## 3 DTV-LMS Algorithm and Analysis
### Performance Analysis  
In a full duplex system, the SI signals at the near-end transceivers and the useful signals at the far-end are independent of each other. In order to make the analysis more convenient, assuming that the error of time delay is 0, the coefficients of the in-phase component and the orthogonal component corresponding to the SI signal \(r_{SI}(\mathrm{t})\) are, respectively, \(\omega_{i}{}^{\star}\), \(\omega_{q}{}^{\star}\). The in-phase and orthogonal components are wide stationary random signals with a mean of 0. Assuming that their power is \(P_{s}\) and the power of the useful signal \(r_{u}(t)\) is \(P_{f}\) at the far end, the mean square of the error signal during \(\Delta t\) can be represented by  
\[ \begin{array}{rl}\mathrm{E}\Big{[}\mathrm{e}^{2}(\mathrm{t})\Big{]}&= \mathrm{E}\Big{[}\Big{(}r_{SI}(\mathrm{t})-r_{1}(t))+r_{u}(t)+n(t)^{2}\Big{]} \Big{\}}\\ &=E\Big{\{}\Big{[}\Big{(}\omega_{i}{}^{\star}-\omega_{i}\ \big{)}\ \big{(}\omega_{q}{}^{ \star}-\omega_{q}\big{)}\Big{]}S(t)\Big{]}^{2}\Big{\}}+\mathrm{E}\Big{[}[r_{u }(t)+n(t)]^{2}\Big{\}}\\ &+2E\big{[}(r_{SI}(\mathrm{t})-r_{1}(t))(r_{u}(t)+n(t))\big{]}\\ &=P_{s}\Big{[}(\omega_{i}{}^{\star}-\omega_{i})^{2}+\big{(}\omega_{q}{}^{ \star}-\omega_{q}\big{)}^{2}\Big{]}+P_{f}+\sigma_{n}^{2}\end{array}  \tag{13}\]  
where \(\sigma_{n}^{2}\) is the mean power of AWGN, and the weight vector formula is further simplified as  
\[ \begin{array}{rl}W(t+\Delta t)&=W(t)+\Big{[}\frac{\mu(t)}{\rho+S^{T}(t)S(t) }\Big{]}\mathrm{e}(t)S(t)\\ &=W(t)+\Big{[}\frac{\mu(t)}{\rho+P_{s}}\Big{]}\Big{[}W^{T}(t)-W^{\star T} \Big{]}P_{s}+\Big{[}\frac{\mu(t)}{\rho+P_{s}}\Big{]}[r_{U}(t)+n(t)]S(t)\\ &=\Big{[}I-\frac{\mu(t)P_{s}}{\rho+P_{s}}\Big{]}(W(t)-W^{\star})+\Big{[}\frac {\mu(t)}{\rho+P_{s}}\Big{]}[r_{U}(t)+n(t)]S(t)\end{array}  \tag{14}\]  
where \(W^{\star}\) is the weight of the real SI signal, and \(\mathrm{I}\) is the identity matrix. Combining Formula (13) and (14), (15) can be obtained as follows:  
\[ \begin{array}{rl}\mathrm{E}\Big{[}\mathrm{e}^{2}(\mathrm{t})\Big{]}&=P_{s} \mathrm{E}\{[W(t-\Delta t)-W^{\star}]^{T}\Big{[}I-\frac{\mu(t)P_{s}}{\rho+P_{s }}\Big{]}^{2}[W(t-\Delta t)-W^{\star}]\}\\ &\qquad\qquad+\Big{[}\frac{\mu(t)P_{s}}{\rho+P_{s}}\Big{]}^{2}P_{s}{}^{2}(P_{f }+\sigma_{n}^{2})+P_{f}+\sigma_{n}^{2}.\end{array}  \tag{15}\]  
The initial value to the weight vector \(W(0)=[00]^{T}\). Assuming \(\mathrm{t}=\Delta\mathrm{nt}\), according to Equation (15), the relationship between the mean square error (MSE) of the nth iteration and the initial MSE can be obtained as follows:  
\[ \begin{array}{rl}\mathrm{E}\Big{[}\mathrm{e}^{2}(\mathrm{t})\Big{]}&=\Big{[} \big{(}\omega_{i}{}^{\star}\big{)}^{2}+\big{(}\omega_{q}{}^{\star}\big{)}^{2} \Big{]}P_{s}\prod\limits_{k=0}^{n-1}[1-\frac{\mu(\Delta\mathrm{t})P_{s}}{\rho+ P_{s}}\Big{]}^{2}+(P_{f}\\ &+\sigma_{n}^{2}\Big{)}\Big{[}1+2\Big{(}\frac{\mu(\Delta\mathrm{t})P_{s}}{\rho +P_{s}}\Big{)}^{2}\Big{]}+2(P_{f}\\ &+\sigma_{n}^{2}\Big{)}\Big{(}\sum\limits_{k=0}^{n-2}P_{n}^{2}\Big{[}\frac{ \mu(\Delta\mathrm{t})P_{s}}{\rho+P_{s}}\Big{]}^{2}\prod\limits_{i=k+1}^{n-1} \Big{[}1-\frac{\mu(\Delta\mathrm{t})P_{s}}{\rho+P_{s}}\Big{]}^{2}\Big{\}}.\end{array}  \tag{16}\]  
It can be seen that the convergence conditions of the above equation are \(\Big{|}1-\frac{\mu(\Delta\mathrm{t})P_{s}}{\rho+P_{s}}\Big{|}<1\), \(\mathrm{k}\in\mathrm{N}^{\star}\). Due to \(\rho\) generally being very small, the condition for the convergence can be simplified as \(\Big{|}1-\mu(\mathrm{k}\Delta\mathrm{t})\Big{|}<1\), and then we can express \(\Big{|}\mu\Big{|}<1\),which is consistent with the convergence conditions of the classical LMS algorithm given above. It can be seen from Equation (16) that the convergence of the algorithm is determined by the trend of the cumulative product value in the error convergence function. Let the convergence factors of the LMS algorithm and the DTVLMS algorithm be \(u_{LMS}(\Delta\mathrm{nt})\).