$a_n(r,\theta) = \mathrm{e}^{\mathrm{j}\hat{\Theta}n }, a(r,\theta) = [a_0(r,\theta),a_1(r,\theta),\dots,a_{N-1}(r,\theta)]^\mathrm{T}$,
where $\hat{\Theta}n = {2\pi(n((2f_0d\sin\theta)/c)-m_n(2\Delta fr/c)) }$, $\theta$ denotes the angle of the target, $r$ is the slant range of the target, $c$ is the speed of light, $m_n$ is random permutations from 0 to N-1, $n = 0,1,...,N-1$
$$A(r,\theta)=a(r,\theta)a(r,\theta)^\mathrm{H}$$
The N − 1 eigenvectors associated with the N − 1 zero eigenvalues of $A(r,\theta)$ span the noise subspace $G(r,\theta)$.
when $a(\hat{r},\hat{\theta})=a(r,\theta)$, where $\hat{r},\hat{\theta}$ denotes estimates of the range and angle of the target, respectively.
\begin{aligned} &\boldsymbol{a}(\hat{r},\hat{\theta})^\mathrm{H}\boldsymbol{G}(r,\theta)\boldsymbol{G}(r,\theta)^\mathrm{H}\boldsymbol{a}(\hat{r},\hat{\theta}) \\ &=2(N-1)-2\cos[(\hat{\Theta}_1-\hat{\Theta}_0)-(\Theta_1-\Theta_0)] \\ &-2\cos[(\hat{\Theta}_2-\hat{\Theta}_0)-(\Theta_2-\Theta_0)]-\cdots \\ &-2\cos[(\hat{\Theta}_{N-1}-\hat{\Theta}_0)-(\Theta_{N-1}-\Theta_0)]=0 \end{aligned} How did it get this result? What is the detailed mathematical derivation.