# A CNN-MPSK Demodulation Architecture with Ultra-Light Weight and Low-Complexity for Communications
## 2 Conventional Modulation and Demodulation of MPSK  
MPSK is one of the most widely used techniques due to its relative simplicity in modulation and demodulation. The modulation of such signals can be represented by  
\[\begin{split}& c_{i}(t)=\sqrt{\frac{2}{T_{c}}}\sin(w_{c}t+\theta), \\ &\theta=2\pi i/M,i=\{0,1,2,\ldots,M-1\},\end{split} \tag{1}\]  
where \(T_{c}\) is the period of modulated signal \(c_{i}(t)\), \(T_{c}=\frac{2\pi}{w_{c}}\), \(w_{c}\) and \(\theta\) denote frequency and phase of the carrier, respectively.  
The MPSK demodulation process typically consists of two BPSK demodulation. Figure 1 illustrates the simple case of coherent demodulation for BPSK [28; 29]. First, the received BPSK signal \(r(t)\) is filtered to eliminate out-of-band noise using a bandpass filter. Afterwards, the filtered output \(z(t)\) is multiplied by a \(\sin\) wave \(2\sin(w_{c}t)\), resulting in an output \(x(t)\) that is twice the frequency of the input signal. The high frequency components in \(x(t)\) are removed by the lowpass filter. The signal is then passed to the decision circuit. Based on the synchronized clock in the timing synchronizer module, we obtain the final result \(o(t)\) which recovers the binary data stream. In particular, the key component in BPSK demodulation is the carrier generator. It needs to yield a local carrier with the identical frequency and phase as the input signal \(r(t)\). However, the local carrier may not be properly generated, leading to a phase difference between the generated carrier and the received carrier, resulting in negative consequences for demodulation [30; 31; 32].  
Figure 2 depicts the two conditional probability density functions (PDF) \(f_{0}(x)\) and \(f_{1}(x)\) for the transmitted bits 0 and 1 with mean values of \(\sqrt{2/T_{c}}\) and \(-\sqrt{2/T_{c}}\), respectively.  
In the event that the bit 1 is transmitted and the receiver determines it as 0, the conditional probability of such an error is defined by  
\[\begin{split} P(0/1)&=\int_{0}^{+\infty}f_{1}(x)dx \\ &=\int_{\frac{\sqrt{2/T_{c}}}{\sqrt{2}n}}^{\infty}\frac{1}{\sqrt{ \pi}}\exp[-u^{2}]du\end{split} \tag{2}\]  
where  
\[u=\frac{(x+\sqrt{2/T_{c}})}{\sqrt{2}\sigma_{n}}. \tag{3}\]  
The lower limit of integration in Equation (2) is simplified to  
\[\frac{\sqrt{2/T_{c}}}{\sqrt{2}\sigma_{n}}=\frac{\sqrt{1/T_{c}}}{\sqrt{N_{0}B}} =\frac{\sqrt{1/T_{c}}}{\sqrt{N_{0}/T_{c}}}=\frac{1}{\sqrt{N_{0}}}=\sqrt{\frac {E_{b}}{N_{0}}}=\sqrt{snr}, \tag{4}\]  
where \(E_{b}\) is already normalized to one when BPSK signals are transmitted. Consequently, in terms of the complementary error function (erfc), assuming bits are equiprobable, the BER for BPSK coherent detection is given by  
\[P_{e}=\frac{1}{2}P(0/1)+\frac{1}{2}P(1/0)=\frac{1}{2}\text{erfc}(\sqrt{snr}). \tag{5}\]  
Similarly, the probability of symbol error of MPSK (\(M\geq 4\)) is overbounded by [33]  
Figure 1: Block diagram of a coherent binary-PSK (BPSK) receiver.  
Figure 2: Conditional bit error probability density functions for BPSK demodulated signals.  
\[\begin{split} P_{e}&=\frac{2}{\sqrt{\pi N_{0}}}\int_{ \sqrt{E_{s}}\sin(\frac{\pi}{M})}^{\infty}\exp[\frac{-x^{2}}{N_{0}}]dx\\ &=\frac{2}{\sqrt{\pi}}\int_{\sqrt{\frac{E_{0}}{N_{0}}\sin(\frac{ \pi}{M})}}^{\infty}\exp[-u^{2}]du\\ &=erfc[\sqrt{\frac{E_{s}}{N_{0}}}\sin(\frac{\pi}{M})],\end{split} \tag{6}\]  
where \(\sqrt{E_{s}}\) is the average energy of the transmitted symbols.