TY - JOUR

T1 - Some Aspects of Band-Limited Signal Extrapolation

T2 - Models, Discrete Approximations, and Noise

AU - Sanz, Jorge L.C.

AU - Huang, Thomas S.

PY - 1983/12

Y1 - 1983/12

N2 - We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise ηis bounded by a known number, max |η (x)| ≤ ɛ. This convergence is obtained by using normal families of entire functions in Cn and some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

AB - We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise ηis bounded by a known number, max |η (x)| ≤ ɛ. This convergence is obtained by using normal families of entire functions in Cn and some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

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U2 - 10.1109/TASSP.1983.1164232

DO - 10.1109/TASSP.1983.1164232

M3 - Article

AN - SCOPUS:0020946921

VL - 31

SP - 1492

EP - 1501

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 6

ER -