Appendix A
1.1 Proof of Lemma 1
Proof
: Considering Assumptions 1–2 and Definitions 1–2, (3) can be rewritten as
$$\begin{aligned} {c_1}(n) = {\bar{r}}(n) - x(n). \end{aligned}$$
(A.1)
Then, the respective cost function in this scenario is expressed as
$$\begin{aligned} E\left[ {{c_1}^2\left( n \right) } \right]&= E\left[ {{{\left( {{\bar{r}}(n) - x(n)} \right) }^2}} \right] \nonumber \\&= E\left[ {{{{\bar{r}}}^{2}}(n)} \right] - 2\,E\left[ {{\bar{r}}(n)\,\,x(n)} \right] + E\left[ {{x^2}(n)} \right] . \end{aligned}$$
(A.2)
The optimization problem for this scenario can be obtained by rewriting (4) for (A.2) as
$$\begin{aligned} {\varvec{s^{*}}}= {\texttt {argmin}}_{{\varvec{s}}}\,\,E\big [{c_{1}}^{2}\left( n\right) \big ]. \end{aligned}$$
(A.3)
Substituting (2) in (A.2), we have
$$\begin{aligned} E\left[ {{c_1}^2\left( n \right) } \right]&={\alpha ^2}\,{\mathbf{{s}}^T}(n)\,\mathbf{{h}}\left( n \right) E\left[ {{x^2}\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \,\mathbf{{s}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s}}^T}(n)\mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) \,{\mathbf{{s}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) \,x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s}}^T}(n)\,E\left[ {{{\varvec{\zeta }}}\left( n \right) \,{{{\varvec{\zeta }}}^T}\left( n \right) } \right] \,\mathbf{{s}}\left( n \right) - 2\alpha \,{\mathbf{{s}}^T}(n)\,\mathbf{{h}}\left( n \right) E\left[ {{x^2}\left( n \right) } \right] \nonumber \\&\quad - 2\left( {1 - \alpha } \right) {\mathbf{{s}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) x\left( n \right) } \right] +E\left[ {{x^2}\left( n \right) } \right] \nonumber \\&= {\alpha ^2}\,{\mathbf{{s}}^T}(n)\,\mathbf{{h}}\left( n \right) \,{r_{xx}}\left( n \right) \,{\mathbf{{h}}^T}\left( n \right) \,\mathbf{{s}}\left( n \right) + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s}}^T}(n)\,\mathbf{{R}}\left( n \right) \,\mathbf{{s}}\left( n \right) \nonumber \\&\quad - 2\alpha \,{\mathbf{{s}}^T}(n)\,\mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) + {r_{xx}}\left( n \right) . \end{aligned}$$
(A.4)
Minimizing the cost function in (A.4) by taking the derivative with respect to the shaping filter coefficients \({s_{_i}}\left( n \right) ,i = 0,1,\ldots ,M - 1\) and equating to zero, we have
$$\begin{aligned} 2\,{\alpha ^2}\,\mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) + 2\,{\left( {1 - \alpha } \right) ^2}\,\mathbf{{R}}\left( n \right) \mathbf{{s^{*}}}\left( n \right) - 2\,\alpha \,\mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) = 0. \end{aligned}$$
(A.5)
Simplifying (A.5), we obtain (5), where \(\mathbf{{s^{*}}}\left( n \right) = \big [ s^{*}\left( n \right) ,s^{*}\left( {n - 1} \right) ,\ldots ,s^{*}( n - M + 1 ) \big ]^T\) such that \(\mathbf{{s^{*}}}\left( n \right) \in {\mathbb {R}}^{M\times 1}\) represent the solution to the optimization problem in (A.3). \(\square \)
1.2 Proof of Lemma 2
Proof
: The signal difference expressed in (3) can be rewritten for this scenario as
$$\begin{aligned} {c_2}(n) = {\bar{u}}\left( n \right) - x\left( n \right) . \end{aligned}$$
(A.6)
The cost function can be expressed as
$$\begin{aligned} E\left[ {{c_2}^2\left( n \right) } \right] = E\left[ {{{{\bar{u}}}^{\,\,2}}\left( n \right) } \right] - 2\,E\left[ {{\bar{u}}\left( n \right) \,x\left( n \right) } \right] + E\left[ {{x^2}\left( n \right) } \right] , \end{aligned}$$
(A.7)
and the optimization problem for this scenario can be obtained by rewriting (4) for (A.7) as
$$\begin{aligned} {\varvec{s^{*}}}= {\texttt {argmin}}_{{\varvec{s}}}\,\,E\big [{c_{2}}^{2}\left( n\right) \big ]. \end{aligned}$$
(A.8)
Combining (2) and (8), and substituting in (A.7), we have
