# A Simple and Robust Equalization Algorithm for Variable Modulation Systems
###### Abstract  
This paper proposes a simple and robust variable modulation-decision-directed least mean square (VM-DDLMS) algorithm for reducing the complexity of conventional equalization algorithms and improving the stability of variable modulation (VM) systems. Compared to conventional adaptive equalization algorithms, known information was used as training sequences to reduce the bandwidth consumption caused by inserting training sequences; compared with conventional blind equalization algorithms, the parameters and decisions of the equalizer were determinate, which was conducive to a stable equalization performance. The simulation and implementation results show that the proposed algorithm has a better bit error rate (BER) performance than that of the constant modulus algorithm (CMA) and modified constant modulus algorithm (MCMA) while maintaining the same level of consumption of hardware resources. Compared to the conventional decision-directed least mean square (DDLMS) algorithm, the proposed algorithm only needs to make quadrature phase shift keying (QPSK) symbol decisions, which reduces the computational complexity. In parallel 11 th-order equalization algorithms, the operating frequency of VM-DDLMS can reach up to 333.33 MHz. variable modulation (VM) systems; mode switching; inter-symbol interference (ISI); data-assisted equalization algorithm; variable modulation-decision-directed least mean square (VM-DDLMS) algorithm  
30 August  
algorithms are generally used to solve the ISI of the channel to improve the accuracy and stability of the satellite communication system. In receivers, channel equalization is necessary to reduce ISI. According to the characteristics of equalization algorithms, it can be divided into adaptive equilibrium and blind equilibrium. The conventional adaptive equalization algorithm needs to adjust the filter coefficients by inserting additional training sequences, which increases the transmission bandwidth and reduces the transmission efficiency. The composition of the frame structure was determined in this study; therefore, it cannot be corrected by traditional adaptive equalization. Although the blind equalization algorithm has a worse steady-state convergence error than the adaptive algorithm, it adjusts the filter coefficients without training sequences. Therefore, the blind equalization algorithm was mainly analyzed in this study. The constant modulus algorithm (CMA), which is simple and easy to implement, was proposed in [1; 2; 3]. However, the convergence speed is slow, the steady-state error is large, and the phase shift cannot be corrected. During the mode switching, the correction parameters need to be modified according to the modulation, which decreases the stability of VM systems. In [4], the modified constant modulus algorithm (MCMA) was proposed to correct the phase shift. Although the MCMA can correct the phase shift, there remains the problem of the convergence speed and steady-state error. The correction parameters need to be modified according to the modulation when the mode is switched, which is not conducive to system stability; hence, it is not suitable for VM systems. In [5], the decision-directed least mean square (DDLMS) algorithm was proposed, which has a fast convergence speed and small steady-state error. However, the DDLMS algorithm needs to switch the decision mode according to the modulation during mode switching, and the implementation of a higher-order modulation is more complex. In [6], the algorithm of DDLMS with aided pre-convergence was proposed, which consists of an improved CMA algorithm with assisted pre-convergence and the DDLMS algorithm, which can further improve the system performance. However, the implementation of this algorithm is more complicated. The variable step size decision-directed LMS (VSS-DD-LMS) algorithm was proposed in [7], which improves the convergence speed by changing the variable-step adjustment algorithm, but it does not apply to systems with large initial errors and a high-order modulation.