# M-ary Phase Position Shift Keying Demodulation Using Stacked Denoising Sparse Autoencoders
## 2 The Improved MPPSK Signaling  
The demodulator of MPPSK systems uses a narrowband SIF [5] which can produce a strong impact at the phase jumping point, resulting in significant improvement in the output SNR. The SIF is followed by a threshold decision device or a geometric characteristic-based nonlinear decision device [19; 20]. There are many variations of the demodulator [21] and they all require a specific threshold to distinguish zero and non-zero symbols. The position information is utilized to decode the non-zero symbols. The threshold is critical for these demodulators to function properly, and it will cause a performance degradation in practice. An issue with the MPPSK signal is that there is no phase jump for an incoming symbol \"0\". Consequently, the SIF response to the symbol \"0\" cannot produce a similar change on the amplitude as the non-zero symbols. Therefore, the detection between the zero and non-zero symbols must exploit the amplitude information.  
The improved MPPSK signaling in [6] allows demodulation by only using the position information of the received signal passed through the SIF. The basic idea is to eliminate the symbol \"0\". That is, the alphabet is changed from [0, 1, 2,..., M\(-\)1] to [1, 2,..., M]. In this improved MPPSK signaling scheme, over the time period [0, \(T\)], the modulation signal is expressed as  
\[g_{k}(t)=\left\{\begin{array}{ll}\sin 2\pi f_{c}t&0\leq t<(k-1)(K+r_{g})T,\\ \sin(2\pi f_{c}t+\theta),(k-1)(K+r_{g})T\leq t<[(k-1)(K+r_{g})+K]T,\\ \sin 2\pi f_{c}t&[(k-1)(K+r_{g})+K]T\leq t<NT,\end{array}\right. \tag{1}\]  
where \(T\) is the period of the carrier signal, \(1\leq k\leq M\), \(f_{c}\) is the carrier frequency, \(T_{s}=NT=N/f_{c}\) is the symbol period, \(N\) is the ratio of the symbol period to the carrier period, \(K\) is the number of carrier periods with a phase jump, \(\theta\) is the modulation angle,\(0\leq\theta\leq\pi\), and \(r_{g}\) is the control factor for the symbol protection interval. For simplicity, set \(\theta=\pi\) in this paper.  
The improved MPPSK modulation given in Equation (1) is not strictly bandwidth-limited signals, since information is embedded into the position of a full carrier signal cycle. Furthermore, each symbol period consists \(N\) cycles of the carrier signal. Many BER bounds are obtained from information theory, assuming strictly bandwidth-limited signals designed according to the Nyquist criterion for zero inter-symbol interference (ISI). It might not be directly applicable. For the improved MPPSK signaling, the relationship between SNR and \(E_{b}/N_{0}\) is  
\[SNR = \left(E_{b}/N_{0}\right)\times\left(R_{b}/W\right) \tag{2}\] \[= \left(E_{b}/N_{0}\right)\times\left[\left(2f_{c}\log_{2}M\right) \right/\left(f_{s}N\right)\right],\]  
where \(R_{b}\) is a symbol rate, \(f_{s}\) is sampling rate, and \(W=f_{s}/2\) is the effective noise bandwidth, although the signal is not strictly bandlimited. \(E_{b}\) is the received energy per bit, \(N_{0}\) is the single-sided noise power spectral density.  
The existing MPPSK demodulators include the decision threshold-based approach [5], the adaptive threshold integral discriminator [20], the coherent processing technique [21], the phase locked loop (PLL) scheme [22] and the matched filtering scheme [23; 24]. As shown in Figure 1, we develop a new demodulation scheme for the improved MPPSK modulation aiming to significantly improve detection performance. In the proposed system, there are the forward error correction (FEC) encoding before improved MPPSK modulation, and the FEC decoding after stacked DSAE demodulator. The stacked DSAE network is used for automatic feature extraction and detection. The architecture is first trained using unlabeled improved MPPSK signals. There are two detection schemes. One is the SIF directly followed by the stacked DSAE. Another is SIF, a coherent processing block and a low pass filter (LPF) followed by the stacked DSAE in the receiver.