We define a symbol transition matrix (STM) as a matrix whose entries are the symbol transition probabilities of a relay in a decode-and-forward (DF) relaying system. Using the STM as a state transition matrix of a discrete-time Markov chain, the source-to-destination STM is shown to be the product of intermediate STMs in multihop DF relaying systems. We show that the probability of correct decision at the destination is the trace of the source-to-destination STM divided by the modulation order. The symbol error probabilities (SEPs) of multihop DF relaying systems with any modulation scheme in any independent and nonidentically distributed (ind) channels are derived using the STM. For multihop DF relaying systems in independent and identically distributed (iid) channels, the eigenvalues of the single link STM are used to simplify the calculation of the SEP for any modulation scheme. Also, the SEP of multihop DF relaying systems with 16-quadrature amplitude modulation (QAM) signals in ind Rayleigh fading channels is derived using the STM. A relay advantage criterion is derived for multihop DF relaying systems in iid Nakagami-m fading channels with both the total transmitted energy and the source-to-destination distance fixed. The relay advantage criterion can be used for deciding whether putting more relays between the source and the destination increases or decreases the SEP.

Novel expressions for the symbol error probability (SEP) of maximal ratio combining (MRC) diversity detection are derived. The case of independent and identically distributed Nakagami-m fading channels is considered. It is shown that expressions for the SEPs of MRC diversity operating in Nakagami-m fading channels can be obtained by both replacing the diversity order, L, with mL and replacing the average signal-to-noise ratio (SNR) per bit, γ̅ with γ̅/m in expressions for the SEPs of MRC diversity in Rayleigh fading channels. An identity which states that the bit error probability (BEP) of L-branch soft-decision MRC detection equals that of (2L - 1)-branch hard-decision detection with majority logic combining, both in independent Rayleigh fading channels and with equal branch SNRs, is recalled. The identity is used to derive new expressions for the SEP of MRC diversity in Nakagami-m fading channels. It is also shown that outage probability and average level-crossing rate for MRC reception in Nakagami-m fading channels can be obtained from corresponding outage probability and average level-crossing rate for MRC reception in Rayleigh fading channels. The relationships established will find application both in deriving solutions to unsolved problems and, where past solutions exist, in finding simpler solutions.

The potential benefits of deploying Matrix C in mobile WiMAX applications are investigated for improving quality of service (QoS). Also, it is shown that Matrix C in the IEEE 802.16e standard is not a Golden code, as previously thought, but a threaded algebraic space-time (TAST) code.

We examine the decoding of Matrix C in the 802.16e WiMAX standard. An exhaustive search and zero-forcing (ES-ZF) decoder and an exhaustive search and nulling canceling (ES-NC) decoder are proposed for uncoded systems. The computational complexity of Matrix C decoding using the ES-ZF decoder is shown to be the same as the complexity of the ZF decoder for Matrix B decoding with twice the number of receive antennas times the complexity of the maximum likelihood (ML) decoder for Matrix B decoding with twice the number of receive antennas. Matrix C can be implemented in a $2 × 2$ multiple-input multiple-output (MIMO) system using the ES-NC decoder with reduced complexity compared to ML decoding with no performance loss. For coded systems, double pruned trees using a zero-forcing (DPT-ZF) algorithm or nulling canceling (DPT-NC) algorithm are proposed. The DPT-NC decoder can be implemented in a $2 × 2$ MIMO coded system with reduced complexity compared to the Max-Log decoding with no performance loss.

Binary weightbooks for multiple-input multiple-output (MIMO) beamforming systems using quantized feedback are designed based on the Grassmannian beamforming criterion. It is shown that the Grassmannian criterion for binary weightbook design is to maximize the minimum Hamming distance of the corresponding block code. Thus, block codes that have large minimum Hamming distance can be used for binary weightbook design for MIMO beamforming systems using quantized feedback. The optimum binary weightbooks when the number of transmit antennas has value from two to six are tabulated.

