Filter Design for Constrained Signal Reconstruction in Linear Canonical Transform Domain

The linear canonical transform (LCT), which includes many classical transforms, has increasingly emerged as a powerful tool for optics and signal processing. Signal reconstruction associated with the LCT has blossomed in recent years. However, many existing reconstruction algorithms for the LCT can only handle noise-free measurements, and when noise is present, they will become ill posed. In this paper, we address the problem of reconstructing an analog signal from noise-corrupted measurements in the LCT domain. A general methodology is proposed to solve this problem in which the analog signal is recovered from ideal samples of its filtered version in a unified way. The proposed methodology allows for arbitrary measurement and reconstruction schemes in the LCT domain. We formulate signal reconstruction in an LCT-based function space, which is the span of integer translates and chirp-modulation of a generating function, with coefficients derived from digitally filtering noise corrupted measurements in the LCT domain. Several alternative methods for designing digital filters in the LCT domain are also suggested using different criteria. The validity of the theoretical derivations is demonstrated via numerical simulation.