& \hspace{0.1cm}\dots, & \hspace{0.1cm} u(t)
    \end{bmatrix}^\top \hspace{-0.1cm},$$ and $S(-B_j,A_j)$ is the Sylvester matrix associated with the polynomials $-B_j(p)$ and $A_j(p)$, whose non-singularity follows from the same analysis done in [@pan2020consistency], where Assumption \[assumption4\] is used. With this, we compute $$\begin{aligned}
    \overline{E}\{\hat{\varphi}_f(t_k)\varphi_f^\top(t_k)\} &= S(-B_j,A_j)\underbrace{\overline{E}\left\{\left[ \dfrac{U_{du}(t)}{A_j^2(p)} \right]_{t_k}\varphi_{f1}^\top(t_k) \right\}}_{=:\Phi} \notag \\
    \label{Psidef}
    &\hspace{-1.5cm}-S(-B_j,A_j)\underbrace{\overline{E}\left\{\left[ \dfrac{U_{du}(t)}{A_j^2(p)} \right]_{t_k}V^\top(t_k)\right\}}_{=:\Psi}.
    \end{aligned}$$ Thus, for showing that $\overline{E}\{\hat{\varphi}_f(t_k)\varphi_f^\top(t_k)\}$ is generically non-singular for a small enough sampling period $h$, it is sufficient to show that $\Psi=0$ and $\Phi$ is generically non-singular for a small enough $h$. The difference between the analysis in [@pan2020consistency Theorem 1] and the proof in the current paper is that the signals of interest are hybrid in nature: some are evaluations of CT signals, whereas others are DT signals interpolated with a reconstruction device, such as a FOH.

The proof of $\Psi=0$ can be found in Lemma \[lemma1\] in the Appendix. Regarding the invertibility of $\Phi$, we will conveniently write $\varphi_{f1}(t_k)$ as $\varphi_{f2}(t_k)+\Delta(t_k)$, where $\Delta(t_k) \in \mathbb{R}^{n+m+1}$ has entries $$\label{deltai}
    \Delta_i(t_k)= \begin{cases}
    \left\{\frac{p^{n+1-i}}{A_j(p)}x(t)\right\}_{t_k}-\frac{p^{n+1-i}}{A_j(p)} x(t_k) &\hspace{-0.2cm},\hspace{0.07cm} i=1,\dots,n \\
    0 &\hspace{-2.2cm},\hspace{0.07cm} i=n+1,\dots,n+m+1,
    \end{cases}$$ and $$\begin{aligned}
    \varphi_{f2}(t_k) &= \Bigg[
    \dfrac{-p^nB^*(p)}{A_j(p) A^*(p)}u(t), \hspace{0.1cm}\dots, \hspace{0.1cm}\dfrac{-pB^*(p)}{A_j(p)A^*(p)}u(t), \notag \\
    &\hspace{2cm} \quad \dfrac{p^m}{A_j(p)}u(t),  \hspace{0.1cm}\dots, \hspace{0.1cm} \dfrac{1}{A_j(p)}u(t)
    \Bigg]^\top_{t_k} \notag \\
    &= S(-B^*,A^*) \left[ \frac{U_{du}(t)}{A_j(p)A^*(p)} \right]_{t_k}, \notag 
    