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 multi- channel 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,
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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 multi-
channel 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,
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 free-
space 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
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
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 self-efficiency 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
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.
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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 τ0=3.6 ms and scintillation σI
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
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 Vmax. 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(2p) 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
Fig. 5 Operations at the check node
Let us take for example, a code over GF (23). The received vector at a
variable node would be r= {p(0), p(1), p(α), … , p(α6)}. 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 Lr 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 Lr, and Vmax number of votes is distributed among these symbols in the list based on their respective probabilities. The outgoing lists L1, L2 and L3 are then made equal to Lr for the initial iteration. This is illustrated in Fig. 4. X1, X2, X3 and X4 denote the most likely symbols chosen from the received vector r. Also
V x V x V x V x V( ) ( ) ( ) ( ) max1 2 3 4+ + + =
Note that not only the list Lr 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 β1, β2, β3, β4 and β5 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 β5 of the outgoing edge over GF(2p). In our
example, we need to check 44 combinations and then see what the resulting symbols are. In general, for any check node of degree dc, we would need to check 4dc-1 combinations (or Mdc-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(2p) since Y may or may not contain all the symbols in the field. From Fig. 5, pi, qi, rk and sl 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
yp q r si j k l
=+ + +β β β β
β1 2 3 4
5
where pi, qj, rk and sl are the symbols from their respective lists that satisfy the above equation. Let ξy denote the set of all possible i,j,k,l indice vectors whose corresponding symbols (pi, qi, rk, sl) satisfy the above equation. Then the number of votes assigned to the symbol y is given by
)()()()()(),,,(
lkjlkji
i sVrVqVpVyVy
∑∈
=ξ
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 y1, y2, y3 and y4 were the 4 symbols chosen from the set Y, then these symbols would form the outgoing list with their corresponding votes as [V(y1),V(y2),V(y3),V(y4)]. However, we must redistribute the votes of the symbols such that the total number of votes in the list is Vmax 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 Vmax where Vmax determines the accuracy of the multiplicities compared to their received probabilities (similar to the notion of having a precision error). When the value of Vmax 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 L2, L3 and the list Lr derived from the received vector r. Let Z denote the set of possible symbols on the outgoing list and Z
C GF(2p). Let l2, l3 and lr denote the set of symbols in the lists L2, L3 and Lr respectively (note that the lists contain the votes of those symbols i.e. L2 = V(l2) and so forth). The set Z is obtained by the condition
Z l l lr= ∩ ∩2 3 This is because we only consider the symbols that are present in all the incoming lists and lr, 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
)()()()( 32 zLzLzLzV r=
where L2(z) denotes the number of votes stored in the list L2 for the symbol z. Similar is the notation for L3(z) and Lr(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 z1, z2, z3 and z4 so that the outgoing list is [V(z1),V(z2),V(z3),V(z4)]. However, we must again redistribute the votes so that total number of votes is Vmax 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.
Coded orthogonal frequency division multiplexing (coded-OFDM) To deal with atmospheric turbulence we proposed to use either: (i) coded-orthogonal 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 ith subgroup containing bi bits, B=∑bi. The bi bits from the ith subgroup are mapped into a complex-valued signal from a 2bi -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 multi-carrier 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 intensity-modulation/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.
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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.
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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 LiNbO3 MZM. In this case, the clipping will be performed by the receiver through the squaring operation inherent in the measurement of optical intensity (by
photodetector). The distortion introduced by photodetector may be reduced by proper filtering. Notice that the U-OFDM scheme will be less power-efficient 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).
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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
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 20.24 dB for QPSK, and 23.38 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 (σR=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 10-5.
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 σX
is set to 0.6 (notice that σX is different from Rytov standard deviation σR used earlier, and for horizontal paths σX∼0.498σR).
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 (σX=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 σX. To generate temporally correlated samples we used two different methods, the first one
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.
τ0= 10 μs, σX=0.1: Uncoded, OOK LDPC(4320,3242), OOK
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
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 xk (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
{ }1 2 1 2, ,..., , , ,...,K Kx x x x x x 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)
1 2
2 1
x xx x⎡ ⎤
= ⎢ ⎥⎣ ⎦
O
corresponds to the Alamouti-like ST code. Here, 1i ix x= − denotes the binary complement of xi.
Sourcebits
Space-time code encoder
Transmitter 1
. . .
Transmitter M
LDPC encoder
Receiver 1
.
.
.
Receiver N
Processor Detectedbits
Atmospheric Turbulence Channel
Sourcebits
Space-time code encoder
Transmitter 1
. . .
Transmitter M
LDPC encoder
Receiver 1
.
.
.
Receiver N
Processor Detectedbits
Atmospheric Turbulence Channel
(a)
Light beam through turbulent channel
optical amp.
Expanding telescope
Detector 1
Fiber
Collimating lens
TA amplifier
Optical source
mth Transmitter
Receiver array
Detector N
…
Light beam through turbulent channel
optical amp.
