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IF Estimation of Multicomponent Nonstationary Signals Based on AFSST

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Abstract

The instantaneous frequency (IF) provides important characteristic information for analyzing signals. To improve the accuracy of IF estimation, time–frequency (TF) analysis with high resolution and high aggregation is necessary. A new IF estimation method is proposed in this paper, which is referred to as ridge path regrouping based on an adaptive short-time Fourier high-order synchrosqueezing transform (AFSST). In the TF analysis of the non-stationary signal, we optimize the window function of the short-time Fourier transform to be adaptive to signal characteristics, and then perform the high-order synchrosqueezing transform. Based on the high-resolution time–frequency distribution (TFD), the frequency ridge of the signal is extracted, and the ridge of the modal aliasing at the intersection is discarded and reorganized according to the fundamentals. Experiments on multicomponent nonstationary signals were performed, and the results show that AFSST has satisfactory TF aggregation and TF resolution. IF estimation experiment based on AFSST, the results indicate that the proposed method can estimate the IF accurately even under a low signal-to-noise ratio (SNR).

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Acknowledgements

This research was funded by the National Natural Science Foundation of China under Grant 61803294 and Natural Science Basic Research Program of Shaanxi (Program No. 2020JQ-684).

Funding

National Natural Science Foundation of China, 61803294, Li Jiang, Natural Science Foundation of Shaanxi Province, 2020JQ-684, Li Jiang.

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  1. College of Information and Control Engineering, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Li Jiang, Wenqing Shang, Shizhao Xiang, Yudong Jiao, Yanni Wang & Junni Zhou

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  1. Li Jiang
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Appendix A

Appendix A

First, we donate

$$ \begin{gathered} \tilde{\omega }(t,f) = X_{N} (t,f)R_{N} (t)^{T} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \left[ {1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x_{2,1} (t,f){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ...{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x_{N,1} (t,f)} \right]\left[ {r_{1} (t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} r_{2} (t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ...{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} r_{N} (t)} \right]^{T} \hfill \\ \end{gathered} $$
(48)
$$ y_{1} = X_{N} R_{N}^{T} $$
(49)

Then, we obtain the new equations:

$$ \left[ \begin{gathered} y_{1} \hfill \\ y_{2} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \hfill \\ y_{N - 1} \hfill \\ y_{N} \hfill \\ \end{gathered} \right] = \left[ \begin{gathered} \begin{array}{*{20}c} 1 \\ 0 \\ \vdots \\ 0 \\ \end{array} \hfill \\ 0 \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}c} {x_{2,1} } \\ 1 \\ \vdots \\ \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} \\ \end{array} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}c} {x_{3,1} } \\ {x_{3,2} } \\ \ddots \\ \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} \\ \end{array} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}c} \cdots \\ \cdots \\ \vdots \\ \begin{gathered} \cdots \hfill \\ \cdots \hfill \\ \end{gathered} \\ \end{array} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. \begin{gathered} x_{N,1} \hfill \\ x_{N,2} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \hfill \\ x_{N,N - 1} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1 \hfill \\ \end{gathered} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ \begin{gathered} r_{1} \hfill \\ r_{2} \hfill \\ {\kern 1pt} {\kern 1pt} \vdots \hfill \\ r_{N - 1} \hfill \\ r_{N} \hfill \\ \end{gathered} \right] $$
(50)

When N = 4,

$$ \begin{gathered} x_{k,1} = \frac{{Z_{{\alpha_{opt} }}^{{t^{k - 1} g}} }}{{Z_{{\alpha_{opt} }}^{g} }},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{for}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1...4, \hfill \\ x_{k,2} = \frac{{\partial_{f} x_{k,1} }}{{\partial_{f} x_{2,1} }} = \frac{{Z_{{\alpha_{opt} }}^{g} Z_{{\alpha_{opt} }}^{{t^{k} g}} - Z_{{\alpha_{opt} }}^{tg} Z_{{\alpha_{opt} }}^{{t^{k - 1} g}} }}{{Z_{{\alpha_{opt} }}^{g} Z_{{\alpha_{opt} }}^{{t^{2} g}} - (Z_{{\alpha_{opt} }}^{tg} )^{2} }} = \frac{{X_{k,2} }}{{X_{2,2} }} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{for}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 2,3,4, \hfill \\ x_{k,3} = \frac{{\partial_{f} x_{k,2} }}{{\partial_{f} x_{3,2} }} = \frac{{X_{k + 1,3} X_{2,2} - X_{k,2} X_{3,3} }}{{X_{4,3} X_{2,2} - X_{3,2} X_{3,3} }} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{for}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 3,4, \hfill \\ x_{k,4} = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{for}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 4, \hfill \\ \end{gathered} $$

