Experiments with extrapolation of band-limited signal

Tsai, Ming‐Jer; O’Connor, Debra J.

doi:10.1109/icassp.1984.1172639

Cited by 2 publications

(3 citation statements)

References 8 publications

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“…Another approach [4,6] requires solution of a linear system of equations with 2M + 1 unknowns, followed by low-pass filtering. Other approaches have been applied to the continuous-continuous problem, including iterative algorithms [ 5,13 ] ; see also [ 9,11,12,17,21,23].…”

Section: Summary Of the Problemmentioning

confidence: 99%

A fast algorithm for extrapolation of discrete-time periodic band-limited signals

Soltanian‐Zadeh

Yagle

1993

Signal Processing

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We present a fast algorithm for exact extrapolation of a discrete-time and periodic band-limited signal from its values in an interval having the same length as the band-width of the signal. Here band-width is the number of non-zero values in one period of the discrete Fourier transform of the signal. Applications to problems in which the number of given values is unequal to the band-width are also given. The procedure is a simple autoregression on the time-domain values of the signal, and is much simpler than previous algorithms for discrete~tiscrete extrapolation, which required computation of a large pseudo-inverse. The procedure is highly parallelizable, and the computational savings are especially significant for multidimensional signal extrapolation. Numerical examples for 1-D and 2-D extrapolation demonstrate (1) that the procedure works perfectly in the absence of noise, and (2) that it works well in the presence of band-limited noise. In the presence of wide-band noise the procedure breaks down, due to ill-posedness of the problem; some regularization techniques are proposed. Zusammenfassung. Wir stellen einen schnellen Algorithmus zur exakten Extrapolation eines zeitdiskreten und periodischen, bandbegrenzten Signals vor, ausgehend von seinen Werten innerhalb eines Intervalls, das die gleiche Lange hat wie die Bandbreite des Signals. Bandbreite bedeutet hier die Zahl der von Null verschiedenen Werte einer Periode der diskreten Fourier-Transformation des Signals. Es werden auch Anwendungen auf Probleme angegeben, bei denen die gegebene Werteanzahl ungleich der Bandbreite ist. Das Verfahren ist eine einfache Autoregression der Zeitbereichswerte des Signals, und ist viel einfacher als vorher angegebene Verfahren ffir die diskrete~tiskrete Extrapolation, die die Berechnung einer grol?~en Pseudoinversen erforderte. Das Verfahren ist hochgradig parallelisierbar, und die rechnerischen Einsparungen sind besonders signifikant bei der multidimensionalen Signalextrapolation. Numerische Beispiele bei l-D und 2-D Extrapolation zeigen (1) dab das Veffahren peffekt arbeitet bei der Abwesenheit von Rauschen, und (2) dag es gut arbeitet bei der St6rung durch bandbegrenztes Rauschen. Bei der Anwesenheit von breitbandigem Rauschen bricht das Verfahren wegen der Schlechtgestelltheit des Problems zusammen; in diesem Fall werden einige Regularisierungsmethoden vorgeschlagen. R6sum6. Nous pr6sentons un algorithme rapide d'extrapolation exact d'un signal p6riodique h temps discret et ~t bande limitde a partir de ses valeurs dans un intervalle ayant la m~me largeur que sa largeur de bande. Par largeur de bande on entend ici le nombre de valeurs non nulles dans une p6riode de la transform6e de Fourier discrbte du signal. Des applications au probl~me dans lequel le nombre de valeurs donn6es est diffdrent de la largeur de bande sont 6galement donn6es. La proc6dure consiste en une simple autor6gression sur les valeurs temporelles du signal, et est beaucoup plus simple que les algorithmes pr6-existants pour I'interpolation discret~li...

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Section: Summary Of the Problemmentioning

confidence: 99%

A fast algorithm for extrapolation of discrete-time periodic band-limited signals

Soltanian‐Zadeh

Yagle

1993

Signal Processing

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“…When the Miller-Tikhonov approach is used, equation The objective of bandlimited extrapolation is the (3) becomes: selection -according to some criterion of optimality -of a unique solution from the many (4) possible solutions to the first order Fredholm integral equation:…”

Section: Introductionmentioning

confidence: 99%

“…Much attention has recently been given to stabilising (3) either by application of Miller-Tikhonov regularising constraints or by the use of the truncated singular value decomposition [3,4].…”

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confidence: 99%

The effects of bandwidth MIS-estimation in bandlimited signal extrapolation

Stewart

Durrani²,

Abbiss³

ICASSP '85. IEEE International Conference on Acoustics, Speech, and Signal Processing

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Farnborough Hampshire UK = = regularisation parameter I = identity matrix = regularised weighting vector The regularised solution is then: ABSTRACT M0 u0 g (3)Expressions are presented which quantify the A major problem encountered in relation to this effects of mis-estimation of the support of the reconstruction method is the ill-conditioning of object function on the regularised minimum-norm equation (3) which is caused by the compact nature solution to the discrete bandlimited extrapolation of the operator of equation (1) or, more specifiproblem.Computational results based upon these cally, by the properties of matrix M0 [2] whose expressions are used to demonstrate the variation condition number (ratio of largest to smallest in the sensitivity of the regularised minimum-norm eigenvalues) becomes very large as the order of solution as a function of the regularisation paraincreases.meter and space-bandwidth product. Much attention has recently been given to stabilising (3) either by application of Miller-Tikhonov regularising constraints or by the use of the truncated singular value decomposition [3,4]. INTRODUCTION When the Miller-Tikhonov approach is used, equation The objective of bandlimited extrapolation is the (3) becomes: selection -according to some criterion of optimality -of a unique solution from the many (4) possible solutions to the first order Fredholm integral equation: g(t) = J f(uj)e327tdw; t e[t1,t2 ... t] (1) where g(t) is a signal sampled at t = t1,t2 tN (with a sampling interval L\T); t1c[-T,Tj whose Fourier transform is compactly supported on [-,D], where """ denotes Fourier transform. Most recent work aimed at inverting this underdetermined system of equations has sought to __ restore a unique mapping from the data space IR1 to fR°(t) eTuR0 the solution space C of smooth continuous functions I -(5) on [-,'] by minimising an energy cost function to ; HI > the constraints of (1) [i]. The effects which noise on the data vector g have Given the data vector = [g(t1), g(t2) ... g(t)] on R0 have been extensively studied [5]. Little nd defining the matrix M0 and vect.r ea H (i,j) effort has been directed, however, towards invest=s(21T%(tj_tj))/1T(ti-tj) and eT = [ii ... igating the sensitivity of the regularised solution e tN] respectively, this minimum energy solution R to mis-estimation of the support [-, Q] of

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Experiments with extrapolation of band-limited signal

Cited by 2 publications

References 8 publications

A fast algorithm for extrapolation of discrete-time periodic band-limited signals

A fast algorithm for extrapolation of discrete-time periodic band-limited signals

The effects of bandwidth MIS-estimation in bandlimited signal extrapolation

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