I am studying the MIMO Radar System from the book MIMO Radar Signal Processing. The derivation about FIM for widely separated antennas system in Chapter 9 (9A.6) makes me confused, since I have no idea how the derivation result is obtained.
The pdf $p_\theta(\mathbf{r})$ is defind as exp{$-\frac{1}{\sigma_w^2}\sum_{\mathbb{\ell}=1}^N \int_T |r_\mathbb{\ell}(t)-\zeta\sqrt{\frac{E}{M}}\sum_{\mathbb{k}=1}^M\rho_\mathbb{\ell k}(X)s_\mathbb{k}(t-\tau_\mathbb{\ell k}(X))|^2 dt$},
where $\sigma_w^2$ is the variance of the noise, $T$ denotes the duration of the radar waveform, $\zeta$ represents the conplex constant caused by unknown reflectivity, $\mathbb{\ell} and \mathbb{k}$ are the index number of receiver and transmitter antenna separately. $\rho_\mathbb{\ell k}(X) = exp(-j2\pi f_c \tau_\mathbb{\ell k}(X))$ is the phase term, $X=(x,y)$ is the unknown target location. $\mathbf{\theta} = [x,y,\zeta_{real}, \zeta_{imag}]$ is the unknown vector.
The appendix 9A of the book uses chain rule to convert the FIM into another form $\mathbf{I}(\theta) = (\nabla_\theta \vartheta^H) \mathbf{I}(\vartheta) (\nabla_\theta \vartheta^H)^H$ where $\vartheta = [\tau_{11}(X),...,\tau_\mathbb{\ell k}(X),...,\tau_{NM}(X),\zeta_{real},\zeta_{imag}]$. Therefore, the core of the FIM calculation is to do the second order derivation on the pdf with $\tau$s and $\zeta$s. I know that for a function $f(x)=ag(x)$, its first order derivation should be $ag(x)\times\frac{\partial g(x)}{\partial x}$, which is similar to the first and second part of (9A.6). However, I have no idea why there should be conjugate transposes and the sum of these two parts. I do not know if I miss some theorems or lemmas which can figure out the derivation.