deterministic inputs will be considered in conjunction with stochastic noise processes, our analysis uses the standard definition of expectation for quasi-stationary signals [@ljung1998system pp. 34], which is $$\overline{E}\{g(t)\} := \lim_{N\to \infty} \frac{1}{N} \sum_{t=1}^N E\{g(t)\}. \notag$$

**.** \[theorem1\] Consider the SRIVC-type estimator with a fixed sampling period $h$ and filtered regressor and instrument vectors given by and respectively, and suppose that Assumptions \[assumption1\] to \[assumption5\] hold. Then, the following statements are true:

1.  There exists a maximum sampling period $h^*>0$ such that, if $h\leq h^*$, the matrix $\overline{E}\{\hat{\varphi}_f(t_k)\varphi_f^\top(t_k)\}$ is generically non-singular.

2.  If $h\leq h^*$, the true parameter $\theta^*$ is the unique converging point.

3.  As the sample size $N$ approaches infinity, $\theta_{j+1}$ converges to $\theta^*$ for $j\geq 1$.

[Proof.]{}

*Proof of Statement 1*. By substituting $$y(t_k) = \left\{\frac{B^*(p)}{A^*(p)}u(t)\right\}_{t_k} + v(t_k) \notag$$ into , we find that $\varphi_f(t_k)=\varphi_{f1}(t_k)-V(t_k)$, where $$\begin{aligned}
    \hspace{-0.21cm}\varphi_{f1}(t_k)\hspace{-0.05cm}:= \hspace{-0.12cm}\Bigg[\dfrac{-p^n}{A_j(p)}&\left\{ \dfrac{B^*(p)}{A^*(p)}u(t) \right\}_{t_k} \hspace{-0.2cm},\hspace{0.1cm} \dots, \hspace{0.1cm}\dfrac{-p}{A_j(p)}\left\{ \dfrac{B^*(p)}{A^*(p)}u(t) \right\}_{t_k} \notag \\
    &\hspace{0.1cm} \left\{\dfrac{p^m u(t)}{A_j(p)} \right\}_{t_k}  \hspace{-0.1cm}, \hspace{0.1cm} \dots,\hspace{0.1cm} \left\{\dfrac{u(t)}{A_j(p)} \right\}_{t_k}
    \Bigg]^\top,
    \end{aligned}$$ and $$\label{Vtk}
V(t_k):= \begin{bmatrix}
\dfrac{p^n}{A_j(p)} v(t_k), & \hspace{0.05cm}\dots, & \hspace{0.1cm} \dfrac{p}{A_j(p)} v(t_k), & 0, & \hspace{0.05cm}\dots, & 0  
\end{bmatrix}^\top.$$ On the other hand, we also have $$\hat{\varphi}_f(t_k)=S(-B_j,A_j) \left[ \dfrac{U_{du}(t)}{A_j^2(p)} \right]_{t_k}, \notag$$ where $$\label{Udu}
    U_{du}(t) := \begin{bmatrix}
    \dfrac{\textnormal{d}^{n+m}}{\textnormal{d}t^{n+m}}u(t), & \dfrac{\textnormal{d}^{n+m-1}}{\textnormal{d}t^{n+m-1}}u(t), 