Computationally efficient signal reconstruction from zero-crossing sampling by the three points DFTs

The DFT (Discrete Fourier Transform) is defined for uniformly sampled signals. The inverse DFT (IDFT) may be interpreted as polynomial representation of the discrete signal, that is uniquely represented by the roots of this polynomial and the scaling factor. It is easy to ensure, by reversible transform, that all this roots are real (not complex) and thus can be observed (measured) in the signal. After measuring the times of signal zero-crossings the DFT is computed by expanding polynomial from its roots, and finally uniformly sampled signal is obtained by IDFT. In the paper basic theory of implicit sampling is reviewed and a new algorithm for efficient signal reconstruction from its zero-crossings, based on three points DFTs, is proposed.