Harmonic Signal Detection and Recovery Based on VMD-GRU

This paper presents a comprehensive approach to extracting weak harmonic signals from a chaotic background. It begins by applying a locally weighted regression model to preprocess observed signals and minimize white noise interference. Subsequently, a hybrid model integrating variational mode decomposition and gated recurrent unit is utilized for signal prediction. The prediction error is then analyzed using the periodogram method to estimate signal frequency. Finally, the least squares method is employed to estimate the remaining parameters of the target signal. Simulation experiments demonstrate successful recovery of a single harmonic signal with a signal-to-noise ratio (SNR) of 51.45 dB. Where the mean square error (MSE) of the estimated signal is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{4. 2 6} \times \mathbf{1 0 - 6}$</tex>, with a mean absolute error (MAE) of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{0. 0 0 0 2}$</tex>. When handling multiple superimposed harmonic signals under a SNR of -54.05 dB, the MSE is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$6.8 \times 10-6$</tex>, with a MAE of 0.0022. The proposed method has been effectively validated using the real sunspot data. In the single-signal recovery scenario, even under the condition of a SNR of -29.89 dB, the MSE is 0.0206, with a MAE of 0.1391. In the multi-signal recovery scenario, when the SNR is -22.45 dB, the MSE is 0.3676, with an MAE of 0.5123. These outcomes affirm the efficacy of the proposed method within chaotic backgrounds.