TY - GEN

T1 - Accuracy of spike-train Fourier reconstruction for colliding nodes

AU - Akinshin, Andrey

AU - Batenkov, Dmitry

AU - Yomdin, Yosef

N1 - Publisher Copyright:
© 2015 IEEE.

PY - 2015/7/2

Y1 - 2015/7/2

N2 - We consider a signal reconstruction problem for signals F of the form F(x) = Σdj=1 ajδ(x-xj) from their Fourier transform F(F)(s) = ∫∞-∞ F(x)e-isxdx. We assume F(F)(s) to be known for each s ε [-Ν,Ν] with an absolute error not exceeding ε > 0. We give an absolute lower bound (which is valid with any reconstruction method) for the 'worst case' reconstruction error of F from F(F) for situations where the xj nodes are known to form an I elements cluster contained in an interval of length h << 1. Using 'decimation' algorithm of [6], [7] we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for h of order 1/N 1/2l-1 the worst case reconstruction error of the cluster nodes is of the same order 1/N 1/2l-1, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order 1/N 1/2l.

AB - We consider a signal reconstruction problem for signals F of the form F(x) = Σdj=1 ajδ(x-xj) from their Fourier transform F(F)(s) = ∫∞-∞ F(x)e-isxdx. We assume F(F)(s) to be known for each s ε [-Ν,Ν] with an absolute error not exceeding ε > 0. We give an absolute lower bound (which is valid with any reconstruction method) for the 'worst case' reconstruction error of F from F(F) for situations where the xj nodes are known to form an I elements cluster contained in an interval of length h << 1. Using 'decimation' algorithm of [6], [7] we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for h of order 1/N 1/2l-1 the worst case reconstruction error of the cluster nodes is of the same order 1/N 1/2l-1, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order 1/N 1/2l.

UR - http://www.scopus.com/inward/record.url?scp=84941086938&partnerID=8YFLogxK

U2 - 10.1109/SAMPTA.2015.7148965

DO - 10.1109/SAMPTA.2015.7148965

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AN - SCOPUS:84941086938

T3 - 2015 International Conference on Sampling Theory and Applications, SampTA 2015

SP - 617

EP - 621

BT - 2015 International Conference on Sampling Theory and Applications, SampTA 2015

PB - Institute of Electrical and Electronics Engineers Inc.

Y2 - 25 May 2015 through 29 May 2015

ER -