Expanding telescope
Detector 1
Fiber
Collimating lens
TA amplifier
Optical source
mth Transmitter
Receiver array
Detector N
…
(b)
Space-time Soft Decoder LDPC Decoder
From FSO linkReceiver array
ProcessorDetected bitsSpace-time
Soft Decoder LDPC DecoderFrom FSO link
Receiver array
ProcessorDetected 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 (σR=3.0) for different number of optical sources M, and photodetectors N, against the electrical SNR ratio per photodetector, denoted by E/N0, 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.
Fig. 12 i.i.d channel capacity for different numbers of optical sources (M), and photodetectors (N) in strong turbulence regime (σR=3.0, α=5.485, β=1.1156) for: (a) binary transmission, and (b) Q-ary PAM.
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 (Es)-to-power-spectral density (N0) ratio. A significant spectral efficiency improvement is possible by using the multi-level schemes.
8 12 16 20 24 28 32
2
4
6
8
10
12
14
16
18
Info
rmat
ion
Rat
e, R
[bits
/cha
nnel
use
]
Electrical SNR, Es/N0 [dB]
M=1, N=1M=2: N=1 N=2 N=4M=4: N=2 N=4
Fig. 13 The MIMO achievable information rates for different number of lasers (M), and photodetectors (N) in strong turbulence regime (σR=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 (σR=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 0.747 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 space-time 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.
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
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.
Fig. 16 BER performance of BI-LDPC-coded PAM with repetition MIMO for different LDPC component codes.
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 code-words 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=2m) 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 Ts a pulse of light of duration T=Ts/Q is transmitted by a laser. (The signaling interval Ts is subdivided into Q slots of duration T.) The total transmitted power Ptot is fixed and independent of the number of lasers so that emitted power per laser is Ptot/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 (σR=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 0.747 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.
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 gamma-gamma 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
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,
receiver state information, channel inversion, and truncated channel inversion, respectively. Here, the values of the gamma-gamma distribution parameters are α = 10, β = 1, which corresponds to the strong turbulence regime. In addition, Fig. 18 shows the upper bound on the capacity Cub. In the case of the positive Gaussian input, the computations show that CSI at the transmitter side may contribute to the increase in the achievable rate by 60% at 10 dB (Rr = 0.2 bits/channel use vs. Rtr = 0.35 bits/channel use), 30% at 1 dB (Rr = 0.5 bits/channel use vs. Rtr = 0.65 bits/channel use), and 17% at 20 dB (Rr = 0.8 bits/channel use vs. Rtr = 0.93 bits/channel use). This comparison points out the difference between FSO fading and RF fading, where the contribution to the capacity of the CSI at the transmitter is negligible. A reason for this might be more detrimental nature of FSO fading comparing to the RF fading. This would mean that in the case of FSO fading, the CSI at the receiver only is not enough to achieve the maximum of the mutual information; in addition, the CSI at the receiver has to be utilized. For higher SNRs there is no benefit of the CSI at the transmitter relative to the case of the CSI at the receiver only. Thus, the feedback in FSO systems can bring benefit at low to moderate SNRs. In the case of the channel inversion, the achievable rate is zero for reasonably high SNRs, as in the case of the Rayleigh RF fading (Nakagami fading with m = 1). The truncated channel inversion shows weaker performance than the case when the CSI is available at the transmitter and the receiver.
Fig. 18 Achievable rate vs. SNR for strong turbulence for “positive” Gaussian input
Comparing to the uniform PAM signaling, we notice that the “positive” Gaussian distribution can have worse performance below some threshold SNR. For instance, for M = 2, this threshold is at 17 dB, and for M = 4, the threshold is at 27 dB. Thus, On-Off Keying (OOK) is not optimal signaling for moderate and high SNRs. To improve capacity, one should consider PAM systems with M = 4, M = 8 or M = 16. Fig. 18 shows that the PAM with M = 16 is 7 dB away from the upper capacity bound at 25 dB, while the positive Gaussian exhibits a constant distance of 9 dB from the capacity bound. To improve the capacity, we obviously need to design new signal constellations that will take into account a positivity condition.
In Fig. 19, the achievable rates are presented for α = 17.13, β = 16.04, corresponding to a weak turbulence regime. In the case of the “positive” Gaussian input, it can be noticed that the CSI at the transmitter does not give any advantage as compared to other strategies. Hence, one can use the channel inversion or the truncated channel inversion policy and employ codes for Gaussian channels. This means that a weak turbulence situation for FSO channels roughly corresponds to the Nakagami fading with a parameter m > 1 of the RF fading channel. On the other hand, the PAM signaling is now even more effective than the Gaussian input for lower SNRs, and the PAM signaling with M = 16 is 4 dB away from the upper capacity bound at around 25 dB. Comparing with the strong turbulence channels, the weak turbulence channel capacity is noticeably larger. For instance, in the weak turbulence case and for M = 2, a required SNR to achieve the capacity of 1 bits/channel use is 32 dB, while in the strong turbulence case, 70 dB is required.
