1. Describe the major research and education activities of the project.

Transcription

1 Activities and Findings This section will serve as your report to your program officer of your project's activities and findings. Please describe what you have done and what you have learned, broken down into four categories: 1. Describe the major research and education activities of the project. This project is primarily concerned with the study of various aspects and channel models of the free-space optics (FSO) communication channel, and the development of novel forward error-correction schemes for such channels. The motivation behind this project is that the FSO communication channel offers an excellent alternative for wireless communications for its wide bandwidth and relatively low cost, compared to RF wireless links. Optical communication through this channel is achieved by a point-to-point connection between two line-of-sight transceivers. We are interested in the FSO systems that are robust in the presence of atmospheric turbulence such as coded orthogonal frequency division multiplexing (coded-ofdm), and coded multiple-input multiple-output (i.e., multi-laser multi-detector or MIMO) concept; both employing low-density parity-check (LDPC) codes. Another important goal of this project is to solve the incompatibility problem that arises from the bandwidth mismatch between RF/microwave and optical channels. We proposed two coded modulation schemes suitable for hybrid microwave-optical communications: (i) coded-ofdm as multiplexing technique, and (ii) Q-ary bit-interleaved coded pulse-amplitude modulation. We are also interested in determining the ultimate channel capacity limits, and to see how close we can approach those limits with proposed concepts. Over the past year, we have pursued several research thrusts related to multi-channel optical communications, temporally correlated FSO channels, novel error-correction schemes, low-complexity decoding algorithms for non-binary LDPC codes, large-girth LDPC code design, design of generalized LDPC codes with component Reed-Muller codes, hybrid wireless-optical communications, quantum communications, and communication over the strong turbulence channel. Specific accomplishments include: 1. Use of rate-less error-correction codes for the temporally correlated FSO channel, 2. Study of orbital angular momentum-based models for multichannel optical communications, 3. Proposal of a list decoding algorithm for non-binary LDPC codes on channels such as additive white Gaussian noise (AWGN) and FSO channels,

2 4. Design of coded orthogonal frequency division multiplexing (coded-ofdm) as an efficient way to deal with the atmospheric turbulence, 5. Design of coded Multiple-Input Multiple-Output (MIMO) communication schemes for data transmission over the optical atmospheric turbulence channels, 6. Study of hybrid RF/microwave-optical communications, 7. Modeling of MIMO free-space optical communication channels, 8. Modeling of free-space optical communication channels with memory by Markov chains, 9. Computation of information theoretic limits for free-space optical communication channels with and without memory, 10. Design of large-girth LDPC codes suitable for use in free-space optical communications, 11. Design of generalized LDPC codes with component Reed-Muller codes for optical communications, 12. Design of quantum LDPC codes, 13. Study of optical coherent state quantum communications, 14. Development of an experimental setup to study the novel freespace optics communication techniques developed by our research team, and 15. FSO communications for intra-chip/inter-chip communications. The educational activities include teaching graduate courses in digital communications and advanced optical communications, and undergraduate courses in signals and systems and digital signal processing as well as presenting tutorial lectures to various groups on and off campus. 2. Describe the major findings resulting from these activities. Orbital angular momentum-based multi-channel communications Orbital angular momentum (OAM) is a property of light associated with the helicity of a photon's wavefront. Optical beams carrying OAM are usually called optical vortices, because they feature a phase discontinuity at their center. The momentum of a vortex field is proportional to the number of turns that this vector completes around the beam's axis after propagating a distance equal to one wavelength. This number is equal to the OAM state. The OAM state of a photon can take any integer value. This infinite set of OAM states forms an orthonormal basis. This property may be exploited in the context of optical communications. The orthogonality among beams with different OAM states allows the simultaneous transmission of information from different users, each on a separate OAM channel. Each orthogonal channel can be perfectly filtered and decoded at the receiver of a free-space

3 optical (FSO) communication link (see Fig. 1). OAM states may also be used for multilevel modulation. For FSO applications, the orthogonality is not maintained in the presence of atmospheric turbulence. As a result of the random turbulence process, part of the energy launched into a single OAM state will be redistributed into other OAM states after turbulent propagation. Consequently, atmospheric turbulence induces a time-varying crosstalk among OAM channels. We have studied the feasibility of a multi-channel OAM terrestrial FSO link and have quantified the channel crosstalk as a function of turbulence strength, number of simultaneous channels, and signal-to-noise ratio [J-11] (see Fig. 2). Through numerical methods, we have simulated the coaxial propagation of Laguerre-Gauss beams each with a distinct OAM state in the range [-50, +50] over a 1-km turbulent path. As expected, these simulations verify that optical turbulence induces OAM crosstalk and that the average crosstalk between channels grows with turbulence strength. Fig.1 Diagram of a free-space optical communication link using multiplexed OAM channels. A number of data-carrying zero-order Gaussian modes each independently modulated by a data stream are shone onto a series of volume holograms, each programmed with a different OAM state. The multiplexed channels are transmitted through a telescope to the receiver, whose de-multiplexing architecture consists of holograms much like that those at the transmitter. For each transmitted OAM state we have (i) quantified the efficiency (% of power remaining in transmitted channel) of each channel in terms of the turbulence strength and (ii) quantified the average crosstalk observed on all channels in the studied range, in terms of turbulence strength. The efficiency in (i) accounts for channel losses due to beam spreading, beam wander, and crosstalk. The averages found above can be understood as the channel matrix of the OAM-multiplexed communication system. Knowledge of this matrix (at each turbulence strength level) is essential to designing

