Vector-valued time-frequency representations

Abstract

Cohen's (1989) class of time frequency distributions (TFDs), which includes the spectrogram (SP), Wigner distribution (WD), and reduced interference distributions (RIDs) has become widely known as a useful signal analysis tool. It has been shown that every real-valued TFD can be written as a weighted sum of SPs. The decomposition has been used to construct fast approximations to desirable TFDs using the SP building block, for which there exist accessible and efficient hardware and software implementations. We introduce a class of linear, vector-valued time-frequency representations (TFRs) that are easily related to associated bilinear TFDs through the SP decomposition. We solve a least-squares signal synthesis problem on modified vector-valued TFRs that are associated with nonnegative TFDs as a weighted sum of least-squares short-time Fourier transform (STFT) signal synthesis schemes. We extend the solution to vector-valued TFRs associated with high-resolution TFDs in order to define a high-resolution alternative to STFT signal synthesis, as demonstrated by desirable properties and examples. The resulting signal synthesis methods can be realized as a weighted sum of STFT synthesis schemes, for which there exist accessible and efficient hardware and software implementations.