Fig. 19 Achievable rate vs. SNR for weak turbulence for “positive” Gaussian input
Fig. 20 shows the capacity of the FSO Markov channel for strong and weak turbulence where the parameter is the quasifrequency ν0. Two values are chosen for the quasifrequency, ν0 = 275 Hz and ν0 = 550 Hz. The values of α and β are the same as before. Rs,550 and Rw,550 are the capacities for strong and weak turbulence when ν0 = 550 Hz, respectively. A similar notation is used when ν0 = 275 Hz. In Fig. 20, a number within parenthesis denotes the number of states of a corresponding Markov chain model. It can be noticed again that the capacity for the weak turbulence is larger than the capacity for the strong turbulence. The smaller the quasifrequency (corresponding to larger channel memory) the larger the capacity; however the difference in the capacity when the quasifrequency is doubled for the same turbulence regime is negligible. This is in accordance with the capacity results for the RF wireless channels that exhibit little difference for the similar change in the Doppler spread. Also, for a fixed ν0 and fixed turbulence regime, the difference between capacity curves decreases as the number of states of the Markov chain increases. Consequently, it can be conjectured that as the number of the Markov chain states increases, the capacity sequence will converge to a true value of the capacity since the resolution (or accuracy) of the Markov model increases. From plots, it can be concluded that 8-state Markov chain model gives a pretty good estimation of the FSO channel capacity. It is assumed that the channel symbol interval is Ts = 10-5 sec, which is much smaller than the reciprocal value of the quasifrequency, such that the FSO channel is slow fading channel.
Fig. 20 Capacity vs. SNR using OOK with quasifrequency as a parameter
Fig. 21 (a) PDF of an experimentally recorded channel waveform with σI
2= 1.04 (strong fluctuations). A gamma-gamma density is fitted to the experimental data to show the suitability of the model. (b) Normalized autocovariance of the channel waveform used in (a). The correlation time is τ0 = 4.8 ms and the quasifrequency is ν0 = 206 Hz. We further evaluate the Markovian model using experimentally recorded FSO channel samples. An experimental apparatus similar consists of a continuous-wave laser beam with wavelength 650 nm, which is expanded and projected toward a reflector 300 m away on a horizontal path.
The reflected beam is collected by a telescope located next to the transmitter, thus completing a 600-m optical path. The collected light is focused on a multimode fiber whose other end is connected to an optical detector. The optical detector delivers a current proportional to the incoming optical signal that is recorded with an oscilloscope. The recorded waveforms deliver temporally correlated channel samples, providing the most realistic way to evaluate the FSO channel.
Fig. 22 (a) PDF of an experimentally recorded channel waveform with σI
2= 0.17 (weak fluctuations). A gamma-gamma density is fitted to the experimental data to show the suitability of the model. (b) Normalized autocovariance of the channel waveform used in (a). The correlation time is τ0 = 7.1 ms and the quasifrequency is ν0 = 140 Hz.
Fig. 23 Capacity vs. SNR for the OOK transmission for a strong and weak turbulence, when the channel models are obtained from experimental data
Two channel waveforms are selected for this evaluation. The first waveform features a scintillation index σI
2= 1.04, in the strong turbulence regime. Fig. 21(a) shows a histogram of the experimental channel samples. A gamma-gamma distribution has been fitted to these experimental data (with parameters α= 2.5 and β = 2.2) to show the suitability of this analytical model. The autocovariance of the channel waveform, normalized to a maximum of one, is shown in Fig. 21(b). From this plot we compute the quasifrequency ν0 = 206 Hz using the Rice formula. The correlation time of the channel τ0, which we define as the reciprocal value of the quasifrequency ν0, is equal to 4.8 ms. In Fig. 22(a), we present the histogram of a channel waveform in the weak turbulence regime, with a scintillation index σI
2= 0.17. In this case, the gamma-gamma density is fitted with α= 13.5 and β = 12. As before, the normalized autocovariance of the channel waveform is computed, giving ν0 = 140 Hz and τ0 = 7.1 ms. This is shown in Fig. 22(b). We have chosen a transmission rate of R = 1 Mbits/sec, which corresponds to a channel symbol interval of Ts = 10-6 sec. The capacity is determined using a Markov model with 8 states for both experimental cases described above. Fig. 23 shows the channel capacities for two studied scenarios, labeled Rs,206 in the strong turbulence case and Rw,140 in the weak turbulence case. The capacity curve in the latter case reaches the maximum value of 1 bits/channel use at approximately 24 dB, while for the former case even 30 dB is not sufficient to reach its saturation point.
Fig. 24 The girth-10 LDPC code against RS, concatenated RS, turbo-product, and girth-8 LDPC codes on an AWGN channel model. Large-girth LDPC codes suitable for optical communications The parity check-matrix H of quasi-cyclic LDPC codes, we proposed in [C-12], can be described as
[ ] [ ] [ ]
[ ] [ ] [ ]
( ) [ ] ( ) [ ] ( ) [ ]
1 2 1
2 1 2 2 2 1
1 1 1 2 1 1
...
...,...
... ... ... ... ...
...
S S S c
S S S c
r S r S r S c
I I I I
I P P PH I P P P
I P P P
−
−
− − − −
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
where I is pxp (p is a prime number) identity matrix, P is pxp permutation matrix (pi,i+1=pp,1=1, i=1,2,…,p-1; other elements of P are zeros), while r and c represent the number of rows and columns in (1), respectively. The set of integers S are to be carefully chosen from the set {0,1,…,p-1} so that the cycles of short length, in corresponding Tanner (bipartite) graph representation of H are avoided. We have shown in [C-12] that large girth (the shortest cycle in bipartite graph), g≥10, LDPC codes provide excellent improvement in coding gain over corresponding turbo-product codes (TPCs).
