## Interpolation and the Discrete Papoulis-Gerchberg Algorithm (1994)

Venue: | IEEE Trans. Signal Processing |

Citations: | 32 - 20 self |

### BibTeX

@ARTICLE{Ferreira94interpolationand,

author = {Paulo Jorge S. G. Ferreira},

title = {Interpolation and the Discrete Papoulis-Gerchberg Algorithm},

journal = {IEEE Trans. Signal Processing},

year = {1994},

volume = {42},

pages = {2596--2606}

}

### OpenURL

### Abstract

In this paper we analyze the performance of an iterative algorithm, similar to the discrete Paponiis-Gerchberg algorithm, and which can be used to recover missing samples in finite-length records of band-limited data. No assumptions are made regarding the distribution of the missing samples, in contrast with the often studied extrapolation problem, in which the known samples are grouped together. Indeed, it is possible to regard the observed signal as a sampled version of the original one, and to interpret the reconstruction result studied herein as a sampling result. We show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data. We give upper and lower bounds for the error as a function of the number of iterations, together with the signals for which the bounds are attained. Also, we analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix. From this analysis we infer the optimum value of the relaxation constant. We also point out, among all sampling sets with the same density, those for which the convergence rate of the recovery algorithm is maximum or minimum. For low-pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer. The worst convergence rates generally occur when the missing samples are contiguous.

### Citations

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Citation Context ... III. RELATED WORK A large percentage of the many published papers concerned with iterative constrained signal restoration is devoted to the band-limited extrapolation problem. Its study goes back to =-=[8]-=-, where an extrapolation method is given, based on the double orthogonality property of the prolate spheroidal wave functions. The introduction of the error-reduction algorithm, also known as the Papo... |

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Citation Context ...e projections onto convex sets may also outperform the Papoulis-Gerchberg iteration, since they allow any number of a priori constraints to be included in the restoration procedure. See, for example, =-=[23]-=--[26]. An iterative algorithm for restoring continuously sampled band-limited L2 functions was proposed in [27]. A continuously sampled signal is obtained by setting to zero a signal except for a neig... |

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Citation Context ...mples is known. The reconstruction algorithm studied herein is an itemtire method which reduces to the discrete finite-dimensional version of the well-known Papoulis-Gerchberg extrapolation algorithm =-=[2]-=-, [31, if the unknown samples are contiguous, and if a relaxation constant/ is unitary. In the latter case, the algorithm may also be described as an alternating projection algorithm of the type found... |

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Citation Context ...1, if the unknown samples are contiguous, and if a relaxation constant/ is unitary. In the latter case, the algorithm may also be described as an alternating projection algorithm of the type found in =-=[4]-=-. Its convergence will be established under no restriction on the distribution of the missing samples, generalizing results found in [5] and providing a theoretical background for the observations rep... |

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Citation Context ...d both one-dimensional and twodimensional reconstruction problems are studied. In fact, it is possible to arrive at sampling expansions for signals which are not band-limited if such a mapping exists =-=[46]-=-. An alternative, which sometimes leads to good results, is to interpolate a set of uniformly spaced points on the nonuniform grid [47]. The Papoulis-Gerchberg algorithm can also be used [48] to obtai... |

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Citation Context ..., [3], brought a new interest to the problem, and, since then, several works dealing with different aspects of the extrapolation problem have appeared. We refer the interested reader to [1], [7], [9]-=-=[19]-=- among many others. These works address topics that include implementation aspects, numerical stability, convergence acceleration, noniterative extrapolation, the sampled analog of the extrapolation p... |

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Citation Context ...E ALGORITHM A number of alternative approaches to (3) do exist. The one that we will use depends upon well-known results in fixedpoint theory. This technique is often used in signal restoration [39], =-=[53]-=- and does not assume the linearity of the operators involved. In fact, other possibly nonlinear nonexpansive constraints can be incorporated in the algorithm without disturbing convergence. Denote the... |

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Citation Context ...s spectral radius. Identity (1) can be established in different ways. One that leads to useful conclusions in the kind of problems that we will consider is the following (for an alternative proof see =-=[60]-=-). If we seek he stationary points of the continuous function (x) -- IlMxll 2, subject to I[xll = we are lead to the equation MHMx = Ax (2) where ,X appears as a Lagrange multiplier. This shows that t... |

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Citation Context ...d integral equations. A number of two-step reconstruction algorithms can be understood in terms of this framework. A noniterative algorithm based on a distinct principle is studied in [35]. Reference =-=[36]-=- deals with iterative least-squares solutions of the linear signal restoration problem, and presents an approach to the solution of this problem based on Bialy's iteration, a general procedure to obta... |

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Citation Context ...ted functions 150]. It is possible to estimate band-limited discrete signals from a finite number of samples, taking into account the length of the interval over which a good extrapolation is desired =-=[51]-=-. Although the result does not have minimum norm, it is optimum for signals concentrated on that interval. The estimation of two-dimensional signals from a finite number of samples was also studied. I... |

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Citation Context ...in this way. It is also known [37] that the Papoulis-Gerchberg algorithm can be obtained from Landweber's iteration, an iterative method for the solution of first-kind Fredholm integral equations. In =-=[38]-=- it is shown that time-limited restoration of shiftinvariant blurred 2 signals can be done iterating an operator equation which defines a contractive mapping. The results are valid for multi-dimension... |

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