# Transient Analysis of a Selective Partial-Update LMS Algorithm
## 3 Stochastic Modelling of the Proposed Algorithm
### Tracking Analysis  
Since adaptive filters that present good performance in stationary environments do not necessarily exhibit good learning behavior in the identification of time-variant plants [21], obtaining a stochastic model that offers guarantees with respect to the algorithm learning capabilities in a tracking scenario is crucial.  
A popular model that accounts for temporal variation in \(\mathbf{w}^{*}(k)\) can be described as a first-order Markov process [22]  
\[\mathbf{w}^{*}(k+1)=\mathbf{w}^{*}(k)+\mathbf{q}(k), \tag{32}\]  
where the statistical characterization of the perturbation vector \(\mathbf{q}(k)\) is established by the assumption stated below.  
**Assumption 1**.: Tracking assumption (_TA_): vector \(\mathbf{q}(k_{1})\) is statistically independent from the input data and from \(\mathbf{q}(k_{2})\), \(\forall k_{2}\
eq k_{1}\). Furthermore, its autocorrelation matrix is given by \(\mathbf{R}_{q}\triangleq\mathbb{E}\big{[}\mathbf{q}(k)\mathbf{q}^{T}(k)\big{]}\sigma_{q}^ {2}\mathbf{I}_{N}\), where \(\mathbf{I}_{N}\) denotes the \(\mathrm{N}\)-order identity matrix._  
_In order to create coefficient error vector in a deviation form, let us consider Equation (32) compared to Equation (16):_  
\[\mathbf{w}^{*}(k+1)-\mathbf{w}(k+1) =\mathbf{w}^{*}(k+1)-\mathbf{w}(k)-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)e(k) \tag{33}\] \[\widehat{\mathbf{w}}(k+1) =\mathbf{q}(k)+\mathbf{w}^{*}(k)-\mathbf{w}(k)-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)e(k)\] \[\widehat{\mathbf{w}}(k+1) =\widehat{\mathbf{w}}(k)-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)e(k)+\mathbf{q}(k),\]  
_By also replacing \(e(k)\) by Equation (2) at Equation (34), after some arrangements, the following recursion deviation vector \(\widehat{\mathbf{w}}(k)\) can be rewritten:_  
\[\widehat{\mathbf{w}}(k+1) =\Big{(}\mathbf{I}-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{x}^{T}(k)\Big{)} \hat{\mathbf{w}}(k)- \tag{34}\] \[-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)\
u(k)+\mathbf{q}(k)\]  
_In order to reach a mean-square performance model (following similar steps than those employed to obtain Equation (25)), taking into account Equation (34), which can be used to derive the non-homogeneous stochastic difference equation by performing the product between Equation (34) and its transposed version, reaching this equation:_  
\[\mathbf{\Phi}(k+1) = \mathbf{\Phi}(k)-\beta\mathbf{\Phi}(k)\mathbf{x}(k)\mathbf{x}^{T}(k)\mathbf{\Gamma}(k)\] \[-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{x}^{T}(k)\mathbf{\Phi}(k)\] \[+\beta^{2}\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{x}^{T}(k)\mathbf{\Phi}(k)\mathbf{x} (k)\mathbf{x}^{T}(k)\mathbf{\Gamma}(k)\] \[+\beta^{2}\
u^{2}(k)\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{x}^{T}(k)\mathbf{\Gamma} (k)\] \[+\mathbf{q}(k)\mathbf{q}^{T}(k)+\mathcal{O}[\
u(k)]+\mathcal{O}[\mu(k)],\]  
_where \(\mathbf{\Phi}(k)\) is as defined by Equation (22), \(\mathcal{O}[\
u(k)]\) denotes noise-related terms that will not interfere with the hereinafter analysis, and \(\mathcal{O}[q(k)]\) is related to the first-order perturbation vector, which would also not be part of that balance. Aiming to make the mathematics tractable, the application of the expectation operator from Equation (36) should be performed alongside some assumptions, which are described below._  
\[\mathbf{v}(k+1)=\mathbf{A}\mathbf{v}(k)+\mathbf{b}+\mathbf{c}, \tag{36}\]  
_where \(\mathbf{c}\triangleq\mathbb{E}\big{\{}\text{vec}\big{[}\mathbf{q}(k)\mathbf{q}^{T}(k) \big{]}\big{\}}\)._  
**Remark 5**.: _Note that under the considered tracking scenario, the stability region of the algorithm remains unaltered, since the time-invariant transition matrix \(\mathbf{A}\) is the same as the stationary configuration [20]._