4 and predicting the performance of such a system. We have determined that multi-channel communications using optical beams with OAM is feasible in weak turbulence conditions with simple encoding/decoding architectures. Communication in strong turbulence conditions is possible at the cost of larger optical transmitters and receivers. Fig.2 Average crosstalk observed on OAM channels [-15, 15] induced by transmit channel {1, 5, 10, 15} in moderate atmospheric turbulence. The channel s selfefficiency is given by the bars whose channel number coincides with the transmit OAM state number. OAM crosstalk spreads to more states as the transmit OAM state number increases. With the assumption that crosstalk and detector noise are mutually independent Gaussian noise sources, we have modeled each OAM mode as a

5 binary symmetric channel whose probability of flip error is a function of the channel efficiency, the crosstalk induced by the other constituent channels, and by detector noise. Using this model we have found the optimal set of OAM mode numbers for a prescribed number of channels, in the sense of maximizing the aggregate capacity. The optimal sets of OAM state numbers are determined at each value of SNR and turbulence strength considered. Raptor codes for temporally-correlated FSO channels The free-space optical (FSO) communication channel is temporally correlated. Signal power fluctuations observed in this channel are very slow compared to the typical data rates of FSO links. Fixed-rate error correction codes can provide significant improvement over uncoded FSO links as long as the temporal correlation can be broken by means of symbol interleavers. However, if the bit time is too short compared to correlation time, the interleaver will be too large to be practical. To overcome this problem, alternatives to the use of an interleaver in the physical layer have been proposed by means of network layer coding. We propose tackling the varying channel quality in the physical layer by means of rate-less codes. Rate-less codes are error-correcting codes that adapt to the quality of the channel by varying their user information rate. Raptor codes are an example of this class of codes. Raptor codes modify their rate by increasing the codeword length if the signal-to-noise ratio decreases. Coded bits are generated continuously by the transmitter until the receiver can decode the transmitted word without error. Raptor codes have been previously proposed to operate on the network layer of radio-frequency (RF) communication systems. As a proof of concept, we have performed an evaluation of a Raptor code on the physical layer of a FSO link using experimentally-recorded FSO channel waveforms, obtained by transmitting a continuous-wave laser over the link and sampling the optical power at the receiver over a period of time. These sampled waveforms are instances of the temporally correlated random channel gains that modulate the user information stream in a terrestrial FSO communication system. The waveforms account for all channel insertion losses including the effects of beam spreading, beam wandering, finite detector size, and particle absorption and scattering. We show that a Raptor code can continuously decoded without errors under the time-varying channel fluctuations, even during long deep channel fades (at which the effective signal-to-noise ratio (SNR) may drop by 15 db with respect to the average channel SNR) by adaptively varying the information data rate (see Fig. 3). This coding scheme is advantageous as it does not require bit interleaving and involves a simple encoding/decoding algorithm.

6 bits/channel use Case σ I 2 = 1.03 Perfect and imperfect CSI using 10K bits C AWGN-OOK Perfect CSI Imperfect CSI SNR [db] Fig. 3 Effective information rate (in bits per channel use) achieved by the Raptor code with inner LDPC code (495, 433) on an experimentally-recorded FSO fading channel with temporal correlation τ 0 =3.6 ms and scintillation σ I 2 =1.03. The segmented curve is the capacity of an AWGN channel using OOK modulation, which serves as an upper bound. The strength of the Raptor code resides on its capability to maintain an information data stream at very low SNR values. List decoding algorithm for non-binary LDPC codes Iteratively decodable codes such as LDPC and turbo codes have generated a considerable amount of research attention over the last decade and these codes have been shown to provide significant coding gains over the AWGN as well as optical communication channels. Unfortunately, capacity-achieving coding gains are obtained only for extremely long LDPC codes and hence are quite complex for practical usage. On the other hand, non-binary LDPC codes (codes over higher order Galois fields) are known to provide larger coding gains even at moderate block lengths compared to binary LDPC codes. However, the decoding algorithm for the non-binary LDPC codes, which utilizes the sum product algorithm (SPA), is extremely complex in terms of computation and requires large amounts of memory, since the messages are vectors consisting of the likelihood ratios (or probabilities) of each and every symbol in the field and hence the analysis of these codes becomes quite complicated. We propose a low-complexity decoding algorithm for non-binary LDPC codes called list decoding, so that the decoding is not only lower in terms of computational complexity but can

7 possibly be more amenable to trapping set or even density evolution type of analysis. In this algorithm, the messages are lists consisting of the most likely symbols in the field (as opposed to the SPA where all the symbols are considered). The rationale behind this idea is that in practice, we have found that as the decoding algorithm starts to converge, certain symbols in the field become much more likely than others for a particular variable node and hence we need to consider only these during computation. We can immediately see that there is a reduction in complexity by considering a list of symbols as the message. Although for this algorithm, any type of channel model can be used (including q-ary symmetric channel), we may gain some advantage by using the practical channel model of transmitting bits of respective symbols across the channel through modulation schemes (like BPSK over AWGN). The list size is chosen based on the modulation scheme and the field size. In addition to using lists as messages, we use multiplicities (or votes) to capture the probabilities of the symbols in the field. The total number of votes in a particular list is fixed, say V max. Thus, a particular list on an edge is specified by two parameters; (i) the mostly likely symbols among all the symbols in the field and (ii) the respective votes of those symbols. Note that by using multiplicities, we are in a way restricting or discretizing the space involving the probabilities of the symbols. As a result, this representation not only reduces the complexity w.r.t computations at the node but also appears to make the analysis tractable. We now explain the operations that we propose to use in the algorithm. As mentioned before, let us assume that we are using a practical channel model where the bits of the symbols over GF(2 p ) are transmitted across the channel (say BSC or BPSK over AWGN). At the decoder, using the likelihoods (or probabilities) of the bits that are received from the channel, we consider 'p' bits at a time and then compute the probabilities of each symbol in the field for a particular variable node. This forms the received vector at a particular variable node. Fig. 4 Operations at the variable node

