## 1 Introduction  
In the process of a ship's movement, propeller blades cut the water flow, creating ship radiated noise with a periodic fluctuation envelope, which is audibly expressed as a rhythmic time-varying loudness characteristic [1]. Therefore, the envelope spectrum of ship radiated noise carries information such as shaft frequency and propeller number [2], which can be used for ship identification with high robustness. In practical applications, the accurate extraction of shaft frequency is a key step in calculating the number of propeller blades, which plays an important role in the field of passive sonar target recognition. At present, researchers have carried out much work concerning shaft frequency feature extraction and have achieved considerable results in mathematical modeling [3, 4, 5]. These studies reveal the physical mechanism of the modulation phenomenon of ship radiated noise, and provide a mathematical basis for the simulation of ship modulation spectrum data. In terms of shaft frequency extraction, a maximum likelihood estimation method of propeller speed is proposed in [6], and a closed-loop detection method for shaft frequency in propeller noise is proposed in [7]. In the work of [8], the application of a Duffing oscillator in extracting line spectrum of weak amplitude demodulation of ship radiated noise is studied. In the work of [9], a new method of envelope line spectrum denoising based on complementary integrated empirical mode decomposition (CEEMD) and wavelet threshold is investigated. The auditory spectrum extraction technology is improved to obtain a significant envelope spectrum at a low signal-to-noise ratio in the work of [10]. In the work of [11], a method for extracting the frequency characteristics of the ship radiated noise envelope using a dual frequency method is proposed. The propeller shaft frequency feature extraction algorithm for underwater acoustic targets based on the average energy accumulation and pulse sequence matching is proposed in [12]. These methods determine the value of shaft frequency by looking for harmonic line spectrum clusters generated by propeller blade modulation, which is better for shaft frequency extraction of high-quality modulation spectra. However, due to the shaft imbalance of propeller blade assembly, the envelope spectrum of ship radiated noise usually has different degrees of missing line spectra [5]. In addition, the envelope spectrum tends to have a low signal-to-noise ratio and is prone to non-harmonic line spectra due to the interference of environmental and other ship radiated noise [13]. These questions are not widely appreciated in existing studies and are the factors that cause inaccuracy with respect to shaft frequency extraction by traditional methods. Deep learning has achieved remarkable results in the fields of speech recognition and computer vision. In recent years, deep learning models such as deep neural networks, auto-encoder networks, and convolutional neural networks have been widely used for line spectrum enhancement and target recognition of ship radiated noise [14; 15; 16; 17; 18; 19]. Deep learning can fit the complex mapping relationship from input to output through data learning to achieve tasks of target classification and regression prediction [20; 21]. It has a powerful nonlinear modeling ability and is an effective means to deal with nonlinear problems in ship shaft frequency extraction. To improve the accuracy of shaft frequency extraction in the case of a missing line spectrum and low signal-to-noise ratio, a ship shaft frequency extraction method based on improved SSDAE is proposed in this paper. In this method, SSDAE is used to enhance the propeller shaft frequency line spectrum and its harmonic line spectrum, and the depth regression network is used to accurately estimate the ship shaft frequency. Experimental results show that the robustness and the accuracy of shaft frequency extraction are improved in the above cases.