At the same time complexity of LDPC codes is lower than that of TPCs, selecting them as excellent candidates for application to systems for beyond 40 Gb/s transmission. For example, by selecting p=1123 and S={0, 2, 5, 13, 20, 37, 58, 91, 135, 160, 220, 292, 354, 712, 830} an LDPC code of rate 0.8, girth g=10, column weight 3 and length N=16845 is obtained.
The results of simulations for an additive white Gaussian noise (AWGN) channel model are given in Fig. 24, where we compare the proposed LDPC codes against RS, concatenated RS, turbo-product, and girth-8 LDPC codes. The girth-10 LDPC(24015,19212) code of rate 0.8 outperforms the concatenation RS(255,239)+RS(255,223) (of rate 0.82) by 3.35 dB, and RS(255,239) by 4.75 dB, both at BER of 10-7. At BER of 10-10 it outperforms lattice based LDPC(8547,6922) of rate 0.81 and girth-8 by 0.44 dB, and BCH(128,113)xBCH(256,239) TPC of rate 0.82 by 0.95 dB. The net effective coding gain at BER of 10-12 is 10.95 dB, which represents the largest net effective coding gain, on an AWGN channel, ever reported in optical communications. GLDPC Codes with Reed-Muller Component Codes Suitable for Optical Communications The generalized LDPC (GLDPC) codes with Reed-Muller (RM) codes as component codes, we proposed in [C-5], is an attractive option for high-speed optical transmission because they provide excellent coding gains, while the RM codes can be decoded using low-complexity maximum a posteriori probability (MAP) decoding based on fast Walsh-Hadamard transform.
To construct a GLDPC code, one can replace each single parity-check equation of a global LDPC code by the parity-check matrix of a simple linear block code, known as the constituent (local) code, and this construction is proposed by Lentmaier and Zigangirov, and we will refer to this construction as LZ-GLDPC code construction. In another construction proposed by Boutros et al., referred here as B-GLDPC code construction, the parity-check matrix, H, is a sparse matrix partitioned into W sub-matrices H1,…,HW. H1 is a block-diagonal matrix generated from an identity matrix by replacing the ones by a parity-check matrix of a local code of codeword-length n and dimension k. Each sub-matrix Hj is derived from H1 by random column permutations. In our proposal, we use the RM codes as constituent codes.
A Reed-Muller code RM(r,m) of order r and codeword length n=2m is the set of all binary vectors associated with coefficients of Boolean polynomials of degree at most r in m variables. An interesting property of RM codes is that they can be defined recursively: RM(r,m)={(a|a+b): a∈RM(r,m-1), b∈RM(r-1,m-1)}, where (x|y) denotes the concatenation operation. The generator matrix of RM(r,m) code, denoted as G(r,m), can be therefore defined recursively by
( ) ( ) ( )( )
, 1 , 1, .
0 1, 1G r m G r m
G r mG r m
− −⎡ ⎤= ⎢ ⎥− −⎣ ⎦
RM(0,m) is a repetition code, RM(m-1,m) is a parity-check code, and RM(m,m) corresponds to 2m-tuples of a vector space. Another interesting property of RM codes is that the dual of RM(r,m) code is another RM(m-r-1,m) code. Therefore, the generator matrix of RM(m-r-1,m) code can be used as the parity check matrix of RM(r,m) code. If the recursion above is applied successively several times the RM(r,m) can be decomposed into several parity-check codes RM(m’-1,m’), repetition codes RM(0,m’), and the first-order RM(1,m’) codes. The MAP decoding of parity-check or repetition codes is trivial, while the first order RM(1,m’) codes can be decoded using an efficient MAP decoding algorithm based on fast Hadamard-Walsh transform (FHWT). The overall complexity of that algorithm is in order n’log2n’ (where n’=2m’), which is significantly lower than complexity of the BCJR algorithm that requires about nn-k+1 operations. Therefore, the complexity of GLDPC codes with RM component codes is of order N log2n. Since the complexity of sum-product algorithm is of order (NLDPC-KLDPC)wr, with wr being the row weight of LDPC code parity check matrix, by proper selection of global LDPC code length N and local RM code length n, the complexity of GLDPC codes is about (NLDPC-KLDPC)wr/[(N/n)∑(n’log2n’)] (n’<n) times lower. For example, RM(4,6) code can be decomposed using the equation above on RM(1,2), RM(1,3), RM(2,2), RM(3,3), and RM(4,4) component codes. Decoding of RM(m’,m’) (m’=1,2,3,4) is trivial while the complexity of RM(4,6) is dominated by complexity of RM(1,3) decoding block and three RM(1,2) blocks, which is of order ∑(n’log2n’) =8log28+3·4log24=48. B-GLDPC code with W=2 and length N=4096 based on RM(4,6) code has therefore complexity 11 times lower than girth-8 column-weight-4 LDPC code of length 8547 (and row weight 21. Notice also that GLDPC decoder for 4096 code contains 4096/64=64 decoder blocks [composed of RM(1,2), RM(2,3) and RM(m’,m’) (m’=1,…,4) decoders], operating in parallel, and this structure is suitable for FPGA or VLSI implementation.
