Localized Kernel Methods for Signal Processing

This dissertation presents two signal processing methods using specially designed localized kernels for parameter recovery under noisy condition. The first method addresses the estimation of frequencies and amplitudes in multidimensional exponential models. It utilizes localized trigonometric polynomial kernels to detect the multivariate frequencies, followed by a more detailed parameter estimation. We compare our method with MUSIC and ESPRIT, which are classical subspace-based algorithms widely used for estimating the parameters of exponential signals. In the univariate case, the method outperforms MUSIC and ESPRIT under low signal-to-noise ratios. For the multivariate case, we develop a coordinate-wise projection and registration approach that achieves high recovery accuracy using significantly fewer samples than other methods. The second method focuses on separating linear chirp components from time-localized signal segments. A variant of the Signal Separation Operator (SSO) is constructed using a localized kernel. Instantaneous frequency estimates are obtained via FFT-based filtering, then clustered and fitted with piecewise linear regression. The method operates without prior knowledge of the number of components and is shown to recover intersecting and discontinuous chirps at SNR levels as low as -30 dB. Both methods share an idea based on localized kernels and efficient FFT-based implementation, and neither requires subspace decomposition or sparsity regularization. Experimental results confirm the robustness and tractability of the proposed approaches across a range of simulated data conditions. Potential extensions include application to nonlinear chirps, adaptive kernel design, and signal classification using extracted features.

Localized kernel method for separation of linear chirps

The task of separating a superposition of signals into its individual components is a common challenge encountered in various signal processing applications, especially in domains such as audio and radar signals. A previous paper by Chui and Mhaskar proposes a method called Signal Separation Operator (SSO) to find the instantaneous frequencies and amplitudes of such superpositions where both of these change continuously and slowly over time. In this paper, we amplify and modify this method in order to separate chirp signals in the presence of crossovers, a very low SNR, and discontinuities. We give a theoretical analysis of the behavior of SSO in the presence of noise to examine the relationship between the minimal separation, minimal amplitude, SNR, and sampling frequency. Our method is illustrated with a few examples, and numerical results are reported on a simulated dataset comprising 7 simulated signals.