where \(X_{k,j} = Z_{{\alpha_{opt} }}^{g} Z_{{\alpha_{opt} }}^{{t^{k} g}} - Z_{{\alpha_{opt} }}^{{t^{j - 1} g}} Z_{{\alpha_{opt} }}^{{t^{k - j + 1} g}} \;{\text{and}}\;\partial_{f} X_{k,j} = X_{k + 1,j} + X_{k + 1,j - 1} - X_{k + 1,2} .\)

Therefore, we can get

$$ \begin{aligned} y_{1} &= \tilde{\omega }_{f} = f - \frac{1}{i2\pi }\frac{{Z_{{\alpha_{opt} }}^{{g^{\prime}}} }}{{Z_{{\alpha_{opt} }}^{g} }} \\ y_{2} &= \frac{{\partial_{f} y_{1} }}{{\partial_{f} x_{2,1} }} = \frac{1}{i2\pi }\frac{{(Z_{{\alpha_{opt} }}^{g} )^{2} + Z_{{\alpha_{opt} }}^{g} Z_{{\alpha_{opt} }}^{{tg^{\prime}}} - Z_{{\alpha_{opt} }}^{tg} Z_{{\alpha_{opt} }}^{{g^{\prime}}} }}{{Z_{{\alpha_{opt} }}^{g} Z_{{\alpha_{opt} }}^{{t^{2} g}} - (Z_{{\alpha_{opt} }}^{tg} )^{2} }} = \frac{{W_{2} }}{{X_{2,2} }} \\ {\text{where, }}W_{2} &= \frac{{(Z_{{\alpha_{opt} }}^{g} )^{2} + Z_{{\alpha_{opt} }}^{g} Z_{{\alpha_{opt} }}^{{tg^{\prime}}} - Z_{{\alpha_{opt} }}^{tg} Z_{{\alpha_{opt} }}^{{g^{\prime}}} }}{i2\pi } \\ y_{3} &= \frac{{\partial_{f} y_{2} }}{{\partial_{f} x_{3,2} }} = \frac{{W_{3} X_{2,2} - W_{2} X_{3,3} }}{{X_{4,3} X_{2,2} - X_{3,2} X_{3,3} }} \\ {\text{where, }}W_{3} &= \partial_{f} W_{2} \\ y_{4} &= \frac{{\partial_{f} y_{3} }}{{\partial_{f} x_{4,3} }} \\ & = \frac{\begin{gathered} (X_{4,3} X_{2,2} - X_{3,2} X_{3,3} )W_{4} \hfill \\ - (W_{3} X_{2,2} - W_{2} X_{3,3} )(X_{5,4} + X_{5,3} - X_{5,2} ) \hfill \\ + (W_{3} X_{3,2} - W_{2} X_{4,3} )(X_{4,4} + X_{4,3} - X_{4,2} ) \hfill \\ \end{gathered} }{\begin{gathered} (X_{4,3} X_{2,2} - X_{3,2} X_{3,3} )(X_{6,4} + X_{6,3} - X_{6,2} ) \hfill \\ - (X_{5,3} X_{2,2} - X_{4,2} X_{3,3} )(X_{5,4} + X_{5,3} - X_{5,2} ) \hfill \\ + (X_{5,3} X_{3,2} - X_{4,2} X_{4,3} )(X_{4,4} + X_{4,3} - X_{4,2} ) \hfill \\ \end{gathered} } \\ {\text{where, }}W_{4} &= \partial_{f} W_{3} . \end{aligned} $$
(51)

