I am trying to understand a line from a textbook which expresses the Fisher information matrix in an alternative form assuming Gaussian noise. In the case of Gaussian noise the log likelihood function is
$$L = -N ln(2\pi\sigma^2)-\frac{1} {2\sigma^2}\sum_{n=0}^{N-1}|x_n-\hat{x}_n|^2$$
and the standard two expressions for the information matrix is
$$I_{ji} = E[ \frac{\partial L} {\partial \theta_j} \frac{\partial L} {\partial \theta_i} ] = -E[\frac {\partial ^2L} {\partial \theta _j \partial \theta _i }]$$
This book then says that by subbing in the expression for L into these standard expressions you can get
$$I_{ij,Gauss}=\frac {1} {\sigma^2}\sum_{n=0}^{N-1}Re[\frac {\partial \hat x_n^*} {\partial \theta_j} \frac {\partial \hat x_n} {\partial \theta_i}]$$
I do not see how this expression comes about. I have tried a few different ways but I get substantially different expression. For thos interested this is from pages 210-211 from this textbook:
https://sci-hub.tw/https://link.springer.com/chapter/10.1007/978-3-642-45697-8_7