Wilfried Gappmair

Further results on the capacity of free-space optical channels in turbulent atmosphere

In recent studies, the average capacity for optimal rate adaptation (ORA) of free-space optical channels in turbulent atmosphere has been derived in closed form, mainly based on the application of Meijers G-function. To this end, the channel was assumed to be memoryless, stationary and ergodic, with independent and identically distributed fading statistics. It was also assumed that scintillations follow a gamma–gamma distribution so as to appropriately describe moderate-to-strong turbulence conditions. In the current contribution, the author will extend this work in two aspects: (i) using the properties of Meijers G-function, it is shown that the average capacity provides also a closed-form solution for adaptation policies other than ORA, namely optimal power and rate adaptation, channel inversion with fixed rate and truncated channel inversion with fixed rate; (ii) if the additional loss caused by a misalignment between transmitter and receiver (pointing error) is taken into account, it is demonstrated that the developed analytical framework applies straightforwardly.

Performance of PPM on terrestrial FSO links with turbulence and pointing errors

In recent work, the error performance of on-off keying has been investigated for free-space optical (FSO) links impaired by both turbulence and pointing loss. In the current letter, the analytical framework is extended to M-ary pulse-position modulation (PPM), providing exact results for M = 2. Since the approach is not directly applicable to PPM signals with M > 2, a closed-form approximation of the average symbol error probability is derived for this case, which is shown via simulation to be tight over a wide SNR range of interest.

Error performance of coded FSO links in turbulent atmosphere modeled by gamma-gamma distributions

For free-space optical (FSO) links in turbulent atmosphere modeled by gamma-gamma distributions, an approximation of the pairwise error probability (PEP) was derived for on-off keying in a recently published paper. It has been assumed that the channel state is perfectly known to the receiver and that the fading samples are independent and identically distributed. Based on these conditions, we show in this letter that the approximation can be replaced by a closed-form solution, disregarding a finite-limit integral which forms also part of the approximation in the reference paper. The PEP is then used to evaluate the union upper bound for convolutional codes with Viterbi decoding. Finally, in order to identify the potential of powerful error correction algorithms, like turbo codes as most prominent example in this respect, the lower bound of the error probability is given via the converse of Shannons coding theorem.

Cramer-Rao Lower Bound for Non-Data-Aided SNR Estimation of Linear Modulation Schemes

Powerful parameter estimators exhibit a jitter variance which is fairly close to the Cramer-Rao lower bound (CRLB) as the theoretical limit. In contrast to symbol timing and carrier frequency/phase, not very much information is available from the open literature with respect to the signal-to-noise ratio (SNR), i. e., the CRLB has been reported only for the data- aided case and some simple M-PSK examples for non-data-aided estimation of the SNR. Motivated by this background, an efficient algorithm is presented which applies to any M-ary one/two- dimensional modulation scheme with axis/halfplane symmetry and a channel distorted by additive white Gaussian noise. Finally, the performance of different SNR estimators is compared to the derived bound.

OOK Performance for Terrestrial FSO Links in Turbulent Atmosphere with Pointing Errors Modeled by Hoyt Distributions

Terrestrial free-space optical links are corrupted by both turbulence and misalignment losses. It is common practice to model turbulence by a gamma-gamma distribution and misalignments by a Rayleigh distribution, where the underlying assumption is that the jitter in vertical and horizontal directions is the same. In the current letter, we generalize this previous approach by modeling misalignments with a Hoyt distribution, which removes the assumption of an identical jitter variance along the horizontal and vertical axes.

Cramer-Rao lower bound and EM algorithm for envelope-based SNR estimation of nonconstant modulus constellations

Signal-to-noise ratio (SNR) estimation for linearly modulated signals is addressed in this letter, focusing on envelope-based estimators, which are robust to carrier offsets and phase jitter, and on the challenging case of nonconstant modulus constellations. For comparison purposes, the true Cramer-Rao lower bound is numerically evaluated, obtaining an analytical expression in closed form for the asymptotic case of high SNR values, which quantifies the performance loss with respect to coherent estimation. As the maximum-likelihood algorithm is too complex for practical implementation, an expectation-maximization (EM) approach is proposed, achieving a good tradeoff between complexity and performance for medium-to-high SNRs. Finally, a hybrid scheme based on EM and moments-based estimates is suggested, which performs close to the theoretical limit over a wide SNR range.

ML and EM algorithm for non-data-aided SNR estimation of linearly modulated signals

The recently published Cramer-Rao lower bound for non-data-aided (NDA) estimation of the signal-to-noise ratio (SNR) reveals a considerable gap, when compared to the jitter performance of NDA algorithms available from the open literature. The maximum-likelihood (ML) solution derived in this paper closes this gap. However, the latter provides a set of two nonlinear vector equations, which might be simplified only for modulation schemes with constant envelope like M-ary PSK. For signals with nonconstant envelope, like 16-QAM as most prominent example in this respect, a much less complex approach based on the expectation-maximization (EM) principle is developed in this paper. In the medium SNR range, this bridges part of the performance gap mentioned previously. Over the full SNR range, we propose a hybrid algorithm, where the EM estimate is replaced by a moment-based method as soon as the true SNR drops below a predefined threshold.

Symbol Rate Estimation with Inverse Fourier Transforms

A two-stage algorithm for blind symbol rate estimation is introduced, which applies to any linear modulation scheme. In the first stage, the inverse Fourier transform of the averaged power spectrum is used to extract a coarse estimate based on the evaluation of the related baseband shape. The accuracy of the result can be improved considerably if the accumulated timing error of the subsequent symbol synchronizer is assessed appropriately and fed back to the resampling unit for further refinement. The performance of the proposed algorithm is verified by simulations and compared to methods published in the open literature

Extended Gardner Detector for Improved Symbol-Timing Recovery of

In a recent paper, the modified Gardner detector has been proposed for advanced symbol-timing recovery of M-ary phase-shift keying signals. Compared with the original algorithm, the self-noise jitter can be considerably reduced for highly bandlimited systems and smaller values of M. In this letter, a completely different approach is investigated, which improves the self-noise performance for larger values of M, as well

M

In a recent study, the error performance of pulse-position modulation (PPM) schemes has been investigated for a terrestrial free-space optical (FSO) link, which was assumed to be impaired by both turbulent atmosphere and pointing errors. Only for the binary case a closed-form solution was achievable, whereas for non-binary conditions just an approximate relationship could be given. The key point in this respect was the approximation of the complementary error function so that the two-fold integral, otherwise required in this context, ended up in a closed form. In the current contribution, the problem is tackled by an additive combination of exponentials whose coefficients are optimised in the least-square sense. It turns out that second-order approximations are a good compromise between accuracy and computational complexity.