The results of simulation for the AWGN channel model are shown in Fig. 25, and are obtained by maintaining the double-precision. GLDPC codes based on BCH(63,57) and RM(4,6) component codes for W=2 perform comparably. The RM(4,6)-based GLDPC code outperforms the BCH(128,113)x BCH(256,239) TPC based with Chase II decoding algorithm on p=3 least reliable bit positions by 0.93 dB at BER of 10-9 (see Fig. 25(a)).
(b) Fig. 25 BER performance on an AWGN channel: (a) GLDPC codes against TPCs and LDPC codes, (b) high-rate codes.
The TPC codeword is significantly longer, and the decoding complexity of GLDPC code based on RM(4,6) is at least 10 lower. During decoding TPC decoder employs 239 Chase II blocks operating in parallel, while GLDPC code on RM(4,6) code requires only 64 low-complexity MAP decoders as explained in Section II. In simulations presented here we have employed an efficient realization of Chase II algorithm proposed. In Fig. 25(b) BER performance of several classes of iteratively decodable codes (TPCs, LDPC and GLDPC codes) of high code rate are compared against conventional RS, and concatenated RS codes. B-GLDPC code of rate 0.88 outperforms concatenated RS code of rate 0.82 by 2.47 dB (also at BER=10-9).
RM(6,8)xRM(5,7) TPC of rate 0.905 outperforms concatenated RS code (R=0.82) by 0.53 dB at BER of 10-9. LDPC code of rate 0.93, designed using the concept of product of orthogonal arrays (OAs), outperforms the same RS concatenated code by 1.15 dB at BER of 10-9. R=0.93 LDPC code outperforms RS code of rate 0.937 by 2.44 dB at BER of 10-9. 3. Describe the opportunities for training and development provided by your project. 1. Two graduate students and one post-doc researcher are working on the
project. Another graduate student is expected to arrive in August. 2. Advanced error control coding techniques, LDPC codes and iterative
decoding are introduced into ECE 535, ECE 537, and ECE 632. 4. Describe outreach activities your project has undertaken.
a. Dr. Djordjevic presented an invited paper at the 41st Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 4-7, 2007.
b. Dr. Djordjevic will present an invited paper, with recent achievements from the project, at The 21st Annual Meeting of the IEEE Lasers & Electro-Optics Society, Newport Beach, CA, 9-13 November 2008.
c. Dr. Djordjevic is serving as an Associate Editor of Research Letters in Optics (see http://www.hindawi.com/journals/rlo/editors.html)
d. Dr. Vasic is serving as a Technical Program Chair Member of 5th International Symposium on Turbo Codes & Related Topics 2008, Lausanne, Switzerland, Sept. 2-5, 2008.
e. Dr. Vasic served as a Technical Program Chair Member of the Communication Theory Symposium of the Global Communications Conference (Globecom 07), which will be held in Washington DC, in November 25-30, 2007.
f. Dr. Vasic served as a Co-Chair of the Communication Theory Workshop in Sedona, Arizona, May 20-22, 2007.
g. In previous year, we published 11 Journal papers, and presented 12 conference papers.
Contributions Now we invite you to explain ways in which your work, your findings, and specific products of your project are significant. Describe the unique contributions, major accomplishments, innovations and successes of your project relative to:
1. The principal discipline(s) of the project;
Orbital angular momentum-based multi-channel communications 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. Simulations that were carried out based on numerical methods verified that optical turbulence induces OAM crosstalk and that the average crosstalk between channels grows with turbulence strength. 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. By modeling an OAM mode as a BSC channel with probability of flip error being a function of the channel efficiency, the crosstalk induced by the other constituent channels, and by detector noise, the optimal set of OAM mode numbers in the sense of maximizing the aggregate capacity were determined for a prescribed number of channels, at each value of SNR and turbulence strength considered. Raptor codes for temporally correlated channels
We have performed an evaluation of a Raptor code on the physical layer of a FSO link using experimentally-recorded FSO channel waveforms and we show that Raptor codes can continuously decode without errors under the time-varying channel fluctuations, even during long deep channel fades (at which the effective signal-to-noise ratio may drop by 15 dB with respect to the average channel SNR) by adaptively varying the information data rate. This coding scheme is advantageous as it does not require bit interleaving and involves a simple encoding/decoding algorithm. We have proposed a simple yet effective scheme to determine the channel state information (i.e., the instantaneous channel gain) at the receiver and have found its optimum operating point in terms of the channel's correlation time. We also find that imperfect (i.e., noisy) knowledge of the channel state information at the receiver can yield a near-ideal performance. List decoding algorithm for non-binary LDPC codes We proposed a list decoding algorithm for non-binary LDPC codes that is much lower in complexity compared to the sum product algorithm. We have determined the operations required for updating the message lists on the edges and we have found that these operations mimic the nature of SPA
reasonably well especially at higher signal-to-noise ratios (SNR). 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). In a way, we can consider this algorithm to be an integer approximation of SPA with the exception that the messages are lists that contain the most likely symbols. The fact that the lists contain only the most likely symbols, reduces computations at the check node to a considerable extent compared to SPA where every message contains all the symbols in a field. The algorithm may be more amenable for carrying out trapping set analysis or even density evolution compared to SPA, since we are dealing with a discretized space where the symbols can take only discrete values (votes). The accuracy at which the multiplicity depicts the probability of a particular symbol depends on the list size and Vmax. Coded orthogonal frequency division multiplexing (coded-OFDM) The coded-OFDM is proposed as an efficient scheme that is able simultaneously to operate under the strong atmospheric turbulence, and to solve the RF/microwave-optical incompatibility problem.
