Signal reconstruction from multiple correlations: Frequency- and time-domain approaches

One-dimensional (1-D) ultrashort laser signals cannot be recorded directly, although it is possible to detect their multiple correlations. The reconstruction of 1-D deterministic sampled signals from their multiple correlations is studied. A computationally efficient, fast-Fourier-transform-based, frequency-domain algorithm is described for simultaneously reconstructing the amplitude and the phase of a finite-duration signal. It is shown that, by modeling the Fourier transform of a discrete sequence as a pole-zero rational function, unique (modulo time shifts) signal recovery is possible from any multiple correlation of order greater than 2. The resulting time-domain algorithm uses all the nonredundant 1-D slices of a multiple-correlation sequence and applies to one- or two-sided, finite- or infinite-duration signals. The signal parameters are obtained in closed form by using a set of linear equations. Noise effects are studied theoretically and experimentally through simulated data. Both frequency- and time-domain algorithms are applicable to modeling and interpolation of raster-scanned images.