# Transient Analysis of a Selective Partial-Update LMS Algorithm
## 2 Proposed SPU-LMS-M-Min for a Single Block  
Consider the following partition of the input vector \(\mathbf{x}(k)\) and the adaptive weights vector \(\mathbf{w}(k)\) into \(M\) equal-length blocks:  
\[\mathbf{x}(k)=\big{[}\mathbf{x}_{0}^{T}(k)\quad\mathbf{x}_{1}^{T}(k)\quad\ldots\quad\mathbf{x }_{M-1}^{T}(k)\big{]}^{T}, \tag{4}\]  
\[\mathbf{w}(k)=\big{[}\mathbf{w}_{0}^{T}(k)\quad\mathbf{w}_{1}^{T}(k)\quad\ldots\quad\mathbf{w }_{M-1}^{T}(k)\big{]}^{T}, \tag{5}\]  
where the vectors \(\mathbf{x}_{i}(k)\) and \(\mathbf{w}_{i}(k)\) contains \(N/M=L\in\mathbb{N}\) coefficients (For simplicity, \(L\) is supposed to be an integer number.) (for \(i\in\{0,1,\ldots,M-1\}\)). In order to reduce the computational burden, under the selective partial-update paradigm, only \(B\) blocks (for \(B\in\{1,2,\ldots,M\}\)) of the adaptive weights are updated in each iteration. For the sake of simplicity, in this section it is assumed that only one block is updated in each iteration (i.e., \(B=1\)). Note that for the existence of a block algorithm, it is necessary that \(M>1\). Therefore, as \(L=N/M\), we have \(N>L\), an inequality valid for any value of \(B\) (including \(B=1\)). Consider that the index of the block that will be updated (\(\mathbf{w}_{i}(k)\)) is denoted by \(i\) (later, the proposed selection procedure of such a index will be derived). The advanced SPU-LMS-M-min algorithm is the resulting solution of the following optimization problem:  
\[\min_{\mathbf{w}_{i}(k+1)}\frac{1}{2}\|\mathbf{w}_{i}(k+1)-\mathbf{w}_{i}(k)\|^{2}\text{ s.t. }e_{p}(k)=\Big{(}1-\beta L\sigma_{x}^{2}\Big{)}e(k), \tag{6}\]  
where \(\sigma_{x}^{2}\) denotes the variance in \(x(k)\), and the posterior error \(e_{\text{p}}(k)\) is  
\[e_{\text{p}}(k) = d(k)-\mathbf{w}^{T}(k+1)\mathbf{x}(k) \tag{7}\] \[= d(k)-\Big{[}\mathbf{w}_{i}^{T}(k+1)\mathbf{x}_{i}(k)+\overline{\mathbf{w}}_{ i}^{T}(k+1)\overline{\mathbf{x}}_{i}(k)\Big{]},\]  
where \(\overline{\mathbf{w}}_{i}(k)\) and \(\overline{\mathbf{x}}_{i}(k)\) are the vectors obtained by removing \(\mathbf{w}_{i}(k)\) and \(\mathbf{x}_{i}(k)\) from \(\mathbf{w}(k)\) and \(\mathbf{x}(k)\), respectively. The equality in Equation (6) imposes a linear constraint on the solution \(\mathbf{w}(k+1)\), which requires that the _a posteriori_ error (dependent on the solution \(\mathbf{w}(k+1)\)) be a fraction of the _a priori_ error \(e(k)\). Using the Lagrange multipliers technique, the constrained optimization problem (6) can be translated into the following _unconstrained_ one:  
\[\mathcal{F}_{i}[\mathbf{w}_{i}(k+1)]=\frac{1}{2}\|\mathbf{w}_{i}(k+1)-\mathbf{w}_{i}(k)\|^{2 }+\lambda\Big{[}e_{p}(k)-\Big{(}1-\beta L\sigma_{x}^{2}\Big{)}e(k)\Big{]}, \tag{8}\]  
where \(\lambda\in\mathbb{R}\) is the Lagrange multiplier. Zeroing \(\
abla_{\mathbf{w}_{i}(k+1)}\mathcal{F}_{i}[\mathbf{w}_{i}(k+1)]\) and using the approximation \(L\sigma_{x}^{2}\approx\|\mathbf{x}_{i}(k)\|^{2},\forall i\in\{0,1,\dots,M-1\}\), yields  
\[\mathbf{w}_{i}(k+1)=\mathbf{w}_{i}(k)+\beta\mathbf{x}_{i}(k)e(k), \tag{9}\]  
whereas the remaining blocks are supposed to be unaltered, that is,  
\[\overline{\mathbf{w}}_{i}(k+1)=\overline{\mathbf{w}}_{i}(k). \tag{10}\]  
**Remark 1**. : _It is worth mentioning that the approximation \(L\sigma_{x}^{2}\approx\|\mathbf{x}_{i}(k)\|^{2},\forall i\in\{0,1,\dots,\)\(M-1\}\) is a feature of the proposed non-normalized SPU scheme, which is not necessary when normalized adaptive filtering algorithms are adopted. For long filters, exactly where one intends to reduce the computational complexity of adaptive filtering algorithms, such an approximation is less critical (assuming a stationary input signal). In the case of non-stationary signals, real-time estimation of the input signal variance can be achieved with a minor increase in computational complexity.