# M-ary Phase Position Shift Keying Demodulation Using Stacked Denoising Sparse Autoencoders
## 1 Introduction  
As wireless applications continue to grow, spectrally efficient modulation schemes will continue to be highly valuable [1; 2; 3]. The classical binary modulation schemes such as amplitude shift keying (ASK), frequency shift keying (2 PSK) and phase-shift keying (2 PSK, or BPSK) are spectrally inefficient. In the past two decades, a number of bandwidth efficient modulation techniques have been proposed. [PERSON] et al. proposed an extended binary phase shift keying (EBPSK) scheme [4] which is much more flexible and has a more compact spectral structure than conventional BPSK modulation. It is shown that, by tuning the modulation parameters, EBPSK could achieve the same bit error rate (BER) performance as the conventional BPSK, but has higher spectrum utilization efficiency than the latter for wireless applications. EBPSK is also extended to M-ary phase position shift keying (MPPSK) [5] to achieve higher transmission rates. An improved MPPSK modulation with a simpler demodulator and a better performance than the original MPPSK scheme is presented [6]. Existing demodulators of MPPSK systems use back propagation (BP) [7], support vector machine (SVM) [8], joint amplitude and waveforms detection (JAWD) [9], and the phase locked loop (PLL) techniques [5]. However, the symbol error rate (SER) performance of MPPSK signaling could be further improved by using more advanced detection techniques. The demodulator for MPPSK signaling described in [5] uses a special impact filter (SIF) which could have a strong impact on the signal amplitude at the phase transition instants to significantly improve the output signal-to-noise ratio (SNR). Deep learning has been successfully applied to a variety of problems [10; 11; 12]. The success of deep learning algorithms generally depends on how data are represented. For communication signals such as MPPSK, since the SIF affects the signal amplitude more appreciably at the phase transition points, the impacted amplitudes of MPPSK waveformscontain abundant features. The features can be learned and used as classifiers to significantly improve detection performance. To date and to our knowledge, there is little research on unsupervised learning for detection in communication systems. Deep learning models include supervised models and unsupervised models. Compared with labeled data in supervised models, unlabeled data in the unsupervised models are easier to obtain when large-scale data are needed. Unsupervised feature learning consists of several basic modules, such as restricted Boltzmann machine (RBM) [13], autoencoder (AE) [10] and sparse coding [11]. It has been employed in the fields of computer visual perception, speech recognition, natural language processing, etc. Feature learning of unsupervised models has a great potential to improve the detection performance in communication systems [13; 14; 15]. For MPPSK signals, the AE algorithm could be employed to find the correlation characteristics among the input signals. In this paper, we developed a detector for MPPSK systems based on the AE learning algorithm, improving the detection performance by learning the useful representations or features. Each possible transmission scenario is considered as a sub-model of deep learning, and a large number of cases are included in its database. We also commonly use training data from the database that contain samples of the modulation signals. The similarity between the training and test data is built by a feature representation to exploit. AE learning networks is categorized into sparse autoencoders (SAEs) and denoising autoencoders (DAEs). The SAE and DAE networks [16; 17; 18] are combined to form the stacked denoising sparse autoencoders (DSAEs) network, further improving the detection performance of MPPSK signals. The remainder of this paper is organized as follows. Section 2 describes the improved MPPSK modulation and the novel demodulator based on the stacked DSAE network. Section 3 describes the principle of the proposed stacked DSAE network.