# On Parameter Estimation for Bandlimited Optical Intensity Channels
## 3 Modified Cramer-Rao Lower Bound  
### Derivation of the Log-Likelihood Function  
The jitter (error) variance of a parameter estimator is a main figure of merit, which is benchmarked by the Cramer-Rao lower bound (CRLB) as the theoretical limit [11; 13]. As already mentioned in the introductory section, little or no information about this topic could be found in the open literature when it comes to the estimation of signal amplitudes and symbol timing for bandlimited OWC links [14].  
According to the signal model specified previously, we are considering a parameter vector \(\mathbf{u}=(u_{1},\,u_{2})=(A,\,\tau)\). Furthermore, we are assuming a data-aided (DA) situation, i.e., the estimation process is based on the knowledge of \(L\) data symbols \(a_{k}\) forming a preamble or pilot sequence, where \(k\in\mathrm{K}=\{k_{0},\,k_{0}+1,\,\,\ldots,\,k_{0}+L-1\}\). Then, if \(\mathbf{r}\) describes the vector representation of (4) over an observation window of \(L\) symbol periods, the CRLB is for unbiased estimation of parameter \(u_{m}\) given by  
\[\mathrm{CRLB}(u_{m})=[\mathbf{J}^{-1}(\mathbf{u})]_{m}, \tag{7}\]  
with \([\cdot]_{m}\) indicating the \(m\)-th diagonal entry of the inverted Fisher information matrix (FIM) expressed by \(\mathbf{J}(\mathbf{u})\). For row \(m\) and column \(n\), the FIM element is formally evaluated as  
\[J_{m,n}\equiv[\mathbf{J}(\mathbf{u})]_{m,n}=-\mathrm{E}\bigg{[}\frac{\partial ^{2}\Lambda(\mathbf{r};\mathbf{u})}{\partial u_{m}\partial u_{n}}\bigg{]}, \tag{8}\]  
where \(\Lambda(\mathbf{r};\mathbf{u})\) is the log-likelihood function (LLF) for the \(L\) observables in \(\mathbf{r}\), and \(\mathrm{E}[\cdot]\) denotes expectation with respect to the AWGN process.  
By detailed inspection of (8), it is clear that the knowledge of the LLF is mandatory for the computation of the CRLB. Nevertheless, if we keep in mind that \(r(t)\) is a white Gaussian process, then the LLF is immediately provided by  
\[\begin{array}{rl}\Lambda(\mathbf{r};\mathbf{u})&=-\frac{1}{2\sigma_{w}^{2}} \int\limits_{-\infty}^{+\infty}\big{[}r(t)-A\cdot s(t-\tau)\big{]}^{2}dt\\ &\sim\frac{1}{2\sigma_{w}^{2}}\int\limits_{-\infty}^{+\infty}\Big{\{}2A\cdot r (t)\,s(t-\tau)-\big{[}A\cdot s(t-\tau)\big{]}^{2}\Big{\}}\,dt\,.\end{array} \tag{9}\]  
Note that immaterial constants and factors not depending on \(\mathbf{u}\) are already omitted in (9). Next, we introduce the definition  
\[z_{k}=\int\limits_{-\infty}^{+\infty}r(t)\,h(t-kT-\tau)\,dt, \tag{10}\]which determines the output of a filter with impulse response \(h(-t)\) evaluated at \(t=kT+\tau\) and fed with \(r(t)\). By detailed inspection of Figure 2, it becomes obvious that this filter is exactly the receiver MF specified by \(q(t)=h(-t)\); for SRC and SDJ shapes, it is clear that \(h(t)=h(-t)\). Moreover, we have  
\[\begin{array}{ll}\int\limits_{-\infty}^{+\infty}\left[s(t-\tau)\right]^{2}dt& =\sum\limits_{k\in\mathbb{K}}\sum\limits_{n\in K}a_{k}a_{n}\int \limits_{-\infty}^{+\infty}h(t-kT-\tau)h(t-nT-\tau)\,dt\\ &=\sum\limits_{k\in\mathbb{K}}\sum\limits_{n\in K}a_{k}a_{n}\,g[(k-n)T]\,,\end{array} \tag{11}\]  
where \(g(t)=h(t)\otimes h(-t)\). Finally, by plugging (10) and (11) into (9), the LLF of our estimation problem is established as  
\[\Lambda(\mathbf{r};\mathbf{u})=\frac{1}{2\sigma_{w}^{2}}\Bigg{(}2A\sum\limits_ {k\in\mathbb{K}}a_{k}z_{k}-A^{2}\sum\limits_{k\in\mathbb{K}}\sum\limits_{n\in \mathbb{K}}a_{k}a_{n}\,g[(k-n)T]\Bigg{)}. \tag{12}\]