## 3 DTV-LMS Algorithm and Analysis  
### DTV-LMS Algorithm  
As shown in Equation (9) above, in a classical LMS algorithm, \(\mu\) is a fixed constant value whose convergence range is between \([0,1/\lambda_{max}]\), and \(\lambda_{max}\) is the maximum eigenvalue of the autocorrelation matrix of the input signal. The convergence speed of the LMS algorithm became faster with an increase of \(\mu\), but at the same time, it produces a problem by increasing SSM. An optimized LMS algorithm called the DTV-LMS algorithm is proposed in this paper. This algorithm makes the value of the step factor change over time (from large to small) to reduce SSM. Based on the iterative formula in the reference [20; 21], we make further improvements the weight and step updating formulas of the DTV-LMS algorithm, as given in Equations (11) and (12), respectively:  
\[\mu(t)=\alpha tan^{-1}\left(-\beta\left|e(t)e(t-\Delta t)\right|\right)+0.4 \mu(t-1) \tag{11}\]  
\[W(t+\Delta t)=W(t)+\left[\frac{\mu(t)}{\rho+S^{T}(t)S(t)}\right]e(t)S(t) \tag{12}\]  
where \(\alpha\), \(\beta\) are constants, which are used to control the step-size in order to accelerate the convergence speed while keeping a low steady-state error; \(\rho\) is also a constant and can solve the numerical problem that a minimal value exists when \(S(t)\) is too small. The autocorrelation of the error signal is used from Formula (11) to reduce the impact of burst impulse noise interference, thereby improving ICR and the cancellation performance. The step updating equation uses the arctangent function to establish a new non-linear relationship with the time parameters. Compared to the sigmoid function and hyperbolic tangent function used in references, the simulation results are shown in Figure 3.  
From the curve shown in Figure 3, it is obvious that when the error factor is close to zero, the sigmoid function and \(\tanh\) function will change dramatically, and the small error changes will result in large step size changes. The optimized algorithm proposed in this paper overcomes this shortcoming very well, and \(\mu(t)\) changes slowly when \(e(t)\) approaches zero, so that the adaptive weight vector value of the algorithm is closer to the actual value when it reaches a steady state. Figure 3: The curves of the step-size characteristics. From Equation (11), we can see that the updating of \(\mu(t)\) is influenced by the statistical characteristics of the near-end transmitting signal, and the adjustment of the weight vector is controlled by the step factor. That is to say, the algorithm uses the statistical characteristics of the transmitting signal to realize an effective adjustment of the weight. In addition, we adjust the coefficients \(\alpha\), \(\beta\) to change the convergence performance. The curves of the relationship between \(\mu(t)\) and \(e(t)\) are shown in Figures 4 and 5, with a change in the values of \(\alpha\) and \(\beta\), respectively. From the two figures, we can see when the slope gets larger, the step causes a mutation just as the error approaches 0, so '\(\alpha\)' should not be too large. Similarly, '\(\beta\)' also needs to be selected reasonably. According to the model of the CCFD system and the structure of RF cancellation mentioned in Section 2, the detailed steps of the SI cancellation method in the RF domain are given as follows:  
1. The group delay time of the transmitted and received signals is estimated by the training sequence. 2. The optimal values of parameters \(\alpha\), \(\beta\) are determined according to various parameters of the CCFD system model. 3. Give the initial value to the weight vector \(W(0)\), assuming e(t) = r\({}_{0}\)(t), when t < 0. 4. The \(\mu\)(t) is calculated at intervals of time \(\Delta\)t, and the weights are updated while adjusting the gains of the two in-phase and orthogonal branches. 5. Calculate the real-time error signal e(t). 6. Return to (4). Figure 4: The variation of curves; when \(\alpha\) = 0.6, \(\beta\) changes. Figure 5: The variation of curves; when \(\beta\) = 0.3, \(\alpha\) changes.