Time–frequency‐based instantaneous frequency estimation of sparse signals from incomplete set of samples

The estimation of time‐varying instantaneous frequency (IF) for monocomponent signals with an incomplete set of samples is considered. A suitable time–frequency distribution (TFD) reduces the non‐stationary signal into a local sinusoid over the lag variable prior to the Fourier transform. Accordingly, the observed spectral content becomes sparse and suitable for compressive sensing reconstruction in the case of missing samples. Although the local bilinear or higher order auto‐correlation functions will increase the number of the missing samples, the analysis shows that an accurate IF estimation can be achieved even if we deal with only few samples, as long as the auto‐correlation function is properly chosen to coincide with the signals phase non‐linearity. In addition, by employing the sparse signal reconstruction algorithms, ideal time–frequency representations are obtained. The presented theory is illustrated on several examples dealing with different auto‐correlation functions and corresponding TFDs.