# An Optimized Differential Step-Size LMS Algorithm
## 3 Autocorrelation Matrix of the Coefficients Error  
In this framework, we assume that the input signal is wide-sense stationary. Therefore, its autocorrelation matrix is given by \(\mathbf{R}=\mathrm{E}[\mathbf{x}(n)\mathbf{x}^{T}(n)]\). The autocorrelation matrix of the coefficients error is  
\[\mathbf{R_{c}}(n)\triangleq\mathrm{E}[\mathbf{c}(n)\mathbf{c}^{T}(n)]. \tag{6}\]  
In addition, we denote \(\mathbf{\Gamma}(n)\triangleq\mathbf{R_{c}}(n)+\sigma_{w}^{2}\mathbf{I}_{L}\) and \(\gamma(n)\triangleq\mathrm{diag}[\mathbf{\Gamma}(n)]\), where \(\sigma_{w}^{2}\) is the variance of the system noise and \(\mathrm{diag}(\cdot)\) defines a vector retaining only the main diagonal elements of \(\mathbf{\Gamma}(n)\). Going forward, Equation (5) is developed by applying the \(\ell_{2}\) norm and the statistical expectation, so that we can introduce the notation:  
\[m(n)=\mathrm{E}\left[\left\|\mathbf{c}(n)\right\|_{2}^{2}\right]=\mathrm{tr}[ \mathbf{R_{c}}(n)], \tag{7}\]where \(\text{tr}(\cdot)\) denotes the trace of a square matrix. Substituting (5) into (6), the autocorrelation matrix of the coefficients error results in  
\[\mathbf{R_{c}}(n) =\text{E}\{[\mathbf{I}_{L}-\mathbf{Y}(n)\mathbf{x}(n)\mathbf{x}^{T }(n)][\mathbf{c}(n-1)-\boldsymbol{\delta_{\text{w}}}(n)]\] \[\quad\times[\mathbf{c}^{T}(n-1)-\boldsymbol{\delta_{\text{w}}}^{ T}(n)][\mathbf{I}_{L}-\mathbf{x}(n)\mathbf{x}^{T}(n)\mathbf{Y}(n)]\] \[\quad+\mathbf{Y}(n)\mathbf{x}(n)\mathbf{x}^{T}(n)\mathbf{Y}(n)v^ {2}(n)\}. \tag{8}\]  
At this point, we consider that \(\boldsymbol{\delta_{\text{w}}}(n)\) is uncorrelated with the a posteriori misalignment at the discrete time index \(n-1\) and with the input signal at the discrete time index \(n\). In addition, we assume that \(\mathbf{c}(n-1)\) and \(\mathbf{x}(n)\) are uncorrelated. The zero-mean measurement noise, \(v(n)\), is independent of the input signal, \(\mathbf{x}(n)\), and of the system noise, \(\boldsymbol{\delta_{\text{w}}}(n)\). The variance of the additive noise is \(\sigma_{v}^{2}=\text{E}[v^{2}(n)]\). With these assumptions, the statistical expectation of the product term in (8) can be written as  
\[\text{E}\{[\mathbf{I}_{L}-\mathbf{Y}(n)\mathbf{x}(n)\mathbf{x}^{ T}(n)][\mathbf{c}(n-1)-\boldsymbol{\delta_{\text{w}}}(n)][\mathbf{c}^{T}(n-1)- \boldsymbol{\delta_{\text{w}}}^{T}(n)][\mathbf{I}_{L}-\mathbf{x}(n)\mathbf{x} ^{T}(n)\mathbf{Y}(n)]\}\] \[=\mathbf{R_{c}}(n-1)+\mathbf{I}_{L}\sigma_{w}^{2}-[\mathbf{R_{c} }(n-1)+\mathbf{I}_{L}\sigma_{w}^{2}]\mathbf{RY}(n)-\mathbf{Y}(n)\mathbf{R}[ \mathbf{R_{c}}(n-1)+\mathbf{I}_{L}\sigma_{w}^{2}]+\mathbf{Y}(n)\mathbf{Q}(n) \mathbf{Y}(n), \tag{9}\]  
where  
\[\mathbf{Q}(n) \triangleq\text{E}\{[\mathbf{x}(n)\mathbf{x}^{T}(n)][\mathbf{c}( n-1)-\boldsymbol{\delta_{\text{w}}}(n)][\mathbf{c}^{T}(n-1)-\boldsymbol{\delta_{ \text{w}}}^{T}(n)][\mathbf{x}(n)\mathbf{x}^{T}(n)]\}\] \[=\text{E}[\mathbf{x}(n)\mathbf{x}^{T}(n)\mathbf{p}(n)\mathbf{p}^ {T}(n)\mathbf{x}(n)\mathbf{x}^{T}(n)]\] \[=\text{E}\Big{\{}[\mathbf{p}^{T}(n)\mathbf{x}(n)]^{2}\mathbf{x}( n)\mathbf{x}^{T}(n)\Big{\}}\] \[m(n)=\mathrm{E}\left[\left\|\mathbf{c}(n)\right\|_{2}^{2}\right]=\mathrm{tr}[ \mathbf{R_{c}}(n)], \tag{7}\]0  
and  
\[m(n)=\mathrm{E}\left[\left\|\mathbf{c}(n)\right\|_{2}^{2}\right]=\mathrm{tr}[ \mathbf{R_{c}}(n)], \tag{7}\]1 \[m(n)=\mathrm{E}\left[\left\|\mathbf{c}(n)\right\|_{2}^{2}\right]=\mathrm{tr}[ \mathbf{R_{c}}(n)], \tag{7}\]2  
Including Equation (11) in (10), we can express the elements of the matrix \(\mathbf{Q}(n)\) as  
\[m(n)=\mathrm{E}\left[\left\|\mathbf{c}(n)\right\|_{2}^{2}\right]=\mathrm{tr}[ \mathbf{R_{c}}(n)], \tag{7}\]3  
for \(r,l=0,1,\ldots,L-1\).