Sparsity-Assisted Signal Denoising and Pattern Recognition in Time-Series Data

We address the problem of signal denoising and pattern recognition in processing batch-mode time-series data by combining linear time-invariant filters, orthogonal multiresolution representations, and sparsity-based methods. We propose a novel approach to designing higher-order zero-phase low-pass, high-pass, and band-pass infinite impulse response filters as matrices, using spectral transformation of the state-space representation of digital filters. We also propose a proximal gradient-based technique to factorize a special class of zero-phase high-pass and band-pass digital filters so that the factorization product preserves the zero-phase property of the filter and incorporates a sparse-derivative component of the input in the signal model. To demonstrate applications of our novel filter designs, we validate and propose new signal models to simultaneously denoise and identify patterns of interest. To denoise or detect a pattern of interest in the signal, our proposed signal models simultaneously combine linear time-invariant (LTI) filters and sparsity-based methods with orthogonal multiresolution representations, such as wavelets and short-time Fourier transform. We illustrate the capabilities of our proposed signal models using sleep-electroencephalography (EEG) data to detect K-complexes and sleep spindles. Reproducible research is available at https://github.com/prateekgv/sasdpr.