Exponential-Reproducing-Kernel-Based Sparse Sampling Method for Finite Rate of Innovation Signal with Arbitrary Pulse Echo Position

Abstract

An exponential reproducing kernel (ERK) has been applied to data sparse sampling for finite rate of innovation (FRI) signals with the characteristic of flexible parameter setting in contrast to other sampling kernels. However, the signal reconstruction process may fail if a pulse echo is present in certain positions. To solve this problem, a novel ERK sparse sampling method was developed for arbitrary pulse echo positions. A constraint relationship between the pulse echo position and reproduced-exponent area was deduced, revealing the cause of invalid signal reconstruction from sparse sampling data. A new sampling time interval calculation algorithm is presented in this paper. Through the proposed method, FRI signals can be reconstructed accurately with a pulse echo present at arbitrary positions without increasing the quantity of sparse sampling data. The signal reconstruction effectiveness and accuracy were verified through simulation experiments. The proposed technique can be used to improve the flexibility of the ERK sparse sampling method in actual applications.

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Acknowledgements

This project is supported by the National Natural Science Foundation of China (Grant No. 51375217). This manuscript has been edited by Mark Darvill at Wallace Academic Editing and is considered to be improved in grammar and native English usage.

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Correspondence to Shoupeng Song.

Appendix: Concept of RE-area

Appendix: Concept of RE-area

The exponential reproducing formula of the \( M \)th-order E-spline function \( \beta_{{\vec{\alpha }}} \left( t \right) \) with \( \beta_{{\vec{\alpha }}} \left( 0 \right) = 0 \), \( \vec{\alpha } = \alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{M} \), can be expressed as [10, 13, 19]

$$ \mathop \sum \limits_{{k \in {\mathbf{Z}}}} c_{m,k} \beta_{{\vec{\alpha }}} \left( {t - k} \right) = e^{{\alpha_{m} t}} , \quad m = 1,2, \ldots ,M $$
(14)

where \( c_{m,k} \) denotes the corresponding exponential reproducing weight coefficient.

Generally, the exponent \( e^{{\alpha_{m} t}} \) can be produced by summing the weighted terms of \( \beta_{{\vec{\alpha }}} \left( {t - k} \right),k \in {\mathbf{Z}} \) in an infinite time shift set. As shown in Fig. 7, in a finite time shift set \( \varvec{K}_{f} = \left\{ {k |k \in \left[ {K_{1} ,K_{2} } \right], k \in {\mathbf{Z}}} \right\} \), the exponential reproducing formula is expressed as

$$ r\left( t \right) = \mathop \sum \limits_{{k \in \varvec{K}_{f} }} c_{m,k} \beta_{{\vec{\alpha }}} \left( {t - k} \right) $$
(15)

where the time support of the reproduced function shifts to \( \left[ {K_{1} ,K_{2} + M} \right). \)

Fig. 7
figure7

Sketch map of reproducing exponent

Clearly, the exponent \( e^{{\alpha_{m} t}} \) cannot be generated in the whole time support of \( r\left( t \right). \) The time support area of the weighted terms \( c_{{m,K_{1} - 1}} \beta_{{\vec{\alpha }}} \left( {t - K_{1} + 1} \right) \) and \( c_{{m,K_{2} + 1}} \beta_{{\vec{\alpha }}} \left( {t - K_{2} - 1} \right) \) must be deleted to reproduce the exponent, or in other words, the areas \( \left( {K_{1} - 1,K_{1} + M - 1} \right) \) and \( \left( {K_{2} + 1,K_{2} + M + 1} \right) \), respectively. Therefore, when \( K_{2} - K_{1} \ge M - 2 \) and \( M \ge 2 \) are both satisfied,

$$ r\left( t \right)\left\{ {\begin{array}{*{20}c} { = e^{{\alpha_{m} t}} ,} & { t \in \varvec{E}} \\ { \ne e^{{\alpha_{m} t}} ,} & {t \in {\complement }_{{\left[ {K_{1} ,K_{2} + M} \right)}} \varvec{E}} \\ \end{array} } \right. $$
(16)

where \( {\complement }_{\text{U}} {\text{A}} \) denotes the supplementary set of set \( {\text{A}} \) in set \( {\text{U}} \) and the time interval \( \varvec{E} \) is defined as the RE-area of the \( M \) th-order E-spline function \( \beta_{{\vec{\alpha }}} \left( t \right) \) under the time shift set \( \varvec{K}_{f} \) and is expressed as

$$ \varvec{E} = \left[ {K_{1} + M - 1,K_{2} + 1} \right] $$
(17)

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Song, S., Shen, J. Exponential-Reproducing-Kernel-Based Sparse Sampling Method for Finite Rate of Innovation Signal with Arbitrary Pulse Echo Position. Circuits Syst Signal Process 38, 1179–1193 (2019). https://doi.org/10.1007/s00034-018-0903-8

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Keywords

  • Finite rate of innovation
  • Sparse sampling
  • Exponential reproducing kernel
  • Pulse echo position