## 2 Methodology
### Steady-State Performance Analysis
#### 2.3.2 Steady-State Performance  
Before the analysis of the steady-state performance of the algorithm, the following assumptions are given.  
**Assumption 1**.: _The additive noise sequence \(v_{k}\) with variance \(\sigma_{v}^{2}\) is independent and identically distributed (i.i.d) and is independent of the input sequence \(\Gamma_{k}\)._  
**Assumption 2**.: _The order of the adaptive filter is long enough that the prior error \(e_{a,k}\) accords with the Gaussian distribution with a mean value of 0 and is independent of the background noise \(v_{k}\)._  
We define prior and posterior errors as  
\[\varepsilon_{a,k}=\tilde{\Phi}_{k}^{T}\Gamma_{k}, \tag{22}\]  
and  
\[\varepsilon_{p,k}=\tilde{\Phi}_{k+1}^{T}\Gamma_{k}. \tag{23}\]  
According to the previous definitions, the relationship between the estimation error and the prior error at the \(k\)th moment can be presented as:  
\[\varepsilon_{k}=\varepsilon_{a,k}+v_{k}. \tag{24}\]  
We define the steady-state mean square error (MSE) as:  
\[MSE=\lim_{n\to\infty}\mathbf{E}\Big{[}\varepsilon_{k}^{2}\Big{]}. \tag{25}\]  
Then the excess mean square error (EMSE) can be presented as:  
\[EMSE=\lim_{n\to\infty}\mathbf{E}\Big{[}\varepsilon_{a,k}^{2}\Big{]}. \tag{26}\]  
Furthermore, the relationship between MSE and EMSE can be presented as:  
\[MSE=EMSE+\sigma_{v}^{2}. \tag{27}\]In the steady-state performance analysis of adaptive filtering, EMSE is widely used as a measure of steady-state performance, because it can filter out the direct influence of noise on the steady-state performance.  
For an adaptive filtering algorithm which can converge to a steady state, when the number of iterations becomes very large, the estimation error of the filter parameters gets very close. In other words, when \(k\rightarrow\infty\), we can get \(\tilde{\Phi}_{k+1}=\tilde{\Phi}_{k}\). According to (17) and (18), one can obtain:  
\[\mu^{2}\mathbf{E}\Big{[}\|\phi_{GMACC}(\varepsilon_{k})\varepsilon_{k}\Gamma_{k}\| ^{2}\Big{]}=2\mu\mathbf{E}\Big{[}\tilde{\Phi}_{k}^{T}\phi_{GMACC}(\varepsilon_{k} )\varepsilon_{k}\Gamma_{k}\Big{]}. \tag{28}\]  
Assume  
\[f(\varepsilon_{k})=\phi_{GMACC}(\varepsilon_{k})\varepsilon_{k}. \tag{29}\]  
Then, the above equation can be written as  
\[\mu^{2}\mathbf{E}\Big{[}\|f(\varepsilon_{k})\Gamma_{k}\|^{2}\Big{]}=2\mu\mathbf{E}[ \varepsilon_{a,k}f(\varepsilon_{k})]. \tag{30}\]  
In steady state, the distribution of \(  
\[f^{\prime\prime}(v)=\left\{\begin{array}{l}|v|^{\alpha-3}\exp\left(-\left|\frac{p}{ \sigma_{+}}\right|^{\alpha}\right)\left(3\alpha\left|\frac{p}{\sigma_{-}} \right|^{\alpha}-3\alpha\right.\\ \left.-3\alpha^{2}\left|\frac{v}{\sigma_{+}}\right|^{\alpha}+\alpha^{2}+\alpha ^{2}\left|\frac{v}{\sigma_{+}}\right|^{2\alpha}+2\right),\qquad\text{ if }\quad v\geq 0\\ \left.|v|^{\alpha-3}\exp\left(-\left|\frac{p}{\sigma_{-}}\right|^{\alpha} \right)\left(3\alpha\left|\frac{v}{\sigma_{-}}\right|^{\alpha}-3\alpha\right. \\ \left.-3\alpha^{2}\left|\frac{p}{\sigma_{-}}\right|^{\alpha}+\alpha^{2}+\alpha ^{2}\left|\frac{p}{\sigma_{-}}\right|^{2\alpha}+2\right),\qquad\text{ if }\quad v<0\end{array}\right.. \tag{37}\]  
According to the above formulas, the closed-form solution of the EMSE can be obtained given the input sample data. It can be seen that the parameters that affect the steady-state EMSE are step size, noise variance, width and shape parameters of the generalized asymmetric Gaussian kernel.