The key idea behind this approach is to lower the symbol rate by using OFDM, and, in combination with interleaving and LDPC codes, to obtain high tolerance to the deep fades that are inherent to a turbulent channel. To enable noncoherent recovery of the QAM symbols, the DC biasing is employed (biased-OFDM, B-OFDM). To improve the power efficiency of B-OFDM we propose two alternative schemes:
– “Clipped-OFDM” (C-OFDM) scheme: based on single-side band (SSB) transmission, and clipping of the OFDM signal after bias addition
– “Unclipped-OFDM” (U-OFDM): To avoid distortion due to clipping the information is imposed by modulating the electrical field (instead of intensity modulation employed in B-OFDM scheme) so that negative portion of OFDM signal is transmitted up to the photodetector.
LDPC-coded OFDM based on SSB U-OFDM and BPSK provides more than 23 dB improvement over LDPC-coded on-off keying when operated in strong turbulence regime. LDPC-Coded MIMO Optical Communication over the Atmospheric Turbulence Channel Coded MIMO communication schemes for data transmission over the optical atmospheric turbulence channels are studied. Two strategies are proposed
and compared. The first is based on repetition coding, and the second on space-time coding. Both approaches employ LDPC codes. The LDPC codes are designed using the concept of pairwise-balanced design, balanced-incomplete block design, and block-circulant (array) codes. To improve the spectral efficiency, we employ a bit-interleaved LDPC-coded modulation based on PAM. A better BER performance is achieved by the iteration of extrinsic information between a demapper and LDPC decoder. The simulations show that the LDPC-coded MIMO schemes can operate under a strong atmospheric turbulence and in the same time provide excellent coding gains comparing with the transmission of uncoded data. To verify the efficiency of proposed coding schemes, achievable information rates and BER are computed when the turbulence is modeled by a gamma-gamma distribution. Coded-MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation We considered the coded Q-ary pulse-position modulation (PPM) as a power-efficient transmission scheme. To enable the transmission under the strong atmospheric turbulence we proposed the use of the MIMO concept. We show that this scheme can also be used as an interface between RF/microwave and optical communications technologies. Information Theoretic Limits for Free-Space Optical Communication Channels with and without Memory The availability of CSI and the effects of channel memory on the capacities and the achievable rates of free-space optical communication channels are investigated. For memoryless channels, the capacities and achievable rates are computed and compared for both uniform and “positive” Gaussian inputs subject to different assumptions on the CSI availability. For the strong turbulence regime, it is shown that the knowledge of CSI both at the transmitter and the receiver increases the achievable rates for low-to-moderate SNRs in comparison to the cases for which the CSI is known only at the receiver. For the weak turbulence regime however, the availability of CSI at both ends of the link does not provide any improvement over a system with CSI known at the receiver alone, and we find that a simple channel inversion technique suffices. In addition, for low SNRs, PAM with
4M ≥ levels outperforms Gaussian-distributed inputs regardless of the knowledge of CSI at the transmitter. For high SNRs, a Gaussian distribution gives superior results, implying the need for new, more efficient positive signal constellations. For channels with memory and without knowledge of CSI, a change in the channel quasifrequency has negligible effects on the capacity for any turbulence regime.
Large-girth LDPC codes suitable for optical communications The state-of-the-art optical communication systems standardized by the ITU employ different concatenated Bose-Chaudhuri-Hocquenghem (BCH)/Reed-Solomon (RS) codes. The iteratively decodable codes, such as turbo codes and LDPC codes, have generated significant research attention, recently. Turbo codes based on recursive systematic convolutional (RSC) codes exhibit the error floor phenomena at bit-error ratio (BER) around 10-4, and as such are not suitable for high-speed optical transmission. Instead, turbo product codes (TPCs) with BCH components codes are proposed for use in optical communications. On the other hand, we have shown recently that TPCs can be matched and outperformed by LDPC codes in terms of coding gain and decoding complexity.
The random LDPC codes provide excellent error-correcting capabilities, but are difficult to implement because of the lack of nice mathematical structure in their parity-check matrices. To overcome this shortcoming, we propose the structured LDPC codes instead, which are easy to implement at the expense of certain performance loss. To improve the performance loss of structured LDPC codes we can use the girth (the shortest cycle in corresponding bipartite graph representation of a parity-check matrix) as the optimization parameter.
