For non-sparse response, we have proposed an IMPNLMS + algorithm [7] by applying time variable parameter *μ*, which does not perform worse than NLMS even for dispersive channels. It can be applied in time-varying environment as well. In this section, the IMPNLMS + algorithm is recalled.

The steepest descent algorithm with step-size control matrix using μ-law [4] in IMPNLMS + algorithm can be written as:

\stackrel{\wedge}{w}\left(k+1\right)=\stackrel{\wedge}{w}\left(k\right)+\frac{\mathit{\beta G}\left(k+1\right)x\left(k\right)e\left(k\right)}{{x}^{T}\left(k\right)G\left(k+1\right)x\left(k\right)+\delta}

(1)

The step-size control matrix (*L* x *L*):

G\left(k+1\right)=\mathrm{diag}\left\{\begin{array}{cccc}\hfill {g}_{{}_{1}}\left(k+1\right)\hfill & \hfill {g}_{{}_{2}}\left(k+1\right)\hfill & \hfill \dots \hfill & \hfill {g}_{{}_{L}}\left(k+1\right)\hfill \end{array}\right\}

(2)

The *l* th coefficient *g*_{
l
} (*k*) has been presented in [6]:

{g}_{l}\left(k\right)=\frac{1-\alpha \left(k\right)}{2N}+\frac{\left(1+\alpha \left(k\right)\right)F\left(\left|{\stackrel{\wedge}{w}}_{{}_{l}}\left(k\right)\right|\right)}{2{\displaystyle {\u2225F\left(\left|{\stackrel{\wedge}{w}}_{{}_{l}}\left(k\right)\right|\right)\u2225}_{1}}+\mathit{\u03f5}}

(3)

Where, logarithmic function in IMPNLMS + differs from [6]:

F\left(\left|{\stackrel{\wedge}{w}}_{{}_{l}}\left(k\right)\right|\right)=\mathrm{l}n\left(1+\mathrm{\mu}\left(k\right)\left|{\stackrel{\wedge}{w}}_{{}_{l}}\left(k\right)\right|\right)

(4)

Here, *μ* (*k*) [8] is a time variable parameter instead of constant:

\mathrm{\mu}\left(k\right)=\frac{1}{\mathit{\u03f5}\left(k\right)}

(5)

\mathit{\u03f5}\left(k\right)=\sqrt{\frac{\gamma \left(k\right)}{L{\displaystyle {\lambda}_{x}^{2}}}}

(6)

\gamma \left(k\right)=\mathit{\eta \gamma}\left(k-1\right)+\left(1-\eta \right){e}^{2}\left(k-1\right)

(7)

The parameter *α* (*k*) in IMPNLMS + can be described as:

\alpha \left(k\right)=2\xi \left(k\right)-1

(8)

\xi \left(k\right)=\left(1-\rho \right)\xi \left(k-1\right)+\rho {\displaystyle {\xi}_{w}}\left(k\right),\phantom{\rule{1.8em}{0ex}}0<\rho \le 1

(9)

{\displaystyle {\xi}_{w}}\left(k\right)=\frac{L}{L-\sqrt{L}}\left(1-\frac{{\left|\left|\stackrel{\wedge}{\mathit{w}}\left(k\right)\right|\right|}_{1}}{\sqrt{L}{\left|\left|\stackrel{\wedge}{\mathit{w}}\left(k\right)\right|\right|}_{2}}\right)

(10)

In IMPNLMS + algorithm, thanks to the adaptation of *μ* (*k*), the algorithm is more flexible to minimize the MSE related to the time-varying *μ* (*k*). The IMPNLMS + algorithm can achieve better convergence even in time-varying environment where the echo path changes obviously. However, the computation of step-size control matrix with μ-law in IMPNLMS + algorithm is expensive. In the next section, we analyze the computational complexity based on IMPNLMS + algorithm. Non-normalized technology is discussed to reduce the computational complexity of IMPNLMS+, which can be also applied to all proportionate NLMS algorithms.