Table 1 The SC-RPAPA algorithm with MRAP

Sparseness

control

\(\hat {\epsilon }(n)=\frac {L}{L-\sqrt {L}}\left (1-\frac {\Vert \hat {\mathbf {h}}(n-1)\Vert _{1}}{\sqrt {L}\Vert \hat {\mathbf {h}}(n-1)\Vert _{2}}\right)\)

\(\mathrm {F}(|\hat {h}_{l}|)=\frac {|\hat {h}_{l}|}{|\hat {h}_{l}|+{\text {max}}\{\hat {\epsilon }(n)-0.4,\epsilon _{\text {min}} \}\sigma _{\text {max}}}\)

\(r_{l}=\text {max}\{\rho \text {max}\{q,\mathrm {F}(|\hat {h}_{0}|),\ldots,\mathrm {F}(|\hat {h}_{L-1}|)\}, \mathrm {F}(|\hat {h}_{l}|)\} \)

\(g_{l}(n-1)=\frac {r_{l}(n-1)}{\frac {1}{L}\sum _{i=0}^{L}r_{l}(n-1)}\)

g(n−1)=[g ₀(n−1),g ₁(n−1),…,g _L−1(n−1)]^T

Memory

update

\(\mathbf {P}^{\prime }(n)=[\mathbf {g}(n-1)\odot \mathbf {x}(n),\mathbf {P}^{\prime }_{-1}(n-1)]\)

\(\mathbf {p}(n)=[\mathbf {x}^{T}(n)\mathbf {P}^{\prime }_{0}(n),\mathbf {p}_{-1}(n-1)]\)

Error output

\(e(n)=d(n)-\mathbf {x}^{T}(n-m)\hat {\mathbf {h}}(n-1)\)

RAP

iteration

\(\hat {\mathbf {h}}^{[0]}=\hat {\mathbf {h}}(n-1)\)

for m=0,1,…,M−1

α(m)=μ/(p _m (n)+δ)

\(\hspace {26pt} e^{[m]}=d(n-m)-\mathbf {x}^{T}(n-m)\hat {\mathbf {h}}^{[m]}\)

\(\hspace {26pt} \hat {\mathbf {h}}^{[m+1]}=\hat {\mathbf {h}}^{[m]}+\alpha (m)\mathbf {P}^{\prime }_{m}(n)e^{[m]}\)

m=m+1

Filter update

\(\hat {\mathbf {h}}(n)=\hat {\mathbf {h}}^{[M]}\)