Mathematical formalism and computational considerations have led to the development of nonstationary filtering concepts that use coherent frames to exploit waveform frequency content that is localized in time. This paper deals with nonstationary filtering concepts that utilize phase space information provided by the Weyl-Heisenberg and wavelet coherent frames. In both cases, the frame formulations possess optimum simultaneous localization in phase space that follows from the use of Gaussian based frame functions. A filtering procedure is presented that first formulates a noise-free waveform signature template in phase space, and then uses this template for nonstationary filtering. The nonstationary filtering operation can be applied either before or after classical matched filtering to obtain improved peak signal-to-root mean square (rms) noise ratio (SNR) performance for enhanced detection or waveform feature extraction. The advantage arises from exploiting the time varying spectrum of the waveform. The manner in which the SNR improvement comes about is examined so that it can be properly interpreted in the context of candidate applications. The procedure is described and the performance is demonstrated using both the Weyl- Heisenberg and wavelet coherent frames applied to examples of linear FM and Barker coded waveforms. The nonstationary filter performance used with the matched filter is compared to classical stationary matched filter performance for the case of additive white Gaussian noise.
Joseph G. Teti,
"Weyl-Heisenberg and wavelet phase space filtering using waveform signature templates", Proc. SPIE 2242, Wavelet Applications, (15 March 1994); doi: 10.1117/12.170040; https://doi.org/10.1117/12.170040