With the help of the back-substitution algorithm, the modulation operators are calculated as:

$$ \begin{gathered} \tilde{q}^{[4,4]} = \frac{\begin{gathered} (X_{4,3} X_{2,2} - X_{3,2} X_{3,3} )W_{4} \hfill \\ - (W_{3} X_{2,2} - W_{2} X_{3,3} )(X_{5,4} + X_{5,3} - X_{5,2} ) \hfill \\ + (W_{3} X_{3,2} - W_{2} X_{4,3} )(X_{4,4} + X_{4,3} - X_{4,2} ) \hfill \\ \end{gathered} }{\begin{gathered} (X_{4,3} X_{2,2} - X_{3,2} X_{3,3} )(X_{6,4} + X_{6,3} - X_{6,2} ) \hfill \\ - (X_{5,3} X_{2,2} - X_{4,2} X_{3,3} )(X_{5,4} + X_{5,3} - X_{5,2} ) \hfill \\ + (X_{5,3} X_{3,2} - X_{4,2} X_{4,3} )(X_{4,4} + X_{4,3} - X_{4,2} ) \hfill \\ \end{gathered} } \hfill \\ \tilde{q}^{[3,4]} { = }\frac{{W_{3} X_{2,2} - W_{2} X_{3,3} }}{{X_{4,3} X_{2,2} - X_{3,2} X_{3,3} }} - \tilde{q}^{[4,4]} \frac{{X_{5,3} X_{2,2} - X_{4,2} X_{3,3} }}{{X_{4,3} X_{2,2} - X_{3,2} X_{3,3} }} \hfill \\ \tilde{q}^{[2,4]} = \frac{{W_{2} }}{{X_{2,2} }} - \tilde{q}^{[3,4]} \frac{{X_{3,2} }}{{X_{2,2} }} - \tilde{q}^{[4,4]} \frac{{X_{4,2} }}{{X_{2,2} }} \hfill \\ \end{gathered} $$
(52)

Finally, we could use \(G_{k}\) and \(G_{j,k}\) to express (52):

$$ \begin{gathered} \tilde{q}^{[4,4]} { = }G_{4} \left( {Z_{{\alpha_{opt} }}^{{t^{0...6} g}} {,}Z_{{\alpha_{opt} }}^{{t^{0...3} g^{\prime}}} } \right){,} \hfill \\ \tilde{q}^{[3,4]} { = }G_{3} \left( {Z_{{\alpha_{opt} }}^{{t^{0...4} g}} {,}Z_{{\alpha_{opt} }}^{{t^{0...2} g^{\prime}}} } \right) - \tilde{q}^{[4,4]} G_{3,4} \left( {Z_{{\alpha_{opt} }}^{{t^{0...5} g}} } \right), \hfill \\ \tilde{q}^{[2,4]} { = }G_{3} \left( {Z_{{\alpha_{opt} }}^{{t^{0...2} g}} {,}Z_{{\alpha_{opt} }}^{{t^{0...1} g^{\prime}}} } \right) - \tilde{q}^{[3,4]} G_{2,3} \left( {Z_{{\alpha_{opt} }}^{{t^{0...3} g}} } \right) - \tilde{q}^{[4,4]} G_{2,4} \left( {Z_{{\alpha_{opt} }}^{{t^{0...4} g}} } \right), \hfill \\ \end{gathered} $$
(53)

where \(G_{k} {(}Z_{{\alpha_{opt} }}^{{t^{0...m} g}} {,}Z_{{\alpha_{opt} }}^{{t^{0...m} g^{\prime}}} {)}\) is a function of \(Z_{{\alpha_{opt} }}^{{t^{l} g}}\) for \(l = 0,...,m\), and \(Z_{{\alpha_{opt} }}^{{t^{l} g^{\prime}}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{for}}{\kern 1pt} {\kern 1pt} {\kern 1pt} l = 0,...,n.\)

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Jiang, L., Shang, W., Xiang, S. et al. IF Estimation of Multicomponent Nonstationary Signals Based on AFSST. Circuits Syst Signal Process 42, 6116–6135 (2023). https://doi.org/10.1007/s00034-023-02388-1

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