Do WSL adaptive filters provide better tracking performance than LMS filters? (Q1596454)

The authors consider the model \[ y(t)= \varphi^T(t)\theta(t)+ v(t), \] where \(\varphi(t)= [y(t-1),\dots, y(t- r),u(t-1),\dots, u(t- p)]^T\) is the vector of measurements comprised of the output and input signals, respectively, \(\theta(t)= [a_1(t),\dots, a_r(t), b_1(t),\dots, b_p(t)]^T\) is the vector of unknown coefficients, and \(\{v(t)\}\) denotes a discrete-time white noise sequence. They analyze two estimators of \(\theta(t)\): the weighted least squares (WLS) estimator which is obtained due to \[ \widehat\theta(t)= \underset{\theta}{\text{arg min}} \sum^{t-1}_{i=0} w(i)[y(t- i)- \varphi^T(t- i)\theta]^2, \] where \(w(i)= \lambda^i\), \(0< \lambda^i< 1\), and the least mean squares (LMS) estimator \[ \widehat\theta(t)= \widehat\theta(t- 1)+ {\mu\varphi(t)\over r(t)} \varepsilon(t), \] where \(\mu> 0\), \(r(t)= \lambda r(t-1)+ \varphi^T(t)\varphi(t)\), \(\varepsilon(t)= y(t)- \varphi^T(t)\widehat\theta(t- 1)\).