Normalized Least Mean Square

$\vec{a}:\text{the unknown vector of the system parameters}$
$\vec{x}:\text{the vector of the input signal }$
$y:\text{the output signal }$

$y=\vec{a}\cdot\vec{x}$

$\vec{a}^{\prime}:\text{the vector of the prior estimated parameters}$

$\text{the estimated output}: y^{\prime}=\vec{a}^{\prime}\cdot\vec{x}$

$\text{error}:e=y-y^{\prime}$

$\vec{a}^*:\text{the vector of the posterior estimated parameters}$

$\text{assuming } y=\vec{a}^*\cdot\vec{x} \text{ and } \vec{a}^*=\vec{a}^{\prime}+\mu\vec{x}$

$e=(\vec{a}^{\prime}+\mu\vec{x})\cdot\vec{x} - \vec{a}^{\prime}\cdot\vec{x}$

$e=\mu\vec{x}\cdot\vec{x}=\mu{\lVert\vec{x}\rVert}^2$

$\mu=\dfrac{e}{{\lVert\vec{x}\rVert}^2}$

$\vec{a}^*=\vec{a}^{\prime}+\dfrac{e}{{\lVert\vec{x}\rVert}^2}\vec{x}$