A sparsity-perspective to quadratic time–frequency distributions

We examine nonstationary signals within the framework of compressive sensing and sparse reconstruction. Most of these signals, which arise in numerous applications, exhibit small relative occupancy in the time–frequency domain, casting them as sparse in a joint-variable representation. We present two general approaches to incorporate sparsity into time–frequency analysis, leading to what we refer to as sparsity-aware quadratic time–frequency distributions. Both approaches exploit time–frequency sparsity under full data and compressed observations. In the first approach, quadratic time–frequency distributions are derived based on optimal multi-task kernel design. In this case, sparseness in the time–frequency domain presents itself as a new design task, adding to the two fundamental tasks of auto-term preservation and cross-term suppression. In the second approach, sparse reconstruction is used, in lieu of the Fourier transform, to obtain quadratic time–frequency distributions from compressed measurements observed in the time domain or the joint-variable domain. It is shown that multiple measurement vector methods and block sparsity techniques play a fundamental role in improving signal local frequency representations. Examples of both approaches are provided. Analysis is supported by results based on simulated data, electromagnetic modeled data, and real Doppler and micro-Doppler data measurements of radar returns associated with human motions.