# An Optimized Differential Step-Size LMS Algorithm
## 4 ODSS-LMS Algorithm
### Simplified Version  
At this stage, we can define the main equations for the optimized differential least-mean-square \"white\" (ODSS-LMS-W) algorithm, in the scenario of a white Gaussian noise signal as input. This algorithm is similar to the OLMS-ICF algorithm presented in [12], which was developed based on the same simplified assumption.  
Considering that \(\mathbf{R}=\sigma_{x}^{2}\mathbf{I}_{L}\), we can rewrite (19), (24), and (26) as  
\[\mathbf{R}_{\mathbf{c}}(n) = \boldsymbol{\Gamma}(n-1)-\sigma_{x}^{2}\mathbf{\Gamma}(n-1) \mathbf{Y}(n)-\sigma_{x}^{2}\mathbf{Y}(n)\boldsymbol{\Gamma}(n-1) \tag{27}\] \[+\sigma_{x}^{4}\mathbf{Y}(n)\{2\boldsymbol{\Gamma}(n-1)+\mathbf{ I}_{L}\text{tr}[\boldsymbol{\Gamma}(n-1)]\}\mathbf{Y}(n)+\sigma_{x}^{2}\sigma_{v}^{2} \mathbf{Y}^{2}(n),\]  
\[m(n) = [m(n-1)+L\sigma_{w}^{2}][1+\sigma_{x}^{4}\mathbf{y}^{T}(n)\mathbf{ y}(n)]-2\sigma_{x}^{2}\boldsymbol{\gamma}^{T}(n-1)\mathbf{y}(n) \tag{28}\] \[+2\sigma_{x}^{4}\mathbf{y}^{T}(n)\text{mdiag}[\boldsymbol{\gamma} (n-1)]\mathbf{y}(n)+\sigma_{v}^{2}\sigma_{x}^{2}\mathbf{y}^{T}(n)\mathbf{y}(n),\]  
and  
\[\mathbf{y}_{\text{o}}(n)=\Big{\{}\sigma_{x}^{2}\mathbf{I}_{L}[m(n-1)+\sigma_{w} ^{2}(L+2)]+2\sigma_{x}^{2}\text{mdiag}[\mathbf{R}_{\mathbf{c}}(n-1)]+\sigma_{v }^{2}\mathbf{I}_{L}\Big{\}}^{-1}\boldsymbol{\gamma}(n-1). \tag{29}\]  
Based on (27), we can define the recursive equation of \(\boldsymbol{\gamma}(n)\) as  
\[\boldsymbol{\gamma}(n) = \boldsymbol{\gamma}(n-1)+\sigma_{w}^{2}\mathbf{I}_{L\times 1}+2 \sigma_{x}^{2}[\sigma_{x}^{2}\mathbf{y}_{\text{o}}(n)-\mathbf{1}_{L\times 1}] \odot\boldsymbol{\gamma}(n-1)\odot\mathbf{y}_{\text{o}}(n) \tag{30}\] \[+\sigma_{x}^{2}\{\sigma_{x}^{2}\text{tr}[\boldsymbol{\Gamma}(n-1) ]+\sigma_{v}^{2}\}\mathbf{y}_{\text{o}}(n)\odot\mathbf{y}_{\text{o}}(n),\]where \(\mathbf{1}_{L\times 1}\) denotes a vector with all its elements equal to one and \(\odot\) represents the Hadamard product. In order to verify the stability condition, let us define  
\[\mathbf{M}(n)\triangleq\sigma_{x}^{2}\mathbf{I}_{L}[m(n-1)+\sigma_{w}^{2}(L+2)]+ 2\sigma_{x}^{2}\text{mdiag}[\mathbf{R}_{\text{c}}(n-1)]+\sigma_{v}^{2}\mathbf{I }_{L}, \tag{31}\]  
which leads to \(\mathbf{y}_{\text{o}}(n)=\mathbf{M}^{-1}(n)\boldsymbol{\gamma}(n-1)\).  
If the system is assumed to be time-invariant, i.e., \(\sigma_{w}^{2}=0\), then  
\[m(n) =m(n-1)+\mathbf{y}^{T}(n)[\sigma_{x}^{4}m(n-1)\mathbf{I}_{L}+2 \sigma_{x}^{4}\mathbf{R}_{\text{c}}(n-1)+\mathbf{I}_{L}\sigma_{x}^{2}\sigma_{v }^{2}]\mathbf{y}(n)-2\sigma_{x}^{2}\boldsymbol{\gamma}^{T}(n-1)\mathbf{y}(n)\] \[=m(n-1)+\sigma_{x}^{2}\mathbf{y}^{T}(n)\mathbf{M}(n)\boldsymbol{ \gamma}(n)-2\sigma_{x}^{2}\boldsymbol{\gamma}^{T}(n-1)\mathbf{y}(n)\] \[=m(n-1)-\sigma_{x}^{2}\boldsymbol{\gamma}^{T}(n-1)\mathbf{M}^{- 1}(n)\boldsymbol{\gamma}(n-1). \tag{32}\]  
Since \(\mathbf{M}(n)\) is a diagonal positive definite matrix, \(m(n)<m(n-1)\), therefore,  
\[m(n)=\sum_{l=0}^{L-1}\gamma_{l}(n-1)[1-\sigma_{x}^{2}\mu_{l}(n-1)\gamma_{l}(n -1)], \tag{33}\]  
where \(\gamma_{l}(n)\) denotes the \(l\)th element of the vector \(\boldsymbol{\gamma}(n)\) and  
\[\mu_{l}(n-1)\triangleq\frac{1}{m(n-1)\sigma_{x}^{2}+2\sigma_{x}^{2}\gamma_{l }(n-1)+\sigma_{v}^{2}}. \tag{34}\]  
Based on (33) and (34), we can outline that  
\[\frac{\sigma_{x}^{2}\gamma_{l}(n-1)}{2\sigma_{x}^{2}\gamma_{l}(n-1)+\sigma_{ v}^{2}+\sigma_{x}^{2}\sum_{i=0}^{L-1}\gamma_{i}(n-1)}<1, \tag{35}\]  
which guarantees the convergence. At the same time, the stability condition is verified, i.e., \(0<m(n)<m(n-1)\).