8 Fig. 5 Operations at the check node Let us take for example, a code over GF (2 3 ). The received vector at a variable node would be r= {p(0), p(1), p(α),, p(α 6 )}. Let us say that the list size is M=4. Let the function V(X) denote the number of votes for a particular symbol X in the list. In the initial iteration, a list L r would be generated based on the received vector at the variable node and this list would serve as the outgoing message on all its respective edges. In our case, the 4 most likely symbols are chosen from r to be in the list L r, and V max number of votes is distributed among these symbols in the list based on their respective probabilities. The outgoing lists L 1, L 2 and L 3 are then made equal to L r for the initial iteration. This is illustrated in Fig. 4. X 1, X 2, X 3 and X 4 denote the most likely symbols chosen from the received vector r. Also V( x1) + V( x2) + V( x3) + V( x4) = Vmax Note that not only the list L r but the received vector r at every variable node is also stored. We will now explain the check node and variable node update rules. Check node update rule: The rule can be explained with the help of Fig. 5. Let us say edges 1,2,3,4 are the incoming edges and we want to generate the outgoing list on edge 5. Also let β 1, β 2, β 3, β 4 and β 5 denote the edge weights. We need to consider all possible combinations of the symbols of the lists on the incoming edges and take their weighted sums (because of the non-binary edge weights) divided by the edge weight β 5 of the outgoing edge over GF(2 p ). In our

9 example, we need to check 4 4 combinations and then see what the resulting symbols are. In general, for any check node of degree d c, we would need to check 4 dc-1 combinations (or M dc-1 for any list size). Let Y denote the set of all possible symbols resulting from the parity check sum, using all possible combinations of the incoming symbols. It is evident that Y C GF(2 p ) since Y may or may not contain all the symbols in the field. From Fig. 5, p i, q i, r k and s l where i,j,k,l = {1,2,3,4}, denote the symbols on the incoming edges 1, 2, 3, 4 respectively. For a particular symbol y є Y, the general parity check sum equation of the symbols from the incoming lists that leads to symbol y is given by β1pi + β2q j + β3rk + β4sl y = β5 where p i, q j, r k and s l are the symbols from their respective lists that satisfy the above equation. Let ξ y denote the set of all possible i,j,k,l indice vectors whose corresponding symbols (p i, q i, r k, s l ) satisfy the above equation. Then the number of votes assigned to the symbol y is given by V ( y) = ( i, j, k, l) ξ V ( p ) V ( q y i j ) V ( r ) V ( s ) In this manner, we determine V(y) for all possible symbols in Y i.e. for all y є Y and then choose the top four symbols (with the highest votes) from the set Y to be in the outgoing list. If say y 1, y 2, y 3 and y 4 were the 4 symbols chosen from the set Y, then these symbols would form the outgoing list with their corresponding votes as [V(y 1 ),V(y 2 ),V(y 3 ),V(y 4 )]. However, we must redistribute the votes of the symbols such that the total number of votes in the list is V max and then this normalized outgoing list is sent to the variable node. In this manner, the checknode update is carried out and we can determine the outgoing lists on all the remaining edges. Note that number of votes for a symbol is supposed to have reasonably captured the probability of that particular symbol (soft information). We have found that this depends on the list size and V max where V max determines the accuracy of the multiplicities compared to their received probabilities (similar to the notion of having a precision error). When the value of V max was chosen to be larger, the accuracy of the multiplicities was found to be higher which is expected. We now look at the variable node update. Variable node update rule: The rule can be explained with the help of Fig. 4. Let us assume that edges 2 and 3 are incoming and we want to determine the outgoing message on edge 1; we need to utilize L 2, L 3 and the list L r derived from the received vector r. Let Z denote the set of possible symbols on the outgoing list and Z k l

10 C GF(2 p ). Let l 2, l 3 and l r denote the set of symbols in the lists L 2, L 3 and L r respectively (note that the lists contain the votes of those symbols i.e. L 2 = V(l 2 ) and so forth). The set Z is obtained by the condition Z = l l l r 2 3 This is because we only consider the symbols that are present in all the incoming lists and l r, and any other symbol that is not common is discarded. However, in practice, we have found that while operating at lower SNR values and during initial iterations, there may be several nodes consisting of edges whose respective lists may not share common symbols or there may be more uncommon symbols among the lists than the common symbols as a result of which the above condition may slow down the rate of convergence considerably. To circumvent around this problem, the above condition may be slightly relaxed to improve rate of convergence, by including the non-common symbols in the set Z but assigning a vote of 1 for each of them (treating them like dummy symbols). Now for a symbol z which belongs to the set Z, the number of votes for the symbol z is given by the product V ( 3 z) = L2 ( z) L ( z) Lr ( z) where L 2 (z) denotes the number of votes stored in the list L 2 for the symbol z. Similar is the notation for L 3 (z) and L r (z). In this manner, we determine the votes for all the symbols in the set Z, i.e. for all z є Z. Then the top 4 symbols with the highest votes are chosen from the set Z to be in the outgoing list, say z 1, z 2, z 3 and z 4 so that the outgoing list is [V(z 1 ),V(z 2 ),V(z 3 ),V(z 4 )]. However, we must again redistribute the votes so that total number of votes is V max and this normalized list will be the final outgoing list on edge 1. We perform similar operations for the remaining edges. The expressions determined for the above update rules can be generalized for any list size or field size and they may be slightly altered to account for other practical considerations but the basic idea essentially remains. We have found that these operations mimic the nature of SPA reasonably well especially at higher signal-to-noise ratios (SNR) but the novelty of this algorithm actually lies in the fact that only multiplicities are used in the various node operations and this is very important since practical hardware implementations involve fixed point arithmetic (as opposed to floating point arithmetic which the SPA requires). It is evident from the above discussion that this algorithm is much lower in complexity compared to the normal SPA since the messages are lists and do not contain all the symbols in the field. The reduction in complexity is much greater for much higher order Galois fields like GF(64). But more importantly, analysis of these codes may now be more tractable.

