# A CNN-MPSK Demodulation Architecture with Ultra-Light Weight and Low-Complexity for Communications
## 3 The Proposed CNN-MPSK Architecture
### Comparison of Multiplications and Additions  
The coherent approach for MPSK involves phase demodulation, requiring linear-phase filters and stable outputs. Therefore, finite impulse response (FIR) filters are preferred. We are concerned with linear-phase FIR filters. The output of an this filter only depends on the present and previous inputs, which can be completely described by  
\[\begin{split} y(n)&=\sum\limits_{k=0}^{L-1}g_{k}x(n- k)\\ &=g_{0}x(n)+g_{1}x(n-1)+\ldots+g_{L-1}x(n-L+1),\end{split} \tag{16}\]  
where \(x(n)\) is the input sequence of length \(N\), \(g_{k}\) denotes filter coefficients and \(L\) represents the FIR filter length. FIRs have no feedback and have stability and freedom from phase distortion. Each coefficient requires a register to hold a delayed input. With the length \(L\) of this filter and the \(N\) input samples, the length of output \(y(n)\) is \(N+L-1\). This process involves \((N+L-1)L\) multiplications and \((N+L-1)(L-1)\) additions [34; 35; 36]. Consequently, the corresponding computational complexity of an FIR filter is described as \(\boldsymbol{O}(n^{2})\).  
According to equation (16), when the length of the received signal is \(N\) and the bandpass filter has \(L_{p}\) coefficients, the operation of the bandpass filter costs \((N+L_{p}-1)L_{p}\) multiplications and \((N+L_{p}-1)(L_{p}-1)\) additions. The carrier multiplier module requires \(N\) multiplications. Moreover, let \(L_{f}\) represent the length of the lowpass filter coefficients, the operation of this filter involves \((N+L_{f}-1)L_{f}\) multiplications and \((N+L_{f}-1)(L_{f}-1)\) additions. As a result, the coherent demodulation involves the total calculations, i.e., \((N+L_{p}-1)L_{p}+N+(N+L_{f}-1)L_{f}=N(L_{p}+L_{f}+1)+L_{p}^{2}+L_{f}^{2}-(L_{p}+ L_{f})\) multiplications, and \((N+L_{p}-1)(L_{p}-1)+(N+L_{f}-1)(L_{f}-1)=N(L_{p}-1)+N(L_{f}-1)+(L_{p}-1)^{2}+(L _{f}-1)^{2}\) additions. Demodulating the same input signal as the coherent demodulation, we utilize the proposed CNN-BPSK architecture. In the architecture, the convolution  
Figure 7: BER comparisons of CNN-MPSK and coherent demodulation for BPSK, 4 PSK, 8 PSK and 16 PSK demodulations.  
operation requires \(3N\) multiplications and \(N\) additions, and the full connection needs \(N/s\) multiplications and \(N/s\) additions, and \(s\) is the length of strides. The architecture involves \(3N+N/s\) multiplications and \(N+N/s\) additions in total.  
In comparison with the conventional demodulation, the number of multiplications and additions of the architecture are greatly reduced, as shown in Table 2. The calculation complexity of the proposed demodulation is \(\mathbf{O}(n)\), while that of the conventional demodulation is \(\mathbf{O}(n^{2})\).  
Table 3 presents a comparison between the proposed deep learning technique and the existing algorithms [7; 11]. We approximate the number of parameters, and the number of operations by orders of magnitude. The last three columns of Table 3 represent the demodulation accuracy for different \(E_{s}/n0\). The proposed technique shows the similar demodulation performance as compared to the other schemes, but benefits from much reduced implementation complexity, i.e., much less operations and parameters to be trained.