## 3 DTV-LMS Algorithm and Analysis
### Performance Analysis
According to Equation (16), we can obtain \(u_{DTVLMS}(\mathrm{n}\Delta\mathrm{t})\) as  
\[u_{LMS}(\mathrm{n}\Delta\mathrm{t})=1-\mu P_{s}{}^{2n} \tag{17}\]\[u_{DTVLMS}(\Lambda\mathrm{\Delta t}) =\prod\limits_{k=0}^{n-1}\left[1-\frac{\mu(\mathrm{k\Delta t})P_{s}} {\rho+P_{s}}\right]^{2} \tag{18}\] \[=\prod\limits_{k=0}^{n-1}\left\{1-\left[\frac{\left(\mathrm{ \Delta t}-1\left(-\beta\right|e(\mathrm{k\Delta t})e(\mathrm{k\Delta t}-\Delta t )\right|+0.4\mu(t-1)\right)\right]P_{s}}{\rho+P_{s}}\right\}^{2}\] \[\approx\prod\limits_{k=0}^{n-1}\left\{1-\left[\mathrm{\Delta t} -\beta\right|e(\mathrm{k\Delta t})e(\mathrm{k\Delta t}-\Delta t)\right|+0.4 \mu(t-1))\right]\right\}^{2}.\]  
It can be seen from the above two formulas, compared with the convergence factors of the traditional fixed-step LMS algorithm, that the convergence speed of the DTV-LMS algorithm is almost independent of the power of the near-end transmitted signal in the RF domain, which can effectively avoid the interference of the gradient noise amplification. In addition, by using the correlation value of the error signal to adjust the step size, the algorithm can take into account the performance of the convergence speed and error. At the same time, the problem of intermittent communication or burst noise interference by relying only on the time parameter can also be effectively solved. According to reference [24], the equation for ICR is \(\mathrm{ICR}=10\mathrm{lg}\left(P_{before}/P_{after}\right)\), where \(P_{before}\) and \(P_{after}\) is the power of the near-end SI signal before and after the SI cancellation in RF domain, respectively. Let \(\mathrm{K}\in N^{*}\). When \(\mathrm{t}\geq\mathrm{K\Delta t}\), the MSE of the DTV-LMS algorithm tends to be stable, and Formula (16) can be simplify as  
\[ \begin{split}\mathbb{E}\left[\mathrm{e}^{2}(\mathrm{t})\right] \approx 2\left[(\omega_{t}^{*})^{2}+\left(\omega_{q}\right)^{2}\right]P_{s} \left[1-\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P_{s}}\right]^{2(n-\mathrm{K})}+2(P _{f}\\ +\sigma_{n}^{2}\right]\left[\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P_ {s}}\right]^{2}\frac{1-\left[1-\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P_{s}} \right]^{2(n-\mathrm{K})}}{1-\left[1-\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P_{s}} \right]^{2}}+(P_{f}+\sigma_{n}^{2}).\end{split}  \tag{19}\]  
Because \(\left|1-\frac{\mu(\mathrm{k\Delta t})P_{s}}{\rho+P_{s}}\right|<1\), when \(\mathrm{K}\)\(\rightarrow\)\(\infty\), \(\lim\limits_{K\rightarrow\infty}\left[1-\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P_{s}} \right]^{2(n-\mathrm{K})}\to 0\), and we can get the formula of MSE as  
\[ \begin{split}\mathbb{E}\left[e^{2}(\infty)\right]\approx(P_{f}+ \sigma_{n}^{2})+2(P_{f}+\sigma_{n}^{2})\left[\frac{\mu(\mathrm{ICR})P_{s}}{ \rho+P_{s}}\right]\left(2-\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P_{s}}\right) \right]\\ =(P_{f}+\sigma_{n}^{2})\left[\left(2+\frac{\mu(\mathrm{ICR})P_{s} }{\rho+P_{s}}\right)/\left(2-\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P_{s}}\right) \right].\end{split}  \tag{20}\]  
Thus, the ICR of the DTV-LMS algorithm can be given as  
\[ \begin{split}\mathrm{ICR}=10\mathrm{lg}\left\{\frac{P_{0}-(P_{f }+\sigma_{n}^{2})}{\mathbb{E}\left[e^{2}(\infty)\right]-(P_{f}+\sigma_{n}^{2} )}\right\}\\ \approx-10\mathrm{lg}\left\{2\frac{\mu(\mathrm{ICR})P_{s}}{\rho+P _{s}}(P_{f}+\sigma_{n}^{2})/P_{SI}\left[2-\frac{\mu(\mathrm{ICR})P_{s}}{\rho+ P_{s}}\right]\right\}.\end{split}  \tag{21}\]  
According to the above equation, ICR will be applied to the simulation experiment in the next section, and its performance will be verified from the simulation results.