Comments on Hilbert Transform Based Signal Analysis

The use of the Hilbert transform for time/frequency analysis of signals is briefly considered. While the Hilbert transform is based on arbitrary continuous signals, most practical signals are digitially sampled and time-limited. To avoid aliasing in the sampling process the signals must also be bandlimited. It is argued that it is reasonable to consider such sampled signals as periodic (this is the basis of the Discrete Fourier Transform [DFT]) since any other interpretation is inconsistent. A simple derivation of the Hilbert transform for a sampled, periodic is then given. It is shown that the instantaneous frequency can be easily computed from the Discrete Fourier Series (or, equivalently, the DFT) representation of the signal. Since this representation is exact, the Hilbert transform representation is also exact.