A signal-plus-noise (S+N) energy selection diversity combining (S+N-ESC) scheme for M-ary phase shift keying (MPSK) in Rayleigh fading channels is mathematically analyzed. Also, the S+N-ESC scheme for MPSK and 16-quadrature amplitude modulation (16-QAM) signaling in Nakagami-m and Rician fading channels is studied by computer simulations. The S+N-ESC selects the branch with the maximum S+N energy for detection. The advantage of the S+N-ESC scheme is that it requires no explicit channel estimation for diversity branch selection. We show that the symbol error probability (SEP) of the S+N-ESC is the same as that of classical selection combining (SC) for MPSK signaling in independent and identically distributed (i.i.d.) Rayleigh fading channels. Also, we show that the SEP of a weighted S+N-ESC scheme is the same as that of classical SC for MPSK signaling in independent and nonidentically distributed (i.n.d.) Rayleigh fading channels. It is shown by computer simulations that the S+N-ESC scheme provides a smaller SEP than classical SC for MPSK signaling in nondispersive fading channels such as Nakagami-m fading channels with Nakagami parameter m >; 1 and Rician fading channels with Rician parameter K >; 0. On the other hand, the S+N-ESC scheme is shown by computer simulations to have a larger SEP than classical SC for 16-QAM signaling in Nakagami-m and Rician fading channels.

The exact average bit error probability (BEP) performance of decode-and-forward (DF) cooperative systems for binary phase shift keying (BPSK) signals in Nakagami-m fading channels for integer values of m is analyzed. Two types of receivers, a λ-maximal ratio combining (λ-MRC) receiver and a piecewise linear (PL) receiver are examined. Closed-form, integral-free, expressions for the BEPs of the λ-MRC receiver and the PL receiver are obtained.

The decoding of Matrix C in the WiMAX standard is investigated. We propose an exhaustive search and zero-forcing (ES-ZF) decoder and an exhaustive search and nulling canceling (ES-NC) decoder for uncoded systems. The computational complexity of the ES-ZF decoder for Matrix C decoding is shown to be the same as the complexity of the ZF decoder for Matrix B decoding with twice the number of receive antennas times the complexity of the maximum likelihood (ML) decoder for Matrix B decoding with twice the number of receive antennas. Matrix C can be implemented in a 2 x 2 multiple-input multiple-output (MIMO) system using the ES-NC decoder with reduced complexity compared to ML decoding with no performance loss. For coded systems, double pruned trees using a zero-forcing (DPT-ZF) algorithm or nulling canceling (DPT-NC) algorithm are proposed. The DPT-NC decoder is shown to be implemented in a 2 x 2 MIMO coded system with reduced complexity compared to the Max-Log decoding, with no performance loss, by computer simulations.

We design binary weightbooks for multiple-input multiple-output (MIMO) beamforming systems using quantized feedback based on the Grassmannian beamforming criterion. We show that the Grassmannian criterion for binary weightbook design is to maximize the minimum Hamming distance of the corresponding block code. Thus, a block code that has a large minimum Hamming distance can be used for binary weightbook design for MIMO beamforming systems using quantized feedback. The optimum binary weightbooks when the number of transmit antennas has value from two to five are tabulated.

A signal-plus-noise (S + N) energy selection diversity combining (S + N-ESC) scheme for M-ary phase shift keying (MPSK) and 16-quadrature amplitude modulation (16-QAM) signaling in Nakagami-m and Rician fading channels is studied. The S + N-ESC selects the branch with the maximum S + N energy for detection. The advantage of the S + NESC scheme is that it requires no explicit channel estimation for diversity branch selection. We show that the symbol error probability (SEP) of the S + N-ESC is the same as that of classical selection combining (SC) for MPSK signaling in independent and identically distributed (i.i.d.) Rayleigh fading channels. Also, we show that the SEP of a weighted S + N-ESC scheme is the same as that of classical SC for MPSK signaling in independent and nonidentically distributed (i.n.d.) Rayleigh fading channels. It is shown that the S + N-ESC scheme provides a smaller SEP than classical SC for MPSK signaling in nondispersive fading channels such as Nakagami-m fading channels with Nakagami parameter m > 1 and Rician fading channels with Rician parameter K > 0. On the other hand, the S + N-ESC scheme is shown to have a larger SEP than classical SC for 16-QAM signaling in Nakagamim and Rician fading channels.

A signal-plus-noise (S + N) energy selection combining (S + N-ESC) scheme and a weighted S + N-ESC scheme for binary phase shift keying (BPSK) signaling are studied. The S + N-ESC selects the branch with the maximum S + N energy for detection. The advantage of the S+N-ESC and the weighted S+N-ESC is that they require no explicit channel estimation for diversity branch selection. The bit error rate (BER) performances of the S + AT-ESC and the weighted S + N-ESC are analyzed for BPSK signaling in independent and nonidentically distributed (i.n.d.) Rayleigh fading channels. We show that the average error rate performance of the S +N-ESC is the same as classical signal-to-noise ratio selection combining (SNR-SC), which requires the estimation of fading amplitudes for branch selection, in independent and identically distributed (i.i.d.) Rayleigh fading channels. We also show that the weighted S + N-ESC has the same BER performance as the classical SNR-SC for i.n.d. Rayleigh fading channels.

