# Transient Analysis of a Selective Partial-Update LMS Algorithm
## 3 Stochastic Modelling of the Proposed Algorithm
### Deficient-Length Analysis  
There are several practical situations to motivate that analysis: Usually we do not know in advance the transfer function to be identified, no how limited the computational resources would be. In addition, the length of the adaptive filter \(N\) is often less than the length \(L\) of the transfer function to be identified. In this deficient-length setting, according to [4], it is possible to model the transfer function coefficients vector and the reference signal as:  
\[\mathbf{w_{o}}^{\star}=[\mathbf{w}^{\star},\mathbf{\bar{w}}^{\star}]^{T}, \tag{37}\]  
\[d_{\text{def}}(k)=[\mathbf{w}^{\star}]^{T}\mathbf{x}(k)+[\mathbf{\bar{w}}^{\star}]^{T}\mathbf{ \bar{x}}(k)+\
u(k), \tag{38}\]  
where  
\[\mathbf{\bar{x}}(k)\triangleq\big{[}x(k-N)\quad x(k-N-1)\quad\cdots\quad x(k-N-L+ 1)\big{]} \tag{39}\]  
and \(\mathbf{\bar{w}}^{\star}\in\mathbb{R}^{L}\) is the \(L\)-length component of the ideal transfer function, \(\mathbf{w}^{\star}\), whose total length is \(N+L\), that the adaptive filter is not able to emulate. Considering Equation (2) and the previous definitions, it is possible to write \(e_{\text{def}}(k)\) in the deficient-length approach:  
\[\begin{array}{ll}e_{\text{def}}(k)&=d_{\text{def}}(k)-\mathbf{\bar{w}}^{T}(k) \mathbf{x}(k)\\ &=(\mathbf{w}^{\star})^{T}\mathbf{x}(k)+(\mathbf{\bar{w}}^{\star})^{T}\mathbf{\bar{x}}(k)+\
u( k)-\mathbf{\bar{w}}^{T}(k)\mathbf{x}(k)\\ &=\mathbf{\bar{w}}^{T}(k)\mathbf{x}(k)+(\mathbf{\bar{w}}^{\star})^{T}\mathbf{\bar{x}}(k)+\
u( k),\end{array} \tag{40}\]  
Subsequently, defining input, reference signals, and error, it is necessary to rewrite Equation (16) for the context of the deficient-length analysis with Equation (41):  
\[\begin{array}{rcl}\mathbf{w}^{\star}(k+1)-\mathbf{w}(k+1)&=&\mathbf{w}^{\star}(k+1)-\bm {w}(k)-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)e_{\text{def}}(k)\\ \mathbf{\bar{w}}(k+1)&=&\mathbf{\bar{w}}(k)-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)e_{\text{def}} (k)\\ \mathbf{\bar{w}}(k+1)&=&\mathbf{\bar{w}}(k)-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)[\mathbf{\bar{w}}^ {T}(k)\mathbf{x}(k)\\ &+&(\mathbf{\bar{w}}^{\star})^{T}\mathbf{\bar{x}}(k)+\
u(k)]\\ \mathbf{\bar{w}}(k+1)&=&\mathbf{\bar{w}}(k)-\beta\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{x}^{T}(k) \mathbf{\bar{w}}(k)\\ &-&\beta\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{\bar{x}}^{T}(k)\mathbf{\bar{w}}^{\star}-\beta \mathbf{\Gamma}(k)\mathbf{x}(k)\
u(k)\end{array} \tag{41}\]  
Calculating the product between Equation (41) and its respective transposed version, applying the same steps as than those employed to obtain Equation (25), also to make more comprehensible the mathematics, the following assumption will be considered in the deficient-length setup:  
**Assumption 2**.: _Whiteness assumption (VA). The excitation data is white._  
**Remark 6**.: _Note that VA is a common assumption in the field of adaptive filtering analysis, even when the input signal is non-stationary (see, e.g., [23]). A more evolved analysis is necessary if such an assumption should be circumvented [24]. Using VA and the previous assumptions, it can be demonstrated that in the suboptimal scenario (38) the vector \(\mathbf{v}(k)\) is updated according to_  
\[\mathbf{v}_{k+1}=\mathbf{A}\mathbf{v}_{k}+\mathbf{b}+\mathbf{d} \tag{42}\]  
_where_  
\[\mathbf{d}\triangleq\beta^{2}\mathbb{E}\Big{\{}\Big{[}\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{ x}^{T}(k)\Big{]}\otimes\Big{[}\mathbf{\Gamma}(k)\mathbf{x}(k)\mathbf{x}^{T}(k)\Big{]} \Big{\}}\mathbf{\bar{v}} \tag{43}\]and \(\overline{v}\triangleq vec\big{[}\overline{w}\overline{w}^{T}\big{]}\)._