$$\begin{aligned} E\left[ {{c_2}^2\left( n \right) } \right]&= E\left[ {{u^{2}}\left( n \right) } \right] + 2\,E\left[ {u\left( n \right) \,{\bar{r}}\left( n \right) } \right] +E\left[ {{{{\bar{r}}}^{2}}\left( n \right) } \right] -2E\left[ {u\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad - 2E\left[ {{\bar{r}}\left( n \right) x\left( n \right) } \right] + E\left[ {{x^2}\left( n \right) } \right] \nonumber \\&= E\left[ {{u^{2}}\left( n \right) } \right] + 2\,\alpha {\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {u\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad + 2\left( {1 - \alpha } \right) {\mathbf{{s}}^T}\left( n \right) E\left[ {u\left( n \right) {{\varvec{\zeta }}}\left( n \right) } \right] \nonumber \\&\quad + {\alpha ^2}\,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^{2}}\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) \,{{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s}}\left( n \right) - 2\,E\left[ {u\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad - 2\alpha {\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^{2}}\left( n \right) } \right] - 2\left( {1 - \alpha } \right) {\mathbf{{s}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) \,x\left( n \right) } \right] \nonumber \\&\quad + E\left[ {{x^2}\left( n \right) } \right] . \end{aligned}$$
(A.9)
Simplifying, we can write
$$\begin{aligned} E\left[ {{c_2}^2\left( n \right) } \right]&= {r_{uu}}\left( n \right) + 2\,\alpha \,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{ux}}\left( n \right) + {\alpha ^2}\,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) {\mathbf{{h}}^T}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s}}^T}\left( n \right) \mathbf{{R}}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad - 2\,{r_{ux}}\left( n \right) - 2\,\alpha \,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) + {r_{xx}}\left( n \right) . \end{aligned}$$
(A.10)
Minimizing the cost function in (A.10) by taking its derivative with respect to the shaping filter coefficients \({s_{_i}}\left( n \right) ,\,i = 0,1,\ldots ,M - 1\) and equating to zero, we have
$$\begin{aligned}&2\,\alpha \,{r_{ux}}\left( n \right) \mathbf{{h}}\left( n \right) + 2\,{\alpha ^2}\,\mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + 2\,{\left( {1 - \alpha } \right) ^2}{} \mathbf{{R}}\left( n \right) \mathbf{{s^{*}}}\left( n \right) - 2\,\alpha \,\mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) = 0. \end{aligned}$$
(A.11)
Simplifying (A.11), we obtain (9), where the optimal set of coefficients \(\mathbf{{s^{*}}}\left( n \right) \) represent the solution to the optimization problem of (A.8), when the acoustic signal from the environment is a time-delayed replica of the wirelessly received signal. \(\square \)
1.3 Proof of Lemma 4
Proof
The cost function for this scenario can be expressed as
$$\begin{aligned} E\left[ {{c_3}^2\left( n \right) } \right]&= E\left[ {{{\left( {{\bar{u}}\left( n \right) - x'\left( n \right) } \right) }^2}} \right] \nonumber \\&= E\left[ {{{{\bar{u}}}^{2}}\left( n \right) } \right] - 2\,E\left[ {{\bar{u}}\left( n \right) x'\left( n \right) } \right] + E\left[ {{{x'}^{2}}\left( n \right) } \right] , \end{aligned}$$
(A.12)
and the optimization problem as
$$\begin{aligned} {\varvec{s^{*}}}= {\texttt {argmin}}_{{\varvec{s}}}\,\,E\big [{c_{3}}^{2}\left( n\right) \big ]. \end{aligned}$$
(A.13)
Combining (2) and (8), and substituting in (A.12), we have
$$\begin{aligned} E\left[ {{c_3}^2\left( n \right) } \right]&= E\left[ {{u^{2}}\left( n \right) } \right] + 2\,E\left[ {u\left( n \right) {\bar{r}}\left( n \right) } \right] + E\left[ {{{{\bar{r}}}^{2}}\left( n \right) } \right] - 2\,E\left[ {u\left( n \right) x'\left( n \right) } \right] \nonumber \\&\quad - 2\,E\left[ {{\bar{r}}\left( n \right) x'\left( n \right) } \right] + E\left[ {{{x'}^{\,\,2}}\left( n \right) } \right] \nonumber \\&= E\left[ {{u^{2}}\left( n \right) } \right] + 2\,\alpha \,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {u\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad + 2\left( {1 - \alpha } \right) {\mathbf{{s}}^T}\left( n \right) E\left[ {u\left( n \right) {{\varvec{\zeta }}}\left( n \right) } \right] \nonumber \\&\quad + {\alpha ^2}\,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^{2}}\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s}}\left( n \right) \nonumber \\&\quad - 2\,E\left[ {u\left( n \right) x'\left( n \right) } \right] + E\left[ {{{x'}^{2}}\left( n \right) } \right] - 2\,\alpha \,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x'\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad - 2\left( {1 - \alpha } \right) {\mathbf{{s}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) x'\left( n \right) } \right] . \end{aligned}$$
