# On Parameter Estimation for Bandlimited Optical Intensity Channels
## 5 Numerical Results  
In the sequel, the mean estimator value (MEV) and the jitter (error) variance are discussed as the main figures of merit, when it comes to the estimation of a transmission parameter \(u_{m}\). The former is defined as the average value of the estimates generated by the algorithm selected for this purpose. From a practical point of view, it is most helpful if this is achieved without bias, i.e., \(u_{m}=\mathrm{E}[\hat{u}_{m}]\). On the other hand, for an efficient recovery algorithm, the related jitter variance should be as close to the MCRLB as possible. For reasons of clarity and readability, analytical and simulation results will be indicated by lines and markers in different styles. Moreover, if not stated otherwise, the following parameterization is employed:  
* The observation window consists of \(L=100\) symbol periods. * We focus on the investigation of 4-PAM constellations, but the basic findings apply to \(M\)-ary PAM signals in general. * The excess bandwidth of SRC or SDJ shapes is given by \(\alpha\in[0.0,\,0.5,\,1.0]\). * An oversampling factor \(N_{s}=4\) is selected; this avoids alias effects and keeps the computational load as small as possible. However, it turned out that this might be problematic for larger values of the excess bandwidth, because in this case the main lobe of the pulse shape becomes smaller (see Figure 1) so that it is not resolved fine enough; this results in a bias effect for timing estimates,as could be verified by simulation results. Hence, for values of \(\alpha\geq 0.5\), it is suggested to apply \(N_{s}=8\) for SRC as well as SDJ shapes. * For convenience reasons, the timing offsets \(\tau\) and amplitude gains \(A\) are normalized to the nominal values of symbol period and signal amplitude, \(T_{0}\) and \(A_{0}\), respectively, as such they will be denoted by \(\varepsilon=\tau/T_{0}\) and \(\rho=A/A_{0}\). * Results achieved by a matched filter or a flat filter receiver are indicated by MF and FF, respectively. Assuming a reference index \(\mu=0\) for symbol timing recovery in the range of the normalized symbol period, i.e., \(\left|\varepsilon\right|\leq 1/2\), Figure 3 shows the MEV for different values of the roll-off factor applied to SRC shapes using an MF receiver. We can see that the estimator exhibits no bias, indicated by the ideal curve in dashed style. Although not shown in Figure 3 to avoid an overload of the diagram, it could be verified that this also holds true for an FF receiver or if pulses are shaped by an SDJ function. Of course, the algorithm for estimating the signal amplitude has been tested as well. Figure 4 illustrates the MEV of the normalized amplitude estimate delivered by (27) in the range of \(\pm 10\) dB for an FF scenario applied to SDJ shapes and different values of \(\alpha\). Again, we observe that the estimation procedure is bias-free, which has also been tested for an MF scenario and SRC shapes. As a function of the electrical SNR value defined in (6), Figures 5 and 6 depict the evolution of the normalized jitter variance for timing estimates using an MF or FF receiver, respectively, and operating  
Figure 4: Mean estimator value for amplitude recovery using SDJ shapes (flat filter (FF) receiver). Figure 3: Mean estimator value (MEV) for timing recovery using SRC shapes (matched filter (MF) receiver). the link with SRC or SDJ shapes. By detailed inspection of Figure 5, we observe that the MF solution is close to the MCRLB in the medium-to-low SNR range, whereas for larger SNRs, the jitter performance ends up in an error floor, which is obviously less severe for higher values of \(\alpha\). This phenomenon is explained by the fact that the receiver MF violates the Nyquist property of the SRC/SDJ shape, producing this way an ISI effect that does not disappear, even if the AWGN component vanishes when \(\gamma_{s}\to\infty\). In addition, it is to be noticed that SDJ pulses exhibit a better jitter performance than do SRC shapes; note that for \(\alpha=0\), both scenarios converge to the MBW case.