Define a unitary matrix $\mathbf{U}=\left[\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_N\right] \in$ $\mathbb{U}(N)$, where $\mathbb{U}(N)$ denotes the unitary group of degree $N$. $$ \mathbf{x}=\mathrm{U}[s_1,s_2,\cdots,s_N]^T $$ where $s_1,s_2,\cdots,s_N$ are independent and equally distributed but not Gaussian random variables with an expectation of 0 , and $$ \mathbb{E}\left(\left|s_n\right|^2\right)=1, \quad \forall n $$ $$ \mathbb{E}\left(s_n\right)=0, \quad \mathbb{E}\left(s_n^2\right)=0, \quad \forall n $$ $$ \mu_4 \triangleq \mathbb{E}\left(\left|s_n\right|^4\right) $$. $$ \tilde{r}_k=\mathrm{x}^H \tilde{\mathbf{J}}_k \mathrm{x}=\tilde{r}_{-k}^*, \quad k=0,1, \ldots, N-1 $$ where $\tilde{\mathbf{J}}_k$ is defined as the $k$ th periodic shift matrix, given as $$ \tilde{\mathbf{J}}_k=\left[\begin{array}{cc} 0 & \mathbf{I}_{N-k} \\ \mathbf{I}_k & 0 \end{array}\right] $$ and $$ \tilde{\mathbf{J}}_{-k}=\tilde{\mathbf{J}}_k^T=\left[\begin{array}{cc} 0 & \mathbf{I}_k \\ \mathbf{I}_{N-k} & 0 \end{array}\right] $$ $$ \mathrm{EISL}=\sum_{k=1}^{N-1} \mathbb{E}\left(\left|\tilde{r}_k\right|^2\right)=\sum_{k=1}^{N-1} \mathbb{E}\left(\left|\mathbf{s}^H \mathbf{U}^H \tilde{\mathbf{J}}_k \mathbf{U s}\right|^2\right) $$ To prove that $$ \sum_{k=1}^{N-1} \mathbb{E}\left(\left|\tilde{r}_k\right|^2\right)=N(N-1)+\left(\mu_4-2\right) N(\left\|\mathbf{F}_N \mathbf{U}\right\|_4^4-1) $$