which reflects the trade-off between the false alarms and misses. Hence, the MWSE given the pseudo data $\q_n$ is $$\label{eq:MWSE}
\begin{split}
    &\text{MWSE}|\q_n=\mathbb{E}[d_{\text{WSE}}(b_n,\widehat{b}_n)|\q_n]=\\
    &\left\{
                \begin{array}{ll}
                 (1-\beta)\Pr(b_n=1|\q_n),&\   \widehat{b}_n=0\ \text{and}\ b_n=1,\\
                 \beta\Pr(b_n=0|\q_n),&\   \widehat{b}_n=1\ \text{and}\ b_n=0,\\
                 0,\quad\quad\quad&\ \widehat{b}_n=b_n,
                \end{array}
            \right.\\
\end{split}$$ where $\Pr(\cdot)$ denotes probability. The optimal estimate $\widetilde{b}_n$ minimizes $\mathbb{E}[d_{\text{WSE}}(b_n,\widehat{b}_n)|\q_n]$ , which implies $$\label{eq:weighted_optB}
    \widetilde{b}_n\!=\!\left\{
                \begin{array}{ll}
                 \!0,&\!   (1-\beta)\Pr(b_n=1|\q_n)\!\leq\! \beta\Pr(b_n=0|\q_n),\\
                 \!1,&\! (1-\beta)\Pr(b_n=1|\q_n)\!>\! \beta \Pr(b_n=0|\q_n).
                \end{array}
            \right.\\$$

Since $f(\q_n|b_n=0)=(2\pi\Delta_v)^{-\frac{J}{2}}\exp\l(-\frac{\q_n\q_n^T}{2\Delta_v}\r)$ and $f(\q_n|b_n=1)=[2\pi(\Delta_v+1)]^{-\frac{J}{2}}\exp\l[-\frac{\q_n\q_n^T}{2(\Delta_v+1)}\r]$, we have $$\Pr(b_n=1|\q_n)=\frac{\frac{\rho}{[2\pi (1+\Delta_v)]^{\frac{J}{2}}}\e^{-\frac{\q_n\q_n^T}{2(1+\Delta_v)}}}{\frac{\rho}{[2\pi (1+\Delta_v)]^{\frac{J}{2}}}\e^{-\frac{\q_n\q_n^T}{2(1+\Delta_v)}}+\frac{(1-\rho)}{(2\pi \Delta_v)^{\frac{J}{2}}}\e^{-\frac{\q_n\q_n^T}{2\Delta_v}}},$$ and $\Pr(b_n=0|\q_n)=1-\Pr(b_n=1|\q_n)$. Plugging $\Pr(b_n=1|\q_n)$ and $\Pr(b_n=0|\q_n)$ into , we have $$\label{eq:b_opt}
    \widetilde{b}_n=\left\{
                \begin{array}{ll}
                 0,&\quad   \q_n\q_n^T\leq\tau,\\
                 1,&\quad   \q_n\q_n^T>\tau,
                \end{array}
            \right.\\$$ where $\tau=2\Delta_v(1+\Delta_v)\ln\l[\frac{\beta(1-\rho)}{(1-\beta)\rho}\l(\frac{1+\Delta_v}{\Delta_v}\r)^{\frac{J}{2}}\r]$, and we remind the reader that $\rho$ is the sparsity rate.

[**Performance limits:**]{} Utilizing  and taking expectation over the pseudo data $\q_n$ for $\mathbb{E}[d_{\text{WSE}}(b_n,\widehat{b}_n)|\q_n]$ , we obtain the minimum MWSE (MMWSE), $$\label{eq:MMWSE_1}
\begin{split}
&\text{MMWSE}=\mathbb{E}[d_{\text{WSE}}(b_n,\widetilde{b}_n)]\\
&\quad =\int_{\q_n\q_n^T>\tau} \beta\Pr(b_n=0|\q_n)f(\q_n)d\q_n +\\
&\quad \quad \int_{\q_n\q_n^T\leq\tau} (1-\beta)\Pr(b_n=1|\q_n)f(\q_n)d\q_n.
\end{split}$$ We have two integrals to simplify in , where we show the first below, and the second can be obtained similarly. To derive the first integral, note that $$\label{eq:oneIntegral}
\small \int_{\q_n\q_n^T>\tau} \Pr(b_n\!=\!0|\q_n)f(\q_n)d\q_n\!=\!\Pr(\q_n\q_n^T\!>\!\tau, b_n\!=\!0).$$ Next, we calculate the pdf of the random variable (RV) $g_n=\frac{\q_n\q_n^T}{\Delta_v}$ given $b_n=0$. Because the entries of $\q_n$ are i.i.d. $\mathcal{N}(0,\Delta_v)$ given $b_n=0$, $g_n$ follows the Chi-square distribution, $f_G(g_n)=\frac{g_n^{\frac{J}{2}-1}\exp(-\frac{g_n}{2})}{2^{\frac{J}{2}}\Gamma(\frac{J}{2})}$, where $\Gamma(\cdot)$ 