(A.14)
Simplifying, we can write
$$\begin{aligned} E\left[ {{c_3}^2\left( n \right) } \right]&= {r_{uu}}\left( n \right) + 2\,\alpha \,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{ux}}\left( n \right) + {\alpha ^2}\,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) {\mathbf{{h}}^T}\left( n \right) \mathbf{{s}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s}}^T}\left( n \right) \mathbf{{R}}\left( n \right) \mathbf{{s}}\left( n \right) - 2\,{r_{ux'}}\left( n \right) - 2\,\alpha \,{\mathbf{{s}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx'}}\left( n \right) + {r_{x'x'}}\left( n \right) . \end{aligned}$$
(A.15)
To reduce the difference between the intended hearing aid output and the actual hearing aid output, the cost function in (A.15) can be minimized by taking its derivative with respect to the shaping filter coefficients \({s_{_i}}\left( n \right) ,i = 0,1,\ldots ,M - 1\) and then equating to zero, we have
$$\begin{aligned}&2\,\alpha \,\mathbf{{h}}\left( n \right) {r_{ux}}\left( n \right) + 2\,{\alpha ^2}\,\mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + 2\,{\left( {1 - \alpha } \right) ^2}\,\mathbf{{R}}\left( n \right) \mathbf{{s^{*}}}\left( n \right) - 2\,\alpha \,\mathbf{{h}}\left( n \right) {r_{xx'}}\left( n \right) = 0. \end{aligned}$$
(A.16)
Simplifying the above equation, we obtain (12), where \(\mathbf{{s^{*}}}\left( n \right) \) represents the solution to the optimization problem in (A.13), when the acoustic inputs are independent of each other and received from two different sources. \(\square \)
Appendix B
1.1 Proof of Lemma 3
Proof
The cost function for \({\bar{G}}\left( k \right) \) optimization can be represented by combining (2), (7) and (8), and substituting in (A.7) as
$$\begin{aligned} E\left[ {{c_2}^2\left( n \right) } \right]&= {\left| {{G_1}} \right| ^2}{{{\bar{g}}}^{2}}\left( n \right) E\left[ {{e^2}\left( n \right) } \right] \nonumber \\&\quad + 2\alpha \left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {e\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad + 2\left( {1 - \alpha } \right) \left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {e\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\alpha ^2}{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^{2}}\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) \,\,x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad - 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {e\left( n \right) x\left( n \right) } \right] - 2\alpha \,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^2}\left( n \right) } \right] \nonumber \\&\quad - 2\left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) x\left( n \right) } \right] + E\left[ {{x^2}\left( n \right) } \right] . \end{aligned}$$
(B.1)
Simplifying the above equation, we can write
$$\begin{aligned} E\left[ {{c_2}^2\left( n \right) } \right]&= {\left| {{G_1}} \right| ^2}{{{\bar{g}}}^2}\left( n \right) \Big \{ E\left[ {v\left( n \right) v\left( n \right) } \right] - 2\,E\left[ {v\left( n \right) y\left( n \right) } \right] + E\left[ {y\left( n \right) y\left( n \right) } \right] \Big \}\nonumber \\&\quad + 2\,\alpha \left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {v\left( n \right) \,x\left( n \right) } \right] {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) \nonumber \\&\quad - 2\,\alpha \left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {y\left( n \right) \,x\left( n \right) } \right] {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) \nonumber \\&\quad + 2\left( {1 - \alpha } \right) \left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {v\left( n \right) \,{{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad - 2\left( {1 - \alpha } \right) \left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {y\left( n \right) \,{{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\alpha ^2}\,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) \,\,x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) \,{{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) \,x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) \,{{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad - 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {v\left( n \right) x\left( n \right) } \right] + 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {y\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad - 2\,\alpha \,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^{\,2}}\left( n \right) } \right] + E\left[ {{x^2}\left( n \right) } \right] \nonumber \\&\quad - 2\left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) \,\,x\left( n \right) } \right] \nonumber \\&= {\left| {{G_1}} \right| ^2}\,{{{\bar{g}}}^2}\left( n \right) \Big [ {{r_{vv}}\left( n \right) - 2\,{r_{vy}}\left( n \right) + {r_{yy}}\left( n \right) } \Big ] \nonumber \\&\quad + 2\,\alpha \left| {{G_1}} \right| {\bar{g}}\left( n \right) {r_{vx}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) \nonumber \\&\quad - 2\,\alpha \left| {{G_1}} \right| {\bar{g}}\left( n \right) {r_{yx}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) \nonumber \\&\quad + {\alpha ^2}\,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{R}}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad - 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) {r_{vx}}\left( n \right) \nonumber + 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) {r_{yx}}\left( n \right) \nonumber \\&\quad - 2\,\alpha \,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) + {r_{xx}}\left( n \right) . \end{aligned}$$
(B.2)
Minimizing the cost function in (B.2) by taking its derivative with respect to the feed-forward path FIR filter coefficients \({{\bar{g}}_{_i}}\left( n \right) ,\,\,i = 0,1,\ldots ,L - 1\) and equating to zero, we have
$$\begin{aligned}&2\left| {{G_1}} \right| ^{2}{\bar{g}}\left( n \right) \Big [ {{r_{vv}}\left( n \right) - 2{r_{vy}}\left( n \right) + {r_{yy}}\left( n \right) } \Big ] + 2\,\alpha \left| {{G_1}} \right| {r_{vx}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) \nonumber \\&\quad - 2\,\alpha \left| {{G_1}} \right| {r_{yx}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) - 2\left| {{G_1}} \right| {r_{vx}}\left( n \right) + 2\left| {{G_1}} \right| {r_{yx}}\left( n \right) = 0. \end{aligned}$$
(B.3)
Simplifying the above equation, we obtain (11), where \({{\bar{\mathbf{g}}}^{*}}\left( n \right) = \Big [ \bar{g}^{*}\left( n \right) ,\bar{g}^{*}\left( {n - 1} \right) ,\ldots ,\bar{g}^{*}\left( {n - L + 1} \right) \Big ]^T\) is the solution to the optimization problem in (10), such that \({{\bar{\mathbf{g}}^{*}}}\left( n \right) \in {\mathbb {R}}^{L\times 1}\), when the acoustic environment input is a delayed version of the signal from the wirelessly transmitting device, and \(\mathbf{{s^{*}}}\left( n \right) \) is the optimal set of coefficients for the shaping filter, as obtained in (9). \(\square \)
1.2 Proof of Lemma 5
Proof
: The cost function for the optimization of \({\bar{G}}\left( k \right) \) can be expressed by combining (2), (7) and (8), and substituting in (A.12) as
$$\begin{aligned} E\left[ {{c_3}^2\left( n \right) } \right]&= \left| {{G_1}} \right| {^2}\,{{{\bar{g}}}^2}\left( n \right) E\left[ {{e^{2}}\left( n \right) } \right] 2\,\alpha \left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {e\left( n \right) x\left( n \right) } \right] \nonumber \\&\quad + 2\left( {1 - \alpha } \right) \left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {e\left( n \right) {{\varvec{\zeta }}}\left( n \right) } \right] \nonumber \\&\quad + {\alpha ^2}\,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^{2}}\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) - 2\,E\left[ {u\left( n \right) x'\left( n \right) } \right] \nonumber \\&\quad - 2\,\alpha \,{\mathbf{{s^{*}}}^T}\,\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x'\left( n \right) x\left( n \right) } \right] - 2\left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) x'\left( n \right) } \right] \nonumber \\&\quad + E\left[ {{{x'}^{2}}\left( n \right) } \right] . \end{aligned}$$
(B.4)
Simplifying the above equation, we can write
