## 1 Introduction
From the discrete-time state space model (1) and (2), the finite measurements \(Z_{M}\) on the most recent window \([i-M,i]\) can be expressed by the following regression form in terms of the state \(x_{i}\) at the current time \(i\):  
\[Z_{M}\ =\ \Gamma_{M}x_{i}+\Lambda_{M}W_{M}+V_{M}, \tag{3}\]  
where \(Z_{M}\), \(W_{M}\), \(V_{M}\) and matrices \(\Gamma_{M}\), \(\Lambda_{M}\) are defined as follows:  
\[Z_{M}\ \ \ \stackrel{{\triangle}}{{=}}\ \ \left[ \begin{array}{c}z_{i-M}\\ z_{i-M+1}\\ z_{i-M+2}\\ \vdots\\ z_{i-1}\end{array} \right],\ \ \ W_{M}\stackrel{{\triangle}}{{=}}\ \left[ \begin{array}{c}w_{i-M}\\ w_{i-M+1}\\ w_{i-M+2}\\ \vdots\\ w_{i-1}\end{array} \right],\ \ \ V_{M}\stackrel{{\triangle}}{{=}}\ \left[ \begin{array}{c}v_{i-M}\\ v_{i-M+1}\\ v_{i-M+2}\\ \vdots\\ v_{i-1}\end{array} \right], \tag{4}\]  
\[\Gamma_{M}\ \ \ \stackrel{{\triangle}}{{=}}\ \ \left[ \begin{array}{c}CA^{-M}\\ CA^{-M+1}\\ CA^{-M+2}\\ \vdots\\ CA^{-1}\end{array} \right],\ \ \ \Lambda_{M}\stackrel{{\triangle}}{{=}}\ \left[ \begin{array}{cccc}CA^{-1}G&CA^{-2}G&\cdots&CA^{-M}G\\ 0&CA^{-1}G&\cdots&CA^{-M+1}G\\ 0&0&\cdots&CA^{-M+2}G\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&CA^{-1}G\end{array} \right], \tag{5}\]  
where notations \(CA^{*}\) and \(CA^{*}G\) are shorthand for matrix multiplications \(C\cdot A^{*}\) and \(C\cdot A^{*}\cdot G\), respectively. The noise term \(\Lambda_{M}W_{M}+V_{M}\) in (3) is zero-mean white Gaussian as follows:  
\[\Lambda_{M}W_{M}+V_{M}\ \ \sim\ \ \mathcal{N}\big{(}Z_{M};0,\Pi_{M}\big{)}, \tag{6}\]where \(\mathcal{N}\big{(}Z_{M};0,\Pi_{M}\big{)}\) denotes the Gaussian probability density function (pdf) evaluated at \(Z_{M}\) with zero-mean and covariance matrix  
\[\Pi_{M} \stackrel{{\triangle}}{{=}} \Lambda_{M}\Big{[}\text{diag}\big{(}\overbrace{Q\,Q\,Q\,\cdots\,Q}^ {M}\big{)}\Big{]}\Lambda_{M}{}^{T}+\Big{[}\text{diag}\big{(}\overbrace{R\,R \,R\,\cdots\,R}^{M}\big{)}\Big{]}, \tag{7}\]  
where \(\text{diag}(Q\,Q\,\cdots\,Q)\) and \(\text{diag}(R\,R\,R\,\,\cdots\,R)\) denote block-diagonal matrices with \(M\) elements of \(Q\) and \(R\), respectively. As shown in Bayesian filtering approaches [15; 16], the FMS filter can be interested in the pdf that is conditional on a finite past measurements \(Z_{M}\) on the most recent window \([i-M,i]\). The most recent window \([i-M,i]\) becomes the averaging window of \(M\) points. Existing FMS filters in [15; 16] have an iterative form, while this paper develops an alternative FMS filter with a simple matrix form. To develop an alternative FMS filter, the conditional density of current state \(x_{i}\) given finite measurements \(Z_{M}\) is derived. **Proposition 1**. : _From the linearity described in (3), the conditional density of current state \(x_{i}\) given finite measurements \(Z_{M}\) has the following expression:_  
\[p\big{(}x_  
Therefore, the conditional density of current state \(x_{i}\) given finite measurements \(Z_{M}\) has the following expression:  
\[p\big{(}x_{i}|Z_{M}\big{)}=\mathcal{N}\big{(}x_{i};\hat{x}_{i}, \Sigma_{M}\big{)}.\]  
This completes the proof. Therefore, from the conditional density (8) of current state \(x_{i}\), the FMS filter with the following simple matrix form  
\[\hat{x}_{i}=\big{(}\Gamma_{M}^{T}\Pi_{M}^{-1}\Gamma_{M}\big{)}^{ -1}\Gamma_{M}^{T}\Pi_{M}^{-1}Z_{M} \tag{12}\]  
provides the state estimate \(\hat{x}_{i}\) conditional on finite measurements \(Z_{M}\). As shown in [13], by applying the best linear unbiased estimation approach, the batch unbiased finite impulse response (FIR) filter with the matrix form was developed with the unbiased constraint when the covariance \(\Pi_{M}\) of the noise term \(\Lambda_{M}W_{M}+V_{M}\) (6) has an identity matrix. On the other hand, by applying the Bayesian approach in this paper, the simple matrix form (12) for the FMS filter is derived in an alternative way using mean \(\hat{x}_{i}\) and covariance value \(\Sigma_{M}\) of Gaussian pdf evaluated at \(x_{i}\).