关于右连左极函数的Gronwall不等式

给出了传统意义下的Gronwall不等式的推广形式,并且给出了在右连左极函数积分下的一种Gronwall不等式。     

A novel noise reduction technique for underwater acoustic signals based on complete ensemble empirical mode decomposition with adaptive noise,minimum mean square variance criterion and least mean square adaptive filter

Underwater acoustic signal processing is one of the research hotspots in underwater acoustics. Noise reduction of underwater acoustic signals is the key to underwater acoustic signal processing. Owing to the complexity of marine environment and the particularity of underwater acoustic channel, noise reduction of underwater acoustic signals has always been a difficult challenge in the field of underwater acoustic signal processing. In order to solve the dilemma, we proposed a novel noise reduction technique for underwater acoustic signals based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), minimum mean square variance criterion (MMSVC) and least mean square adaptive filter (LMSAF). This noise reduction technique, named CEEMDAN-MMSVC-LMSAF, has three main advantages: (i) as an improved algorithm of empirical mode decomposition (EMD) and ensemble EMD (EEMD), CEEMDAN can better suppress mode mixing, and can avoid selecting the number of decomposition in variational mode decomposition (VMD); (ii) MMSVC can identify noisy intrinsic mode function (IMF), and can avoid selecting thresholds of different permutation entropies; (iii) for noise reduction of noisy IMFs, LMSAF overcomes the selection of decomposition number and basis function for wavelet noise reduction. Firstly, CEEMDAN decomposes the original signal into IMFs, which can be divided into noisy IMFs and real IMFs. Then, MMSVC and LMSAF are used to detect identify noisy IMFs and remove noise components from noisy IMFs. Finally, both denoised noisy IMFs and real IMFs are reconstructed and the final denoised signal is obtained. Compared with other noise reduction techniques, the validity of CEEMDAN-MMSVC-LMSAF can be proved by the analysis of simulation signals and real underwater acoustic signals, which has the better noise reduction effect and has practical application value. CEEMDAN-MMSVC-LMSAF also provides a reliable basis for the detection, feature extraction, classification and recognition of underwater acoustic signals.