Algorithm 1 Summary of proposed HSAF–APA–Fair algorithm with adaptive step-size approach

Initial value: $\tilde{\mathbf{w}}\left(0\right)={[10\cdots 0]}^T$ ${\tilde{\mathbf{q}}}_{i,n}\left(0\right)={[10\cdots 0]}^T$ and $\varrho =1\times {10}^{-7}$ Fixed parameter: $0<\alpha <1$ For $n=1,2,\dots ,N$ To calculate ${\mathbf{s}}_n$ as $$\begin{array}{cc}\hfill s_n& ={\mathbf{u}}_{n,1}^T\mathrm{C}{\tilde{\mathbf{q}}}_{i,n},\hfill \\ \hfill {\mathbf{s}}_n& ={\left[s_n{s}_{n-1}\dots s_{n-N+1}\right]}^T\hfill \end{array}$$ To compute ${\mathbf{u}}_n$ and i as $$\begin{array}{cc}\hfill {\mathbf{u}}_n& ={\left[u_n^3{u}_n^2{u}_{n}1\right]}^T\hfill \\ \hfill u_n& ={\displaystyle \frac{x_n}{\Delta x}}-⌊{\displaystyle \frac{x_n}{\Delta x}}⌋\hfill \\ \hfill i& =⌊{\displaystyle \frac{x_n}{\Delta x}}⌋+{\displaystyle \frac{Q-1}{2}}\hfill \end{array}$$ To compute ${\tilde{e}}_n$ as $$\begin{array}{c}\hfill {\tilde{e}}_n=d_n-{\tilde{\mathbf{w}}}_{n-1}^T{\mathbf{s}}_n\end{array}$$ To determine ${\tilde{\mathbf{q}}}_{i,n}$ as: $$\begin{array}{cc}\hfill {\tilde{\mathbf{q}}}_{i,n}& ={\tilde{\mathbf{q}}}_{i,n-1}+{\displaystyle \frac{{\tilde{\mu }}_{q_n}}{{\mathbf{X}}_n{\mathbf{X}}_n^T}}\left\{{\displaystyle \frac{\alpha {\mathbf{u}}_n^T\mathbf{C}{\tilde{\mathbf{w}}}_n{\tilde{e}}_n}{\alpha + |{\tilde{e}}_n|}}\right\}\hfill \\ \hfill {\mathbf{X}}_n& =\left[{\mathbf{x}}_n{\mathbf{x}}_{n-1}\dots {\mathbf{x}}_{n-N+1}\right]\hfill \end{array}$$ To obtain ${\tilde{\mu }}_{q_n}$ as $$\begin{array}{cc}\hfill {\tilde{\mu }}_{q_n}& ={\beta }_q{\tilde{\mu }}_{q_{n-1}}+{\rho }_q{\mathbf{g}}_{q_{n-1}}^T{\mathbf{g}}_{q_n}{\zeta }_n^2\hfill \\ \hfill {\mathbf{g}}_{q_n}& =\psi {\mathbf{g}}_{q_{n-1}}+(1-\psi ){\mathbf{u}}_n^T\mathbf{C}{\tilde{\mathbf{w}}}_n{\tilde{e}}_n\hfill \\ \hfill {\zeta }_n& =\sigma {\zeta }_{n-1}+(1-\sigma ){\tilde{e}}_n^2\hfill \end{array}$$ To calculate ${\tilde{\mathbf{w}}}_n$ $$\begin{array}{cc}\hfill {\tilde{\mathbf{w}}}_n& ={\tilde{\mathbf{w}}}_{n-1}+{\displaystyle \frac{{\tilde{\mu }}_{w_n}}{{\mathbf{S}}_n{\mathbf{S}}_n^T}}\left\{{\displaystyle \frac{\alpha {\mathbf{s}}_n{\tilde{e}}_n}{\alpha + |{\tilde{e}}_n|}}\right\}\hfill \\ \hfill {\mathbf{S}}_n& =\left[{\mathbf{s}}_n{\mathbf{s}}_{n-1}\dots {\mathbf{s}}_{n-N+1}\right]\hfill \end{array}$$ To obtain ${\tilde{\mu }}_{w_n}$ as $$\begin{array}{cc}\hfill {\tilde{\mu }}_{w_n}& ={\beta }_w{\tilde{\mu }}_{w_{n-1}}+{\rho }_w{\mathbf{g}}_{w_{n-1}}^T{\mathbf{g}}_{w_n}{\zeta }_n^2\hfill \\ \hfill {\mathbf{g}}_{w_n}& =\psi {\mathbf{g}}_{w_{n-1}}+(1-\psi ){\mathbf{s}}_n{\tilde{e}}_n\hfill \end{array}$$ end