# On Parameter Estimation for Bandlimited Optical Intensity Channels
## 2 Signal and Channel Model  
In the sequel, we assume that the real-valued data symbols \(a_{k}\), \(k\in\mathbb{Z}\), are independent and identically distributed (i.i.d.) elements of an \(M\)-ary pulse amplitude modulation (PAM) alphabet A. For convenience reasons, the alphabet is organized such that the symbols are normalized to unit energy, i.e., \(a_{k}\in\mathrm{A}=\{\,0,\ 1,\ \ldots,\ M-1\,\}/\sqrt{\eta_{M}}\), where \(\eta_{M}=\frac{1}{6}(M-1)(2M-1)\). This means that the average value is given by  
\[\mu_{a}=\frac{M-1}{2\sqrt{\eta_{M}}}=\sqrt{\frac{3\,(M-1)}{2\,(2M-1)}}. \tag{1}\]  
Furthermore, it is assumed that the pulses in the transmitter station are either shaped by a squared raised cosine (SRC) or a squared double jump (SDJ) function [6] specified as  
\[h(t)=\frac{1}{\sqrt{T}}\ \left\{ \begin{array}{ccc}\left(\sin c(t/T)\ \frac{\cos(\pi\,\mathrm{d}t/T)}{1-(2\mathrm{d}t/T)^{2}}\right)^{2}&\ldots& \mathrm{SRC}\\ \left(\frac{1-\alpha}{2}\sin c[(1-\alpha)t/T]\ +\frac{1+\alpha}{2}\sin c[(1+ \alpha)t/T]\right)^{2}&\ldots&\mathrm{SDJ}\end{array} \right. \tag{2}\]  
where \(\mathrm{sinc}(x)=\sin(\pi x)/(\pi x)\), \(0\leq\alpha\leq 1\) is the excess bandwidth (roll-off factor), and \(T\) stands for the symbol period the OWC link is operated with. Note that SRC and SDJ shapes satisfy both the non-negativity and the Nyquist criteria, exemplified in Figure 1 for \(\alpha\in\{0.0,\,0.5,\,1.0\}\). Note also that SRC and SDJ are equivalent for \(\alpha=0\), which represents the minimum bandwidth (MBW) case. By detailed inspection of the plot, we observe that the width of the main lobe decreases when \(\alpha\) is increased and that this effect is more pronounced for SDJ scenarios. No matter which sort of pulse shape is finally implemented, the generated transmitter signal can be written as  
\[s(t)=\ \sum_{k}a_{k}\,h(t-kT). \tag{3}\]By introduction of the electro-optical conversion factor \(K_{0}\), the irradiated signal at the output of the intensity modulator is given as \(I_{0}(t)=K_{0}\,s(t)\). If we adopt an observation window for the estimation procedure, which is small enough so that the channel state does not change significantly over this period of time, then fading effects need not be taken into account. This means that the optical signal of interest at the receiver is only affected by some propagation loss and a delay in time, henceforth denoted by \(K_{l}\) and \(\tau\), respectively; as a result, the transfer function of the channel is obtained as \(\lambda(t)=K_{l}\,\delta(t-\tau)\), where \(\delta(\cdot)\) symbolizes [PERSON]'s delta function. Therefore, the optical signal intensity at the input of the receiver terminal is simply described as \(I_{1}(t)=\lambda(t)\,\otimes\,I_{0}(t)=K_{l}\,K_{0}\,s(t-\tau)\), where \(\otimes\) denotes the convolutional operator. Specifying in the sequel the detector responsivity as \(R_{d}\), the electrical signal at the detector output is determined by \(x(t)=R_{d}\,\,I_{1}(t)=A\cdot s(t-\tau)\), where \(A=R_{d}\,\,K_{l}\,\,K_{0}\) is the gain factor characterizing the OWC link. Finally, we have to consider that \(x(t)\) is distorted by additive white Gaussian noise (AWGN), in the following denoted as \(w(t)\), with zero mean and variance \(\sigma_{w}^{2}\). Putting these pieces together, the receiver signal can be expressed as  
\[r(t)=x(t)+\,w(t)=A\cdot s(t-\tau)+\,w(t). \tag{4}\]  
However, before being treated in further stages of operation, the signal in (4) has to pass the receiver filter \(q(t)\), whose output \(z(t)=q(t)\,\otimes\,r(t)\) is appropriately sampled to avoid alias effects. For convenience reasons, the signal and channel model used for analytical and simulation work is summarized in Figure 2.  
Before we proceed to the computation of the Cramer-Rao lower bound in the next section, it makes sense to introduce the average optical power as \(P_{0}=\mu_{a}\,\overline{h}\), where  
\[\overline{h}=\frac{1}{\sqrt{\overline{f}}}\int\limits_{-\infty}^{\infty}h(t) \,dt=\left\{