Given a tempered distribution $s \in \mathcal{S}'(\mathbb{R}^{2d})$, define the Weyl pseudodifferential operator of symbol $s$ as the mapping $\mathcal{S}(\mathbb{R}^{d}) \rightarrow \mathcal{S}'(\mathbb{R}^{d})$ determined by $$\left(\textrm{Op}^{\textrm{w}}(s)u\right)(x) := \frac{1}{(2\pi)^{d}} \int_{\mathbb{R}^{d}} \int_{\mathbb{R}^{d}} s\left(\frac{x+x'}{2},\xi\right)e^{i(x-x')\cdot \xi} u(x') dx'd\xi, \quad x \in \mathbb{R}^{d},\ u \in \mathcal{S}(\mathbb{R}^{d}).$$

I understand that if the symbol $s$ belongs to $L^{1}(\mathbb{R}^{2d})$, then the above operator extends to a compact operator in $L^2(\mathbb{R}^{d})$. However, I haven't been able to prove it or to find the proof in any reference. Could you help me prove this statement?