$$\begin{aligned} E\left[ {{c_3}^2\left( n \right) } \right]&= {\left| {{G_1}} \right| ^2}{{{\bar{g}}}^{2}}\left( n \right) \Big \{ E\left[ {v\left( n \right) v\left( n \right) } \right] - 2\,E\left[ {v\left( n \right) y\left( n \right) } \right] + E\left[ {y\left( n \right) y\left( n \right) } \right] \Big \}\nonumber \\&\quad + 2\,\alpha \,\left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {v\left( n \right) \,x\left( n \right) } \right] \nonumber \\&\quad - 2\,\alpha \,\left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {y\left( n \right) \,x\left( n \right) } \right] \nonumber \\&\quad + 2\left( {1 - \alpha } \right) \left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {v\left( n \right) \,{{\varvec{\zeta }}}\left( n \right) } \right] \nonumber \\&\quad - 2\left( {1 - \alpha } \right) \left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {y\left( n \right) \,{{\varvec{\zeta }}}\left( n \right) } \right] \nonumber \\&\quad + {\alpha ^2}\,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {{x^2}\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + \alpha \left( {1 - \alpha } \right) {\mathbf{{s^{*}}}^T}(n)E\left[ {{{\varvec{\zeta }}}\left( n \right) x\left( n \right) } \right] {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) {{{\varvec{\zeta }}}^T}\left( n \right) } \right] \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad - 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {v\left( n \right) x'\left( n \right) } \right] + 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) E\left[ {y\left( n \right) x'\left( n \right) } \right] \nonumber \\&\quad - 2\,\alpha \,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) E\left[ {x\left( n \right) \,\,x'\left( n \right) } \right] - 2\,\alpha \,{\mathbf{{s^{*}}}^T}\left( n \right) E\left[ {{{\varvec{\zeta }}}\left( n \right) x'\left( n \right) } \right] \nonumber \\&\quad + E\left[ {{{x'}^{2}}\left( n \right) } \right] \nonumber \\&= {\left| {{G_1}}\right| ^2}\,{{{\bar{g}}}^2}\left( n \right) \Big [ {{r_{vv}}\left( n \right) - 2{r_{vy}}\left( n \right) + {r_{yy}}\left( n \right) } \Big ] \nonumber \\&\quad + 2\,\alpha \,\left| {{G_1}}\right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{vx}}\left( n \right) \nonumber \\&\quad - 2\,\alpha \left| {{G_1}} \right| {\bar{g}}\left( n \right) {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{yx}}\left( n \right) \nonumber \\&\quad + {\alpha ^2}\,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx}}\left( n \right) {\mathbf{{h}}^T}\left( n \right) \mathbf{{s^{*}}}\left( n \right) \nonumber \\&\quad + {\left( {1 - \alpha } \right) ^2}{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{R}}\left( n \right) \mathbf{{s^{*}}}\left( n \right) - 2\left| {{G_1}} \right| {\bar{g}}\left( n \right) {r_{vx'}}\left( n \right) \nonumber \\&\quad + 2\,\left| {{G_1}} \right| {\bar{g}}\left( n \right) {r_{yx'}}\left( n \right) - 2\,\alpha \,{\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{xx'}}\left( n \right) + {r_{x'x'}}\left( n \right) . \end{aligned}$$
(B.5)
Minimizing the cost function in (B.5) by taking its derivative with respect to the feed-forward path FIR filter coefficients \({{\bar{g}}_{_i}}\left( n \right) ,i = 0,1,\ldots ,L - 1\) and equating to zero, we have
$$\begin{aligned}&2\left| {{G_1}} \right| {^2}{\bar{g}}\left( n \right) \left[ {{r_{vv}}\left( n \right) - 2{r_{vy}}\left( n \right) + {r_{yy}}\left( n \right) } \right] + 2\,\alpha \left| {{G_1}} \right| {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{vx}}\left( n \right) \nonumber \\&\quad - 2\,\alpha \left| {{G_1}} \right| {\mathbf{{s^{*}}}^T}\left( n \right) \mathbf{{h}}\left( n \right) {r_{yx}}\left( n \right) - 2\left| {{G_1}} \right| {r_{vx'}}\left( n \right) + 2\left| {{G_1}} \right| {r_{yx'}}\left( n \right) = 0. \end{aligned}$$
(B.6)
Simplifying the above equation, we obtain (13), where \({{\bar{\mathbf{g}}}^{*}}\left( n \right) = \Big [\bar{g}^{*}\left( n \right) ,\bar{g}^{*}\left( {n - 1} \right) ,\ldots ,\bar{g}^{*}\left( {n - L + 1} \right) \Big ]^T\) is the solution to the optimization problem in (10) for \({\bar{G}}\left( k \right) \) . \(\square \)
