Fast Signal Completion Algorithm with Cyclic Convolutional Smoothing

Recently, signal completion methods using delay-embedding transforms (DT) have been actively studied. Since the DT is an operation to transform a signal into a Hankel matrix, the high computational cost associated with the increase in data size is an issue. In this study, we consider modeling smooth signals based on inverse delay-embedding instead of delay-embedding. We propose a new algorithm that incorporates the properties of the delay-embedding-based methods while reducing the computational cost. The proposed algorithm takes advantage of the inverse delay-embedding being a cyclic convolution, and the computational complexity can be reduced to $\mathcal{O}(NlogN)$ by transforming the optimization problem to Fourier space. Numerical experiments with typical signals and audio data show the effectiveness of the proposed algorithm in signal declipping and completion problems.