11 Coded orthogonal frequency division multiplexing (coded-ofdm) To deal with atmospheric turbulence we proposed to use either: (i) codedorthogonal frequency division multiplexing (OFDM) scenario [J-1],[J-2],[C- 1],[C-2], or (ii) coded-multiple-input multiple output (MIMO) scenario [J- 3],[C-1],[C-2]. The coding in both scenarios is based on the best known codes-low-density parity-check (LDPC) codes. The first approach that is able to operate under strong atmospheric turbulence and at the same time to enable hybrid RF/microwave-optical communications over the atmospheric turbulent channel is based on coded- OFDM. The block diagrams of the proposed transmitter and receiver configurations are shown in Fig. 6 (a) and (b), while the transmission system based on FSO communication is shown in Fig. 6(c). The data streams from L different RF channels are combined using OFDM and encoded using an LDPC encoder. The LDPC encoded data stream is then parsed into groups of B bits. The B bits in each group (frame) are subdivided into K subgroups with the i th subgroup containing b i bits, B= b i. The b i bits from the i th subgroup are mapped into a complex-valued signal from a 2 b i -point signal constellation such as QAM. The complex-valued signal points from all K subchannels are considered as the values of the discrete Fourier transform (DFT) of a multicarrier OFDM signal. After D/A conversion and RF up-conversion, the OFDM signal drives a Mach-Zehnder modulator (MZM) for transmission over the FSO link. The DC component facilitates recovering the QAM symbols incoherently. At the receiver, an optical system collects the light, and focuses it onto a detector, which delivers an electrical signal proportional to the incoming optical power. After the RF down-conversion, carrier suppression, A/D conversion and cyclic extension removal, the transmitted signal is demodulated using the FFT algorithm. The soft outputs of the FFT demodulator are used to estimate the symbol reliabilities, which are converted to bit reliabilities, and provided as input to an LDPC iterative decoder. Since the bipolar signals cannot be transmitted over an intensitymodulation/direct detection (IM/DD) link, and OFDM signals must include a DC bias in order to allow incoherent detection. The most straightforward method of DC bias addition is to add sufficient DC bias so that the resulting OFDM signal is non-negative. This scheme is referred to as the biased- OFDM (B-OFDM) scheme. For illustrative purposes the MZM RF input signal associated with a B-OFDM scheme is shown in Fig. 7(a). The main disadvantage of the B-OFDM scheme is its poor power efficiency. To improve the power efficiency we propose two alternative schemes.

12 RF channels 1 L Laser diode LDPC QAM P/S D/A RF to MUX IFFT MZM encoder mapper converter converter upconverter FSO link (a) Carrier Symbol Bit from RF suppression LDPC PD FFT reliability reliability FSO link downconverter and decoder calculation calculation A/D converter (b) FSO-OFDM Expanding telescope FSO-OFDM transmitter Fiber Detector receiver Input data Output data optical amp. Compressing Light beam through telescope turbulent channel Collimating lens (c) N G /2 samples N G /2 samples Original N FFT samples DEMUX RF users 1 L Preffix Suffix OFDM symbol after cyclic extension, N FFT + N G samples T G /2 (d) T FFT T G /2 Effective part T win T=T FFT +T G +T win OFDM symbol duration kt (e) Fig. 6 LDPC-coded OFDM system: (a) transmitter configuration, (b) receiver configuration, (c) FSO link, (d) an OFDM symbol after cyclic extension, and (e) an OFDM symbol after windowing.

13 MZM RF input, v RF [V] 0.06 (a) B-OFDM Time, t [ps] (c) MZM RF input, v [V] MZM RF input, v [V] (b) U-OFDM Time, t [ps] C-OFDM Time, t [ps] Fig. 7 Waveforms of the SSB OFDM signal with 64 sub-carriers at MZM RF input in a back-to-back configuration for: (a) B-OFDM, (b) C-OFDM, and (c) U-OFDM. The first alternative scheme, which we shall refer to as the clipped- OFDM (C-OFDM) scheme, is based on single-side band (SSB) transmission and clipping of the OFDM signal after adding a bias. The clipping can be either symmetric or asymmetric. Our initial studies have shown that in symmetric clipping the optimum bias should be selected such that ~50% of the total electrical signal energy before clipping is allocated for transmission of a carrier. The MZM RF input signal for the clipped-ofdm scheme is shown in Fig. 7(b). We note that clipping introduces inter-modulation distortion that may degrade BER performance. However, because C-OFDM allocates more energy per information bit than B-OFDM a tradeoff results. The optimum choice of system parameters and their dependence on FSO channel conditions is an important issue, however, due to space limitations this study will be omitted. In order to avoid distortion due to clipping at the transmitter, the information-bearing signal may be mapped into the optical domain by modulating the electrical field of the optical carrier using a LiNbO 3 MZM. In this case, the clipping will be performed by the receiver through the squaring operation inherent in the measurement of optical intensity (by

