slow decay can mask weak desired features of the complex envelope. In order to suppress these undesired behaviors of the mathematically defined complex envelope, a filtered version is suggested and investigated in terms of its singularity rejection capability and better decay rate. Finally, numerical computation of the complex envelope, as well as its filtered version, by means of a fast Fourier transform (FFT) is investigated and the effects of aliasing are assessed quantitatively. A program for the latter task, using an FFT collapsing procedure, is furnished in BASIC.