Gaussian noise removal with exponential functions and spectral norm of weighted Hankel matrices

Exponential functions are powerful tools to model signals in various scenarios, such as magnetic resonance spectroscopy/imaging, radar, and concatenative text-to-speech synthesis. Exponential signals, however, are usually corrupted by Gaussian noise in practice, raising difficulties in sequential analysis and quantification of the signals. In this work, we propose a denoising method based on low-rank Hankel matrices for exponential signals corrupted by Gaussian noise. An accurate estimate of the spectral norm of weighted Hankel matrices is provided as theoretical guidance to set the regularization parameter. The bound can be efficiently calculated since it only depends on the standard deviation of the noise and a constant. Aided by the bound, one can easily obtain a good regularization parameter to produce promising denoised results. Our experiments on simulated and magnetic resonance spectroscopy data demonstrate a superior denoising performance of our proposed approach in comparison with the typical Cadzow and the state-of-the-art QR decomposition methods, especially in the low signal-to-noise ratio regime.