14 photodetector). The distortion introduced by photodetector may be reduced by proper filtering. Notice that the U-OFDM scheme will be less powerefficient that the C-OFDM scheme, but is still expected to be better than the B-OFDM scheme. The MZM RF input signal for the U-OFDM scheme is shown in Fig. 7(c). (a) (b) (c) (d) Fig. 8 Received constellation diagrams of QPSK (a)-(c) and 16-QAM (d) SSB FSO- OFDM systems with electrical SNR per bit of 18 db under the weak turbulence for: (a),(d) U-OFDM scheme, (b) C-OFDM scheme, and (c) B-OFDM scheme. The influence of both atmospheric turbulence and receiver electronic noise (AWGN) on QPSK and 16-QAM SSB FSO-OFDM systems is illustrated in Fig. 8. Results obtained from a SSB OFDM system with 64 sub-carriers are shown. The average launched power is set to 0dBm, the electrical SNR is set to 18 db, and the received signal constellation diagrams are obtained assuming weak atmospheric turbulence. We note that turbulent propagation changes the symmetry of these signal clusters from circular (i.e., for a pure AWGN channel) to elliptical (see Fig. 8) for the FSO channel. Both C-OFDM and U-OFDM schemes are more immune to atmospheric turbulence than is

15 the B-OFDM scheme. The U-OFDM system performs only slightly better than C-OFDM. It appears that the better power efficiency of C-OFDM compensates the distortion introduced by clipping. Higher energy per bit associated with C-OFDM may result in improved immunity to electric noise. Higher immunity to electrical noise may result in slightly better BER performance of C-OFDM scheme when compared to U-OFDM scheme. Simulation results of an LDPC coded SSB U-OFDM system under the strong turbulence regime are given in Fig. 9(a). The coding gain improvement of the LDPC-coded OFDM system over the LDPC-coded on-off keying (OOK) system is db for QPSK, and db for BPSK. The 16- QAM FSO-OFDM system is not able to operate in the regime of strong turbulence. The comparison of different LDPC coded SSB OFDM schemes in weak turbulence (σ R =0.6), is given in Fig. 9(b). The C-OFDM scheme slightly outperforms the U-OFDM scheme. Both C-OFDM and U-OFDM schemes outperform the B-OFDM scheme by approximately 1.5dB at BER of The numerical results shown in Figs. 9 are obtained adopting the Gamma-Gamma probability density function (PDF), and assuming that the received intensity samples are independent and uncorrelated. In reality, especially at high bit rates, the channel has temporal correlation, and consecutive bits that propagate experience similar channel conditions. In many OFDM systems this approach is reasonable for the following reasons: (i) when the channel conditions do not vary, a simple channel estimation techniques based on pilot signals can be used to overcome the temporal correlation, and (ii) the immunity to temporal correlation can further be improved by using interleaving. The interleaving can be visualized as the forming an Lxn (n is the codeword length) array of L LDPC codewords (the parameter L is known as interleaving degree) written row by row, and transmitting the array entries column by column. If the original code can correct a single error burst of length l or less, then the interleaved code can correct a single error burst of length ll. Therefore, interleaved OFDM can successfully eliminate temporal correlation introduced by the FSO channel. To illustrate the applicability of LDPC-coded OFDM in the presence of temporal correlation we performed simulations by employing the joint temporal correlative distribution model, which describes the fading in an FSO channel at a single point of space at multiple instances of time. This method is based on the Rytov method to derive the normalized log-amplitude covariance function for two positions in a receiving plane perpendicular to the direction of propagation. The results of simulations are shown in Fig. 10. The standard deviation σ X is set to 0.6 (notice that σ X is different from Rytov standard deviation σ R used earlier, and for horizontal paths σ X 0.498σ R ).

16 Bit-error rate, BER σ R =3.0 BPSK SSB OFDM: Decoder input LDPC(4320,3242) QPSK SSB OFDM: Decoder input LDPC(4320,3242) 16-QAM SSB OFDM: Decoder input LDPC(4320,3242) OOK: Decoder input LDPC(4320,3242) Bit-error rate, BER Electrical SNR, E b /N 0 [db] (a) σ R =0.6 QPSK SSB Unclipped-OFDM: Decoder input LDPC(4320,3242) QPSK SSB Clipped-OFDM: Decoder input LDPC(4320,3242) QPSK SSB Biased-OFDM: Decoder input LDPC(4320,3242) OOK: Decoder input LDPC(4320,3242) Electrical SNR, E b /N 0 [db] (b) Fig. 9 (a) BER performance of LDPC-coded SSB U-OFDM system with 64-subcarriers under the strong turbulence. Block-circulant LDPC code (4320,3242) of rate 0.75 is employed. (b) Comparison of different LDPC coded SSB FSO-OFDM systems with 64-subcarriers under the weak turbulence. The BER performance can further be improved by using the interleaver with larger interleaving degree than that used in Fig. 10, at the expense of increasing encoder/decoder complexity. Notice the OOK modulation scheme enters BER floor for this value of standard deviation (σ X =0.6), and even advanced FEC is not able to help too much. However, LDPC-coded OOK is able to operate properly at lower standard deviations σ X. To generate temporally correlated samples we used two different methods, the first one

