is the Gamma function. Let $r_n=\Delta_v g_n=\q_n\q_n^T$, then we obtain $$f(r_n)=\frac{1}{\Delta_v}f_G(\frac{r_n}{\Delta_v})=\frac{r_n^{\frac{J}{2}-1}\exp\l[-\frac{r_n}{2\Delta_v}\r]}{(2\Delta_v)^{J/2}\Gamma(J/2)},$$ which helps to simplify . Therefore,  can be simplified, $$\label{eq:MMWSE_final}
\begin{split}
& \text{MMWSE}=\beta(1-\rho)\underbrace{\int_{r_n=\tau}^{\infty}\frac{r_n^{\frac{J}{2}-1}\exp\l[-\frac{r_n}{2\Delta_v}\r]}{(2\Delta_v)^{J/2}\Gamma(J/2)}dr_n}_{\Pr(\text{false alarm})}\\
& \quad +(1-\beta)\rho\underbrace{\int_{r_n=0}^{\tau}\frac{r_n^{\frac{J}{2}-1}\exp\l[-\frac{r_n}{2(1+\Delta_v)}\r]}{[2(1+\Delta_v)]^{J/2}\Gamma(J/2)}dr_n}_{\Pr(\text{miss})}.
\end{split}$$

[**Hamming distance**]{}: In digital wireless communication systems, the signal only takes discrete values. A useful error metric is the (per-entry) Hamming distance [@Cover06], which equals 1 if the estimate of an entry of the signal differs from the true value. (Section \[sec:app\] will present an example in wireless communication that minimizes the Hamming distance.) The reader can verify that Hamming distance can be interpreted as a special case of weighted support set error, where $\beta=0.5$ provides equal weight to both errors . That said, we provide more insights about this particular case, which is ubiquitous in communication systems.

For the jointly sparse model in , we define the Hamming distance as $$\label{eq:Hamming}
d_{\text{HD}}(\x_n,\widehat{\x}_n)=\mathbbm{1}_{\x_n\neq \widehat{\x}_n},$$ where $\mathbbm{1}_{\mathcal{A}}$ is the indicator function. If ([*i*]{}) the pdf $\phi(\x_n)$ in  is a $J$-dimensional Dirac-delta function $\delta(\x_n-\mathbf{1})$, where $\mathbf{1}$ is an all-one row vector, and ([*ii*]{}) the estimate satisfies $\widehat{x}_n^{\{1\}}=\cdots =\widehat{x}_n^{\{J\}}, \forall n\in \{1,\ldots,N\}$, then the weighted support set error  with weight $\beta=0.5$ is equal to half of the Hamming distance  for super symbols. In –, 