## 2 Problem Formulation
These assumptions are not fundamental and they can be modified or extended to other noise models. For the non-stochastic noise, with reference to the methods in [PERSON] et al. [30], it is possible to generalize the methods of this paper to the case of deterministic noise by adjusting the model structure. Parameter Estimation Algorithm and Its Convergence: The Scenario with Known Probability of Packet Loss  
Assume that the system input \(\{u_{k}\}\) is n-periodic, i.e., \(u_{k+n}=u_{k}\), \(\forall k\geq 1\). Define \(\pi_{1}=\phi_{1}^{T},\ldots,\pi_{n}=\phi_{n}^{T}\). Then, the circulant matrix formed by \(\{u_{k}\}\) is  
\[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]  
Based on the input period \(n\), the data length \(N\) is divided as follows:  
\[L_{N}=\left\lfloor\frac{N}{n}\right\rfloor, \tag{7}\]  
where \(\lfloor\cdot\rfloor\) denotes the floor function, representing the greatest integer less than or equal to the number \"\(\cdot\)\". For the data length \(N\), when the probability of packet loss \(p\) is known, an algorithm is designed to estimate the unknown parameter \(\theta\):  
\[\widehat{\theta}_{N} = \Phi^{-1}\Big{[}C-F^{-1}(\xi_{N,1}),\ldots,C-F^{-1}(\xi_{N,n}) \Big{]}^{T}, \tag{8}\] \[\xi_{N,i} = \frac{m}{L_{N}}\sum_{l=1}^{L_{N}}\frac{\gamma_{(l-1)n+i}}{1-p}, \ i=1,2,\ldots,n, \tag{9}\]  
where \(F^{-1}(\cdot)\) is given by Assumption 1. The effectiveness of the algorithm is predicated on the assumption that \(\Phi\) is invertible. By integrating the designability of inputs with the characteristics of the circulant matrix, a condition is provided to ensure the invertibility of \(\Phi\). Specifically, it is required that \(u_{1}+u_{2}x+u_{3}x^{2}+\cdots+u_{n}x^{n-1}=0\) does not have any unit roots. For ease of presentation, for \(i=1,2,\ldots,n\), the following notations are introduced:  
\[F_{i} = F(C-\pi_{i}\theta), \tag{10}\] \[f_{i} = f(C-\pi_{i}\theta),\] (11) \[\mu(l,i) = (l-1)n+i, \tag{12}\]  
where \(f(\cdot)\) is the probability density function of the noise. **Theorem 1**. : _Under uncertain communication (5), consider the system (1) that introduces CSTCS (3) and (4) and binary quantization (2). Provided that Assumptions 1 and 2 are valid, the estimate \(\widehat{\theta}_{N}\) derived from (8) and (9) strongly converges to the true value, i.e.,_  
\[\widehat{\theta}_{N}\to\theta,\;\;\text{as}\;N\to\infty\;\;\text{w.p.1}.\]  
Proof. : Under Assumptions 1 and 2, it can be seen that  
\[\text{P}(s_{k}=1) = \text{P}(y_{k}\leq C)\] \[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]0 \[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]1 \[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]2 \[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]3  
Then, we have  
\[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]4 \[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]5  
By the Kolmogorov law of large numbers [31], it can be ascertained that  
\[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]6 \[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]7  
From (3) and (4), we have \(\frac{1}{L_{N}}\sum_{l=1}^{L_{N}}s_{\mu(l,i)}=\frac{m}{L_{N}}\sum_{l=1}^{L_{N} }\gamma_{\mu(l,i)}^{e}+\frac{G_{N,i}}{L_{N}}\), where  
\[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]8  
Noting that \(G_{N,i}\in\{0,1,2,\ldots,m-1\}\), it follows that  
\[\Phi=[\pi_{1},\ldots,\pi_{n}]^{T}. \tag{6}\]9  
Denote \(\mathscr{F}_{k}\) as the \(\sigma\)-algebra formed by \(d_{1},d_{2},\ldots,d_{k}\). Then, from (14) we have  
\[L_{N}=\left\lfloor\frac{N}{n}\right\rfloor, \tag{7}\]0  
Define \(\widetilde{\gamma}_{\mu(l,i)}=\gamma_{\mu(l,i)}-(1-p)\gamma_{\mu(l,i)}^{e}\). According to the definition of \(\gamma_{\mu(l,i)}^{e}\), it can be seen that \(\gamma_{\mu(l,i)}^{e}\) is \(\mathscr{F}_{\mu(l,i)}\)-measurable and finite.