In our recent paper [C-12] we are presented the design of the structured LDPC codes of girth at least 8 with parity-check matrices represented in a quasi-cyclic (block-circulant or array) fashion, to facilitate the implementation at high-speed. We have noticed that the codes with girth 12 require unacceptable long code words if the code rate is to be kept reasonable high (≥0.8), so that we restricted our attention to girth-10 LDPC codes. We have shown that girth-10 BC LDPC(24015,19212) code of code rate 0.8 outperforms longer BCH(128,113)xBCH(256,239) TPC of code rate 0.82 by 0.95 dB at BER of 10-10 on an AWGN channel model. It provides the net effective coding of 10.95 dB at BER of 10-12 on the same channel. GLDPC Codes with Reed-Muller Component Codes Suitable for Optical Communications GLDPC codes with RM component codes are considered as possible options for high-speed optical transmission. It is demonstrated by simulation that the RM based GLDPC codes are able outperform their turbo product counterparts in terms of coding gain with lower complexity in decoding algorithm. On the other hand, they require larger number of iterations. GLDPC codes perform comparably to girth-8 LDPC codes, but have lower complexity. Several classes of GLDPC codes with RM component codes suitable for use in both free-space optical communications and fiber-optics
communications were presented. GLDPC coding with RM component codes is an attractive option for high-speed optical transmission because it can utilize an efficient low-complexity MAP decoding algorithm based on the fast Walsh-Hadamard transform instead of high-complexity BCJR decoder.
2. Other disciplines of science or engineering;
None 3. The development of human resources; None 4. The physical, institutional, or information resources that form the infrastructure for research and education; and None
5. Other aspects of public welfare beyond science and engineering,
such as commercial technology, the economy, cost-efficient environmental protection, or solutions to social problems.
None
Publications and Products In this section, you will be asked to describe the tangible products coming out of your project. Specifically: 1.What have you published as a result of this work?
Journal publications [J-1] I. B. Djordjevic, and B. Vasic, “LDPC-coded OFDM in fiber-optics
communication systems [Invited],” J. Opt. Netw., Vol. 7, pp. 217-226, 2008.
[J-2] I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC-coded OFDM for optical communication systems with direct detection,” IEEE/LEOS Journal of Selected Topics in Quantum Electronics, vol. 13, no. 5, pp. 1446 - 1454, Sept.-Oct. 2007.
[J-3] I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric
turbulence channel,” IEEE/OSA J. Lightwave Technol., vol. 26, no. 5, pp. 478-487, March 1, 2008.
[J-4] I. B. Djordjevic, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation,” Optics Express, vol. 15, no. 16, pp. 10026-10032, Aug. 2007.
[J-5] I. B. Djordjevic, “LDPC-coded optical coherent state quantum communications,” IEEE Photon. Technol. Lett., vol. 19, no. 24, pp. 2006-2008, Dec. 15, 2007.
[J-6] I. B. Djordjevic, “Quantum LDPC codes from balanced incomplete block designs,” IEEE Comm. Lett., accepted for publication.
[J-7] S. Z. Denic, I. Djordjevic, J. Anguita, B. Vasic, and M. Neifeld, “Information theoretic limits for free-space optical channels with and without memory,” to appear in IEEE J. Lightwave Tech.
[J-8] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” OSA Appl. Opt., vol. 46, no. 26, pp. 6561-6571, Sept. 2007.
[J-9] H. G. Batshon, I. B. Djordjevic, L. L. Minkov, L. Xu, T. Wang and M. Cvijetic, "Proposal to Achieve 1 Tb/s per Wavelength Transmission Using 3-Dimensional LDPC-Coded Modulation," IEEE Photon. Technol. Lett., vol. 20, no. 9, pp. 721-723, May 1, 2008.
[J-10] I. B. Djordjevic, M. Cvijetic, L. Xu, and T. Wang, "Using LDPC-coded modulation and coherent detection for ultra high-speed optical transmission," IEEE/OSA J. Lightwave Technol., vol. 25, pp. 3619-3625, Nov. 2007.
[J-11] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free- space optical link,” to appear in OSA Appl. Opt. (2008). Papers Submitted for Journal Publication [S-1] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Raptor codes for the temporally correlated FSO channel,” submitted for publication.
Books or other non-periodical, one-time publications
[B-1] Adriaan J. van Wijngaarden and B. Vasic (editors), “Theoretical Advances in Information Recording,” The American Mathematical Society, AMS, vol. in the DIMACS Series, in preparation.
[B-2] William Shieh and Ivan Djordjevic, Orthogonal Frequency Division Multiplexing for Optical Communications. Elsevier, in preparation.
Conference Publications [C-1] I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld,
“LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” in Proc. Globecom 2007, Paper no. F02-4, pp. 2220 - 2225, 26-30 November 2007 - Washington, D.C.
[C-2] I. B. Djordjevic, “LDPC-coded optical communication over the atmospheric turbulence channel,” in Proc. 41st Asilomar Conference on Signals, Systems, and Computers, Nov. 4-7, 2007, pp. 1903 - 1909, 2007. (Invited paper.)
[C-3] I. B. Djordjevic, “Coding for free-space optical communications,” The 21st Annual Meeting of the IEEE Lasers & Electro-Optics Society, Newport Beach, CA, 9-13 November 2008. (Invited paper.)