17 is based on the Levinson-Durbin algorithm, and the second one is based on an algorithm due to Wood and Chan. Both methods gave identical plots τ 0 = 10 μs, σ X =0.6: Uncoded, QPSK-OFDM LDPC(4320,3242), QPSK-OFDM Uncoded, 16-QAM-OFDM LDPC(4320,3242), 16-QAM-OFDM 16-QAM-OFDM, LDPC(4320,3242), L=30 Uncoded, OOK τ 0 = 10 μs, σ X =0.1: Uncoded, OOK LDPC(4320,3242), OOK 10-2 Bit-error rate, BER Electrical SNR, E b /N 0 [db] Fig. 10 BER performance in the presence of temporal correlation LDPC-Coded MIMO Optical Communication over the Atmospheric Turbulence Channel The performance of FSO communication systems can be improved by using MIMO communication techniques. In the case of FSO communications, the MIMO concept is realized by employing multiple optical sources at the transmitter side and multiple detectors at the receiver side, as shown in Fig. 11. Although this concept is analogous to wireless MIMO concept, the underlying physics is different, and optimal and sub-optimal configurations for this channel are needed. In several recent publications, the MIMO scheme alone and its concatenation with different coding techniques are studied assuming an ideal photon-counting receiver. In this part of our research, we show that LDPC-coded repetition MIMO is an excellent

18 candidate, capable of enabling the communication over the strong atmospheric turbulence channels [J-3], [C-1]. We have two goals: (i) to study different techniques for coded FSO MIMO communication, and (ii) to evaluate the performance of proposed techniques in terms of the achievable information rates and the channel capacity. Two types of information theoretic bounds are determined: (i) the independent identically distributed (i.i.d.) channel capacity of the MIMO optical atmospheric channels using an approach proposed by Ungerboeck, and (ii) the MIMO achievable information rates using Telatar s approach. The atmospheric optical channel is modeled by adopting the gamma-gamma probability density function due to Al-Habash, which is valid for a wide range of turbulence strengths. A photodetection is assumed to be non-ideal. We study two approaches as possible candidates to achieve these theoretical limits. The first approach is based on the LDPC-coded repetition MIMO principle. The second is based on the LDPC Space-Time (ST) coding MIMO scheme. The LDPC codes employed are designed using the combinatorial objects known as balanced incomplete block designs (BIBDs), and pairwise balanced designs (PBDs), accompanied by block-circulant (array) codes. Both schemes are able to operate under strong atmospheric turbulence and provide excellent coding gains. To improve the spectral efficiency of proposed schemes, we employed a bit-interleaved LDPC-coded modulation based on the pulse-amplitude modulation (PAM). In order to improve BER performance, we iterate the extrinsic log-likelihood ratios (LLRs) between a posteriori probability (APP) demapper and LDPC decoder. The selection of LDPC codes suitable for iterative demmaping-decoding is performed by the use of extrinsic information transfer (EXIT) chart. To facilitate the implementation at high speeds, structured LDPC codes are employed in simulations. We assume an on-off keying (OOK) transmission over the atmospheric turbulence channel using incoherent light sources and direct detection. The information bearing signal is LDPC encoded. A ST encoder accepts K encoded bits x k (k=1,2,,k) from an LDPC encoder. The ST encoder maps the input bits into the TxM matrix O whose entries are chosen from x, x,..., x, x, x,..., x { 1 2 K 1 2 K} so that the separation of decision statistics is possible at the receiver side. T denotes the number of channel uses required to transmit K input bits. Notice that case K=T=M=2 (M is the number of optical sources introduced in subsection A) x x 1 2 O = x2 x1 corresponds to the Alamouti-like ST code. Here, complement of x i. x i = 1 x i denotes the binary

19 Receiver 1 Source bits LDPC encoder Space-time code encoder Transmitter 1... Transmitter M... Processor Detected bits Receiver N (a) Atmospheric Turbulence Channel mth Transmitter Fiber Expanding telescope TA amplifier Detector 1 Optical source optical amp. Collimating lens Light beam through turbulent channel Detector N Receiver array (b) Processor From FSO link Space-time Receiver array LDPC Decoder Soft Decoder Detected bits (c) Fig. 11 (a) Atmospheric optical LDPC-coded MIMO system with space-time block codes, (b) mth transmitter and receiver array configurations, and (c) processor configuration. In Fig. 12(a), we plotted the i.i.d. capacity for binary transmission in strong turbulence regime (σ R =3.0) for different number of optical sources M, and photodetectors N, against the electrical SNR ratio per photodetector, denoted by E/N 0, in the presence of scintillation. A slightly better improvement is obtained by increasing the number of optical sources than by increasing the number of photodetectors. The MIMO FSO systems with M=N=2 and M=4, N=1 are comparable. In Fig. 12(b), we plotted the i.i.d. capacity for the Q-ary PAM. A significant i.i.d. channel capacity improvement is obtained by employing the MIMO concept relative to the single-source single-detector technique.

20 i.i.d. capacity, R [bits/channel use] i.i.d. capacity, R [bits/channel use] Q=2 0.6 M=1, N=1 0.5 M=2: N=1 0.4 N=2 0.3 N=4 M=4: 0.2 N=1 N= Electrical SNR, E/N 0 [db] (a) Q=16 Q=8 Q= Electrical SNR, E/N 0 [db] (b) Fig. 12 i.i.d channel capacity for different numbers of optical sources (M), and photodetectors (N) in strong turbulence regime (σ R =3.0, α=5.485, β=1.1156) for: (a) binary transmission, and (b) Q-ary PAM. Q= 4: Q= 8: Q=16: M=1, N=1 M=2, N=2 M=4, N=4 M=2, N=2 M=4, N=4 M=2, N=2 M=4, N=4 MIMO achievable information rates using Telatar s approach are calculated by Monte Carlo simulations, and they are shown (in bits/channel use) in Fig. 13 against electrical average symbol energy (E s )-to-powerspectral density (N 0 ) ratio. A significant spectral efficiency improvement is possible by using the multi-level schemes.

