# On Parameter Estimation for Bandlimited Optical Intensity Channels
## 1 Introduction  
There is no doubt that optical wireless communication (OWC) systems have some remarkable benefits compared to radio frequency (RF) solutions [1; 2; 3; 4]: no regulatory and license problems, rapid deployment and inexpensive operation, as well as data security and throughput, just to cite briefly the most important aspects. However, even for OWC links, the available bandwidth is not infinite, but subject to limitations because of different reasons, e.g., physical restrictions of electrical and optical elements in transmitters and receivers, dispersion effects introduced by the channel, or impairments due to multipath propagation in indoor applications. For an optical intensity link [5], this means that the baseband pulse has to be shaped appropriately in the transmitter unit to avoid negative signal components. On the other hand, it is still appreciated that the waveform satisfies the Nyquist criterion in order to keep the detection process in the receiver module as simple as possible.  
For a bandlimited OWC link, characterized by intensity modulation and direct detection (IM/DD), it was shown in [6] that both the non-negativity and Nyquist requirements are satisfied for a squared sinc pulse; for practical reasons admitting some amount of excess bandwidth, this results in a squared raised cosine (SRC) function. It was proved in [6] as well that a suitable root-Nyquist solution does not exist in the context of limited bandwidth, although this would be necessary to implement a matchedfilter (MF) in the receiver station to maximize the signal-to-noise ratio (SNR) at the detector output. As an alternative, the authors in [7; 8] suggested that the non-negativity constraint is achieved by an offset component, although this is less efficient in terms of power, but it needs only half of the bandwidth compared to an SRC approach. Finally, a two-dimensional signal space was studied in [9; 10] to increase the spectral efficiency of optical links.  
Powerful estimation of the most important transmission parameters is key for any receiver terminal [11; 12], which does not only hold true for radio, but also for optical scenarios. Regarding the latter, knowledge of both symbol timing and signal amplitude is of paramount importance; otherwise, subsequent stages, like detector units or error correction schemes, cannot be operated reliably. Scanning the open literature in this context, numerous contributions about channel estimation and/or symbol timing recovery are available, but they are not or only marginally related to bandwidth constraints and pulse shaping. From the author's point of view, it is the first time that this specific problem is analyzed and discussed in a structured and non-heuristic manner for the current contribution.  
The remainder of the paper is organized as follows: The signal and channel model used for analytical and simulation work is introduced in Section 2. In Section 3, we derive the Cramer-Rao lower bound as the theoretical limit of the error performance. Next, a maximum likelihood estimator is developed in Section 4; since this requires a receiver MF violating the Nyquist property, a modified solution is investigated based on the implementation of a flat receiver filter. Numerical results are presented in Section 5, and Section 6 concludes the paper.