Consistent Sampling and Efficient Signal Reconstruction

We propose efficient signal reconstruction methods such that the reconstructed signal has the same measurements as the underlying original signal if the former was observed by the same system as that for the latter. The reconstructed signal is a linear combination of reconstruction functions, and the main task is to compute the coefficients from the measurements. This generally involves an inversion of the cross-correlation matrix between the sampling functions and the reconstruction functions. We show that the coefficients are computed efficiently using the discrete Fourier transform (DFT) when sampling is shift-invariant and when the reconstruction functions are shift-invariant functions, Fourier basis functions, or Fourier cosine basis functions. Because of the DFT, the proposed methods do not require matrix inversion computation; rather these methods make it possible to obtain the coefficients with an O(N log N) computation, where N stands for the number of samples. We show the effectiveness of the proposed method by experiments using an ECG signal.