21 Information Rate, R [bits/channel use] M=1, N=1 M=2: N=1 N=2 N=4 M=4: N=2 N= Electrical SNR, E s /N 0 [db] Fig. 13 The MIMO achievable information rates for different number of lasers (M), and photodetectors (N) in strong turbulence regime (σ R =3.0, α=5.485, β=1.1156). The BER vs. electrical SNR in the presence of scintillation (per photodetector), for a strong turbulence regime (σ R =3.0, α=5.485, β=1.1156), are shown in Fig. 14. The BER is shown for a different number of optical sources, and photodetectors, by employing an (6419,4794) irregular girth-6 LDPC code of a rate designed using the concept of the PBD. The Alamouti-like ST code performance is comparable to the repetition MIMO, while T=4 ST performs worse than the corresponding repetition MIMO. The reason for such a behavior comes from the fact that we operate with non-negative real signals rather than with complex, so that the spacetime codes from orthogonal designs are not optimal in an FSO channel. The LDPC-coded MIMO with Alamouti-like code (M=2) and N=4 photodetectors provides about 20 db improvement over LDPC-coded OOK with single optical source and single photodetector. Further performance improvements can be obtained by iterating between LDPC decoder and soft ST decoder, at the expense of the increased decoding delay. Although a significant coding gain is obtained, from the channel capacity curves, it is obvious that we are still several dbs away from the channel capacity. This suggests that neither the coded repetition MIMO nor the wireless space-time codes are channel capacity approaching techniques. In order to come closer to the channel capacity, novel ST codes taking the underlying free-space optical physics into account are needed, but still not known. One possible option would be the use of Bell Labs Layered Space-Time Architecture (BLAST) to deal with space interference, in combination with long LDPC codes.

22 Bit-error rate, BER MLMD (known CSI): M=1, N=1 M=2, N=1 M=2, N=2 M=4, N=1 Alamouti-like code (M=2): N=1 N=2 N=4 ST code T=4 (M=4): N=1 N=2 Bit-error ratio, BER Electrical SNR, E/N 0 [db] (a) LDPC-coded: MLMD (known CSI): M=1, N=1 M=2, N=1 M=2, N=2 M=4, N=1 Alamouti-like code (M=2): N=1 N=2 N=4 ST T=4 code (M=4): N=1 N= Electrical SNR, E/N 0 [db] (b) Fig. 14 BERs of binary LDPC(6419,4794)-coded MIMO ST coding scheme against LDPC-coded repetition MIMO: (a) uncoded case, and (b) coded case. The results of simulations for bit-interleaved LDPC(6419,4794)-coded PAM are shown in Fig. 15 for different MIMO configurations and different number of signal constellation points employing the Gray mapping rule. Once more, although excellent BER performance improvement is obtained (about 23 db for M=N=4, Q=4 over M=N=1, Q=4), there is still some space for improvement to come closer to the channel capacity, which was left for further research. The comparison for different component LDPC codes is given in Fig. 16. The scheme employing a girth-6 (g-6) irregular PBD-based

23 LDPC code of rate 0.75 performs comparable to a girth-8 regular BC-LDPC code of the same rate. The scheme based on a girth-8 regular BIBD code of rate 0.81 performs worse than 0.75 codes. However, the difference is becoming less important as the constellation size grows. Bit-error ratio, BER BI LDPC-coded PAM: M=1, N=1 Q=2 Q=4 M=2, N=2 Q=4 Q=16 M=4, N=4 Q=2 Q=4 Q=8 Q= Electrical SNR, E/N 0 [db] Fig. 15 BER performance of BI-LDPC(6419,4794)-coded PAM with repetition MIMO. Bit-error ratio, BER g-6 LDPC(6419,4794)-coded PAM: (irregular code) M=1, N=1, Q=4 M=2, N=2, Q=4 M=2, N=2, Q=4 g-8 LDPC(4320,3242)-coded PAM: (regular code) M=1, N=1, Q=4 M=2, N=2, Q=4 M=2, N=2, Q=4 g-8 LDPC(8547,6922)-coded PAM: (irregular code) M=1, N=1, Q=4 M=2, N=2, Q=4 M=2, N=2, Q= Electrical SNR, E/N 0 [db] Fig. 16 BER performance of BI-LDPC-coded PAM with repetition MIMO for different LDPC component codes.