[C-4] I. B. Djordjevic, G. T. Djordjevic, "Bit-Interleaved LDPC-Coded Modulation Suitable for Free-Space Optical Communication over the Atmospheric Turbulence Channel," in Proc. 8th International Conference on Telecommunications in Modern Satellite, Cable and Broadcasting Services, 2007 (TELSIKS 2007), pp. 234-237, 26-28 Sept. 2007.
[C-5] I. B. Djordjevic, “Generalized LDPC codes and turbo-product codes with Reed-Muller component codes," in Proc. 8th International Conference on Telecommunications in Modern Satellite, Cable and Broadcasting Services, 2007 (TELSIKS 2007), pp. 127 - 134, 26-28 Sept. 2007. (Invited paper.)
[C-6] I. B. Djordjevic, L. Xu, T. Wang, "PMD Compensation in Multilevel Coded-Modulation Schemes with Coherent Detection using Alamouti-Type Polarization-Time Coding," IEEE LEOS Summer Topicals 2008, accepted for publication.
[C-7] I. B. Djordjevic, "Mitigation of Linear and Nonlinear Impairments in High-Speed Optical Networks by Using LDPC-Coded Turbo Equalization," to be presented at IEEE LEOS Summer Topicals 2008, invited paper.
[C-8] I. B. Djordjevic, L. Xu, T. Wang, "Generalized LDPC Codes with Component Reed-Solomon Codes for Beyond 100 Gb/s Optical Transmission," IEEE LEOS Summer Topicals 2008, accepted for publication.
[C-9] I. B. Djordjevic, "Beyond 100-Gb/s Optical Transmission based on Coded Modulation and Coherent Detection," to be presented at 2008 Coherent Optical Technologies and Applications (COTA) Topical Meeting, invited paper.
[C-10] I. B. Djordjevic, "LDPC-coded modulation for beyond 100-Gb/s optical transmission," in Proc. Optical Transmission Systems and Equipment for Networking VI-SPIE Optics East Conference, vol. 6774,
pp. 677409-1-677409-14, 9 - 12 September 2007, Boston, MA, USA. (Invited Paper.)
[C-11] I. B. Djordjevic, M. Cvijetic, L. Xu, T. Wang, "Ultra high-speed optical transmission based on LDPC-coded modulation and coherent detection for employment in all-optical network scenario," in Proc. 9th International Conference on Transparent Optical Networks (ICTON 2007), pp. 171 - 174, July 1-5, 2007 - Rome, Italy.
[C-12] I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24-28 2008, Paper no. JWA53.
2.What Web site or other Internet site have you created? http://www.ece.arizona.edu/~fso-comm/index.html 3.What other specific products (databases, physical collections, educational aids, software, instruments, or the like) have you developed? Participants: 1. Ivan B. Djordjevic
Over the past year, Dr. Djordjevic was working on temporally correlated FSO channels, novel error-correction schemes, large-girth LDPC code design, design of generalized LDPC codes with component Reed-Muller codes, hybrid wireless-optical communications, quantum LDPC codes design, optical coherent state quantum communications, coded-OFDM, coded-MIMO, and communication over the strong atmospheric turbulence channel.
2. Bane Vasic 3. Mark A. Neifeld 4. Jaime Anguita
Theoretical work on free-space optical communications systems. Channel modeling, simulation, and analysis of error control techniques.
5. Shiva Planjery
Working on high-speed iterative decoders for non-binary LDPC codes. Such codes provide larger coding gains even at moderate block lengths compared to binary LDPC codes but traditional decodres have very complexity. We are investigating fixed precision algorithms suitable for FSO applications.
6. Ruchicka Verma (co-supervised with Prof. Akoglu) In spring 2008, she was working 10 hours per week on implementation of LDPC codes in Verilog.
7. Nariman Rahimian This PhD student is expected to arrive in August 2008. His research will be on coded-MIMO and coded-OFDM for FSO communications.
Have You had Other Collaborators or Contacts?
1. Lei Xu, NEC Labs, Princeton, NJ 2. Ting Wang, NEC Labs, Princeton, NJ 3. Franko Kueppers, Optical Sciences, University of Arizona 4. Milorad Cvijetic, NEC America, Inc. 5. Raymond Kostuk, Department of Electrical and Computer Engineering,
Univesity of Arizona, Tucson, Arizona 6. Milos Ivkovic, LSI Corporation, Allentown, Pennsylvania 7. Olgica Milenkovic, Department of ECE, University of Illinois at Urbana-
Champaign, Urbana, Illinois 8. Ildar Gabitov, Department of Mathematics, University of Arizona 9. Nikola Alic, University of California-San Diego 10. Robert Indik, Department of Mathematics, University of Arizona 11. Sundararajan Sankaranarayanan, Seagate Technology, Pittsburgh 12. Siyuan Yu, University of Bristol, UK 13. Takashi Mizuochi, Mitsubishi Electric Corporation, Kanagawa, Japan 14. Sergei K. Turitsyn, Aston University, Birmingham, UK 15. Ali Akoglu, Department of ECE, University of Arizona 16. Stojan Denic, Toshiba Wireless, Bristol, UK 17. Goran T. Djordjevic, University of Nis, Serbia
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