Periodicity-Aware Signal Denoising Using Capon-Optimized Ramanujan Filter Banks and Pruned Ramanujan Dictionaries

We propose a 'periodicity-aware' hybrid analysis-synthesis framework for denoising discrete-time periodic signals. Our method uses Ramanujan filter banks (RFB) for analysis and dictionaries for synthesis. The synthesis dictionary retains appropriate subspaces for signal reconstruction, by pruning the Ramanujan dictionary based on the outputs of the RFB. Unlike the other existing denoising methods, a unique advantage of the proposed method is that the denoised output is guaranteed to be composed of integer-periodic components with periods smaller than a pre-selected value. Our method works well even when the signal length is small, and has a high SNR gain over a wide range of input signal SNRs. Furthermore, we propose to adapt each filter in the analysis bank to the incoming data, by optimizing the filter coefficients through a multi-band Capon formulation. This helps in suppressing the spurious energy peaks generated from higher period filters in the analysis bank, further improving the denoising performance. Implementing multiband Capon filters requires inverses of several autocorrelation matrices. To reduce computations, a way to recursively compute these inverses based on Levinson's recursion is discussed. Next, we prove several multirate properties of Ramanujan subspace signals. An important property among these is that after decimation, a $q$-th Ramanujan subspace signal still remains in the $q$-th Ramanujan space, if and only if the decimation rate $M$ is coprime to $q$. This is helpful to further reduce computations required in the analysis part of the denoising framework by downsampling the filter outputs. Extensive Monte-Carlo simulations comparing different variants of the proposed method and several existing denoising methods are also provided.