Error rates of optically pre-amplified PPM wireless systems with coding and arbitrary optical filter response

We present novel results for the uncoded and coded bit-error probability (BEP) of optically pre-amplified pulse-position modulation (PPM) wireless systems. For uncoded systems, a novel analytic method for the evaluation of the BEP is derived. The method takes into account the non-ideal optical filter response and utilizes a finite Karhunen–Loève series expansion to calculate the BEP. Using the proposed approach, it is possible to accurately evaluate the PPM BEP for arbitrarily shaped filters where the well-established ${\chi }^2$ method only provides approximate results. Considering a Lorentzian filter response, the discrepancy between the two methods amounts to 0.5 dB in a variety of filter bandwidths and PPM modulation orders. The Lorentzian filter response was chosen as an illustrative practical example whose series can be calculated analytically. The proposed method is also valid for any type of optical filter for which the Karhunen–Loève series expansion can be calculated analytically or numerically. Due to the finite number of terms that are required irrespective of the signal energy level, the proposed method can also be applied without loss of accuracy to assess the system performance under the effects of turbulence and adverse weather conditions. For coded systems with Lorentzian filters, Monte-Carlo simulations are utilized to evaluate the BEP performance of the 5G LDPC codes, and it is demonstrated that they impart an energy gain up to 3.3 dB for 4–PPM and 2.3 dB for 16–PPM at a target BEP of $10^{-5}$. The optimal code rates are also discussed for several combinations of the optical filter bandwidth and PPM modulation order and it is shown that in almost all of the cases the optimal code rate is 11/13. Moreover, the sum-product and min-sum decoders perform within 0.1 dB from each other for the best code rates, which points towards the utilization of the min-sum decoder in all settings, since its operation does not require knowledge of the filter parameters. Finally, the comparison between the coded systems with Lorentzian and ideal passband filters exhibits the same 0.5 dB discrepancy that was observed for uncoded systems.