# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 2 System Model  
The signal model that shall be used throughout the paper is given by [11]  
\[d(n) =\mathbf{h}^{T}(n)\mathbf{X}(n)\mathbf{g}(n)+v(n)\] \[=y(n)+v(n), \tag{1}\]  
where \(d(n)\) is the zero-mean desired (or reference) signal at the discrete-time index \(n\), \(\mathbf{h}(n)\) and \(\mathbf{g}(n)\) are the two impulse responses of the system of lengths \(L\) and \(M\), respectively, the superscript \({}^{T}\) is the transpose operator, \(\mathbf{X}(n)=[\mathbf{x}_{1}(n)\quad\mathbf{x}_{2}(n)\quad\ldots\quad \mathbf{x}_{M}(n)]\) is the zero-mean multiple-input signal matrix, \(\mathbf{x}_{m}(n)=[x_{m}(n)\quad x_{m}(n-1)\quad\ldots\quad x_{m}(t-L+1)]^{T}\) is a vector containing the \(L\) most recent time samples of the \(m\)th (\(m=1,2,\ldots,M\)) input signal, \(y(n)=\mathbf{h}^{T}(n)\mathbf{X}(n)\mathbf{g}(n)\) is the output signal and it represents a bilinear form, and \(v(n)\) is a zero-mean additive noise (with the variance \(\sigma_{v}^{2}\)). It is assumed that all signals are real-valued, and \(\mathbf{X}(n)\) and \(v(n)\) are independent. As we can notice, \(y(n)\) is a bilinear function of \(\mathbf{h}(n)\) and \(\mathbf{g}(n)\), because for every fixed \(\mathbf{h}(n)\), \(y(n)\) is a linear function of \(\mathbf{g}(n)\), and for every fixed \(\mathbf{g}(n)\), it is a linear function of \(\mathbf{h}(n)\). We can rewrite the matrix \(\mathbf{X}(n)\), of size \(L\times M\), as a vector of length \(ML\), by using the vectorization operation:  
\[\text{vec}[\mathbf{X}(n)]=[\mathbf{x}_{1}^{T}(n)\quad\mathbf{x}_{2}^{T}(n) \quad\ldots\quad\mathbf{x}_{M}^{T}(n)]^{T}=\widetilde{\mathbf{x}}(n). \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Significance** \\ \hline \(a\) & scalar \(a\) \\
**A** & matrix \(\mathbf{A}\) \\ \(T\) & transpose operator \\ \(\otimes\) & Kronecker product operator \\ \(\|\mathbf{b}\|\) & Euclidean norm of the vector \(\mathbf{b}\) \\ \(\mathbb{E}(\cdot)\) & mathematical expectation \\ \(\mathbf{I}_{L}\) & identity matrix of size \(L\times L\) \\ \(\text{tr}(\mathbf{B})\) & trace of the square matrix \(\mathbf{B}\) \\ \(\text{vec}(\mathbf{C})=\mathbf{c}\) & vectorization operation, i.e., conversion of a matrix (\(\mathbf{C}\) of size \(L\times M\)) into a vector (\(\mathbf{c}\) of length \(ML\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation used throughout the paper. Therefore, the output signal can be expressed as  
\[y(n) = \mathbf{h}^{T}(n)\mathbf{X}(n)\mathbf{g}(n) \tag{3}\] \[= \left[\mathbf{g}(n)\otimes\mathbf{h}(n)\right]^{T}\widetilde{ \mathbf{x}}(n)=\mathbf{f}^{T}(n)\widetilde{\mathbf{x}}(n),\]  
where \(\otimes\) denotes the Kronecker product between the individual impulse responses, while the vector \(\mathbf{f}(n)=\mathbf{g}(n)\otimes\mathbf{h}(n)\), of length \(ML\), represents the spatiotemporal (i.e., global) impulse response of the system. Consequently, the signal model in (1) becomes  
\[d(n)=\mathbf{f}^{T}(n)\widetilde{\mathbf{x}}(n)+v(n). \tag{4}\]  
The major difference compared to a general MISO system is yielded by the fact that in this bilinear context \(\mathbf{f}(n)\) is formed with only \(M+L\) different elements, despite being of length \(ML\). The goal is the identification of the two impulse responses \(\mathbf{h}(n)\) and \(\mathbf{g}(n)\), and, in this way, of the spatiotemporal impulse response \(\mathbf{f}(n)\). For this aim, we can use two adaptive filters, \(\widehat{\mathbf{h}}(n)\) and \(\widehat{\mathbf{g}}(n)\); hence, the global impulse response can be evaluated as  
\[\widehat{\mathbf{f}}(n)=\widehat{\mathbf{g}}(n)\otimes\widehat{\mathbf{h}}( n). \tag{5}\]  
Let \(\eta\
eq 0\) be a real-valued constant. We can see from (1) that  
\[\left[\frac{1}{\eta}\mathbf{h}^{T}(n)\right]\mathbf{X}(n)\left[\eta\mathbf{g} (n)\right]=\mathbf{h}^{T}(n)\mathbf{X}(n)\mathbf{g}(n)=y(n), \tag{6}\]  
meaning that the pairs \([\mathbf{h}(n)/\eta,\eta\mathbf{g}(n)]\) and \([\mathbf{h}(n),\mathbf{g}(n)]\) are equivalent in the bilinear form.