# A Simple and Robust Equalization Algorithm for Variable Modulation Systems
## 4 Equalization Algorithm Simulation and Hardware Implementation  
### Performance Simulation of Different Equalization Algorithms  
To verify the performance of the proposed algorithm in VM systems, four sources, QPSK, 8 PSK, 16 APSK, and 32 APSK, were constructed according to the parameters shown in Table 1. The DDLMS algorithm was not applicable to the VM system. The CMA, MCMA, and VM-DDLMS adaptive equalization algorithms were simulated, respectively, under different signal-to-noise ratio (SNR) conditions, where the channel was a multipath channel plus Gaussian white noise channel. \(h(n)=[0.2,\ 0.5,\ 1,-0.1]\) was the channel impulse response.  
Under the 32 APSK modulation with an SNR of 25 dB, the constellation diagram received after different equalization algorithms is shown in Figure 5. The equalizer input signal constellation had an obvious ISI after passing through the satellite communication channel. It can be seen from Figure 5b-d that the signal constellation after equalization was more concentrated, and the performance was much better than the signal without the equalization algorithm. Figure 6 shows the residual error diagrams of different equalization algorithms. It can be seen from Figure 6a that the steady-state error of the CMA algorithm stabilized after about 9000 symbols, and the steady-state error value was concentrated below 0.35. From Figure 6b, it can be seen that the steady-state error of the MCMA algorithm stabilized after about 8500 symbols. The error value was concentrated below 0.4. From Figure 6c, it can be seen that the steady-state error of the VM-DDLMS algorithm stabilized after about 6000 symbols, and the steady-state error value was concentrated below 0.15. The part where the steady-state error exceeded 0.15 is mainly because the known information was used as the training sequence to update the tap coefficients.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Modulation Method** & **Code Length** & **Frame Length** \\ \hline QPSK & 8160 & 4242 \\
8 PSK & 8160 & 2846 \\
16 APSK & 8160 & 2166 \\
32 APSK & 8160 & 1758 \\ \hline \hline \end{tabular}
\end{table}
Table 1: VM system simulation parameters.  
Figure 4: Structure diagram of the VM-DDLMS algorithm. \(e^{*}(k)\) is the conjugate of \(e(k)\).  
As shown in Figure 7, the VM-DDLMS algorithm and CMA had a better equalization performance for the QPSK and 8 PSK modulation. When the BER was \(10^{-5}\), the performance loss of all algorithms was less than 1 dB. In the 16 APSK and 32 APSK modulation, the VM-DDLMS algorithm outperformed both CMA and MCMA. When the BER was \(10^{-5}\), only the performance loss of VM-DDLMS and CMA was less than 1 dB. Therefore, the VM-DDLMS algorithm simulation performance was better compared to CMA and MCMA. The proposed algorithm retained the advantages of the DDLMS algorithm, which reduced the computational complexity without a dynamic adjustment of the correction parameters when the mode switching was performed in the VM system. The proposed algorithm does not require the real-time switching of the symbol decision mode according to the modulation. In addition, the QPSK symbol decision is a low-order modulation so that the proposed algorithm can converge when the decision error of the signal is large. Therefore, the VM-DDLMS algorithm is simple to implement and has a high practicability.  
Figure 5: (**a**) Equalizer input signal constellation; (**b**) CMA equalizer output signal constellation; (**c**) MCMA equalizer output signal constellation; (**d**) VM-DDLMS equalizer output signal constellation.  
Figure 6: (a) CMA algorithm error convergence curve; (b) MCMA algorithm error convergence curve; (c) VM-DDLMS algorithm error convergence curve.