This research will develop a new mathematical framework for nonlinear signal processing and wireless communication channels.
The nonlinear signal processing problem addressed here is signal reconstruction from the magnitude of a redundant linear representation. When the linear redundant representation is associated to a group representation (such as the Weyl-Heisenberg group), the relevant Hilbert-Schmidt operators inherit this invariance property. Thus a fast (nonlinear) reconstruction algorithm seems possible. A wireless communication channel is modeled as a linear operator that describes how transmitted signals propagate to a receiver. For ultrawide band (UWB) signals, the Doppler effect no longer can be modeled as a frequency shift. Instead it is captured as a time dilation operator. A continuous superposition of time-scale shifts is used to model a UWB communication channel, and consequences to pseudo-differential operator theory are analyzed.
The investigator takes up two problems related to signal processing. In the first he considers how to represent signals in ways that allow more effective reconstructon of them from limited information. In the second he analyzes properties of wireless communication channels, aiming at improvements in transmission. The solutions to these problems have a strong impact in the strategic area of information technology. Important applications such as signal processing, X-ray crystallography, and quantum computing are affected by solutions to the first problem. High-impact applications related to the second problem include ultrawide-band through-wall imaging systems, higher-throughput 802.15 Wireless Personal Area Networks, and wireless sensor networks.
Nonlinear Signal Processing and Wireless Communications using Frames and Operators Theory is a four-year, $177K grant.