24 Note that in the simulations above, we assume that the channel is uncorrelated. As we mentioned earlier, we assumed that a temporal correlation can be overcome by means of interleavers, and possibly by orthogonal-frequency division multiplexing (OFDM). Unfortunately, the temporal correlation is difficult to simulate, especially under strong turbulence regimes. Coded-MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation In order to achieve MIMO FSO transmission, M laser sources and N photodetectors can be employed as described in previous Section. The laser sources and photodetectors have to be positioned so that different transmitted symbols from different channels experience different atmospheric turbulence conditions. For aggregation of RF/microwave channels and a conversion into optical domain, in our recent article [J-4], we have described the following scheme coded pulse-position modulation (PPM). The source bit streams coming from L RF/microwave sources are multiplexed together and encoded using an (n,k) LDPC code of code rate r=k/n (k-the number of information bits, n-the codeword length). The m n block-interleaver, collects m codewords written row-wise. The mapper accepts m bits at a time from the interleaver column-wise and determines the corresponding slot for Q-ary (Q=2 m ) PPM signaling using a Gray mapping rule. With this BICM scheme, the neighboring information bits from the same source are allocated into different PPM symbols. In each signaling interval T s a pulse of light of duration T=T s /Q is transmitted by a laser. (The signaling interval T s is subdivided into Q slots of duration T.) The total transmitted power P tot is fixed and independent of the number of lasers so that emitted power per laser is P tot /M. This technique improves the tolerance to atmospheric turbulence, because different Q-ary PPM symbols experience different atmospheric turbulence conditions. The BER results of simulations for strong turbulence regime (σ R =3.0, α=5.485, β=1.1156) are shown in Fig. 17, for different number of lasers, photodetectors and number of slots, by employing an (6419,4794) irregular girth-6 LDPC code of rate designed using the concept of pairwise balanced designs, introduced in [J-4]. The BICM scheme with spectral efficiency of 3 bits/symbol combined with MIMO scheme employing 2 lasers and 4 photodetectors provides about 20 db improvement over LDPC coded binary PPM employing one laser and one photodetector.

25 Bit-error rate, BER Electrical SNR, E b /N 0 [db] Uncoded: M=1, N=1: Q=2 Q=4 Q=8 M=2, N=2: Q=2 Q=4 Q=8 M=2, N=4: Q=2 Q=4 Q=8 LDPC(6419,4794,0.747)-coded: BICM: M=1, N=1, Q=2 M=2, N=2, Q=4 M=2, N=4, Q=4 M=2, N=4, Q=8 MLC: M=2, N=4, Q=4 (1.194 bits/symbol) M=2, N=4, Q=8 (2.241 bits/symbol) Fig. 17 BER performance of bit-interleaved LDPC-coded modulation against MLC for different MLMD configurations. Information Theoretic Limits for Free-Space Optical Communication Channels with and without Memory The performance of any communication system can be significantly affected by the channel memory and the availability of the Channel State Information (CSI). The goal of our research is to study the capacities and achievable rates for FSO communication channels subject to different assumptions on the channel memory and the knowledge of CSI [J-7]. The focus is on Intensity Modulation/Direct Detection (IM/DD) FSO systems subject to different degrees of optical turbulence (inducing intensity fluctuation on the received signal) and Additive White Gaussian Noise (AWGN) introduced by the receiver electronics. To cover both the strong and the weak turbulence regimes, the received signal intensity fluctuations are modeled by a gammagamma distribution. With respect to the channel memory assumption, two scenarios are investigated: 1) intensity fluctuations are temporally Independent and Identically Distributed (IID) and 2) intensity fluctuations are described by a Markov model. It should be noticed that the results for an IID model can be applied to the so-called block-fading channels under the assumption that the channel fluctuations correspond to a stationary and ergodic random process. For the IID scenario, two input distributions are considered: 1) discrete uniform and 2) positive Gaussian (i.e., a Gaussian distribution that generates inputs which are positive with probability close to 1). Such input

26 distributions are chosen for two reasons. First, an input distribution that maximizes the mutual information under the average optical power constraint for the positive input is not known in general. Secondly, a positive Gaussian input (for which the mutual information can be computed) will help us gain intuition on the behavior of the achievable rates under different CSI assumptions. As we will show, for the strong turbulence regime and low-to-moderate SNRs, the knowledge of CSI at both the transmitter and the receiver gives a higher achievable rate than that of the case for which the CSI is present at the receiver only. This means that adaptive communication strategies (i.e., those using feedback and adaptive coding), can be beneficial. For weak turbulence regimes, knowing the CSI at the transmitter is no longer beneficial. In this case a simple technique of channel inversion is possible, enabling the use of codes for AWGN channels. In both regimes, for low SNRs, the positive Gaussian inputs yields lower achievable rates than M=4 PAM. However, for high SNRs, the Gaussian input distribution is more efficient than the PAM. Thus, it follows that larger multilevel (than the currently used M=2) and more efficient signal constellations have to be designed. In the case of FSO channels with memory, a Markov model is used (which is a generalization of the Gilbert-Elliot model) that assumes no knowledge of CSI at either the transmitter or the receiver. To extract a transmitted signal, the receiver uses the knowledge of the communication channel distribution, which is commonly referred to as Channel Distribution Information (CDI). The channel capacity is computed for strong and weak turbulence regimes and for different values of the channel quasifrequency. Quasifrequency is a parameter analogous to Doppler spread found in Radio- Frequency (RF) fading channels, and it represents a measure of the effective bandwidth of the turbulence-induced signal fluctuations. From extensive measurements, it is determined that a practical range for the quasifrequency is between 100 Hz and 500 Hz. Numerical results also show little change in the capacity as a function of quasifrequency. The capacity computations are carried out for a simple modulation format such as PAM. Because FSO channel is envisioned as the solution to the connectivity bottleneck problem and as a supplement to wireless links, the complexity of transmitter and receiver must be low. Therefore, IM/DD is proposed as a reasonable choice for FSO links. Since the negative signal cannot be transmitted over an FSO link with direct detection, PAM is a viable modulation format. Other multilevel schemes, such as those based on quadrature amplitude-modulation require the use of DC bias, and the power efficiency of such schemes is low. Fig. 18 shows the achievable rates for uniform PAM signaling when M = 2, 4, 8 and 16, and for the positive Gaussian input, when four different transmitter/receiver strategies are applied. Rtr, Rr, Rinv and Rtinv denote the achievable rates for the transmitter and receiver state information case,