We propose the Hadamard matrix as a codebook for multiple-input multiple-output (MIMO) beamforming systems, and show that a quantized equal gain transmission (QEGT) scheme using the Hadamard codebook has the same symbol error probability performance as a receiver selection diversity combining scheme operating in independent generalized Rayleigh fading channels. We show that a MIMO beamforming system using quantized feedback over Rician fading channels can be analyzed by considering a corresponding receiver selection combining scheme operating in single-input multiple-output (SIMO) systems over correlated generalized Rician fading channels. Also, we propose a greedy cophasing scheme for MIMO beamforming systems using quantized feedback. The greedy cophasing scheme provides 3.2 dB power gain over an orthogonal space-time block code when the number of transmit antennas is four, the number of receive antennas is one, and the number of feedback bits is three.

A new symbol error probability (SEP) formula for selection combining (SC) with M-ary phase shift keying (MPSK) signals in i.i.d. Nakagami-m fading channels is derived.

A l-Bayesian erasure test (BET) based on order statistics with diversity reception for noncoherent M-ary frequency shift keying (FSK) signaling is examined. In the l-BET, the M-ary FSK symbol is erased whenever the a posteriori probability of correct decision, which is a function of the l-largest sums of energy detector outputs, is less than a threshold. For binary FSK signaling in Rayleigh flat fading channels, the 2-BET is found to be the same as the difference threshold test (DTT), which declares an erasure whenever the difference between the largest and the second largest sums of energy detector outputs is less than a threshold. The output threshold test (OTT), which declares an erasure whenever the largest sum of energy detector outputs is less than a threshold, is proved equivalent to the 1-BET for the Rayleigh flat fading channel. Closed-form expressions for the probabilities of correct decision and symbol error for the DTT, Viterbi's ratio threshold test (RTT), and the OTT are derived when square-law combining (SLC) diversity is used. Several erasure tests including the Z-BET, the RTT, the OTT, and the DTT are compared in terms of probability of codeword error. The 2-BET provides almost the same performance as the BET that requires all the sums of energy detector outputs. The 2-BET provides power gains of 0.9 dB and 0.7 dB over the RTT for (7, 3) and (15, 7) Reed-Solomon (RS) codes, respectively, when the probability of codeword error is 10-3and the diversity order is one.

We propose the Hadamard matrix as a codebook for multiple-input multiple-output (MIMO) beamforming systems, and show that a quantized equal gain transmission (QEGT) scheme using the Hadamard codebook has the same symbol error probability performance as a receiver selection diversity combining scheme operating in independent generalized Rayleigh fading channels. Also, we propose a greedy cophasing scheme for MIMO beamforming systems using quantized feedback. The greedy cophasing scheme provides 3.2 dB power gain over an orthogonal space-time block code when the number of transmit antennas is four, the number of receive antennas is one, and the number of feedback bits is three.

A signal-plus-noise (S + JV) energy selection combining (S+JV-ESC) scheme for binary phase shift keying (BPSK) signaling is proposed. The S + JV-ESC selects the branch with the maximum S + JV energy for detection. The advantage of the S + JV-ESC is that it requires no channel estimation for diversity branch selection. The bit error rate (BER) performance of the S +JV-ESC is analyzed for BPSK signaling in independent and identically distributed (i.i.d.) Rayleigh fading channels. We show that the performance of the S + JV-ESC is the same as conventional signal-to-noise ratio selection combining (SNR-SC), which requires estimation of the fading amplitudes for branch selection.

A Z-Bayesian erasure test (BET) based on order statistics with diversity reception for noncoherent M-ary frequency shift keying (FSK) signaling is examined. In the Z-BET, the M-ary FSK symbol is erased whenever the a posteriori probability of correct decision, which is a function of the Z-largest sums of energy detector outputs, is less than a threshold. For binary FSK signaling in Rayleigh flat fading channels, the 2-BET is found to be the same as the difference threshold test (DTT), which declares an erasure whenever the difference between the largest and the second largest sums of energy detector outputs is less than a threshold. The output threshold test (OTT), which declares an erasure whenever the largest sum of energy detector outputs is less than a threshold, is proved equivalent to the 1-BET for the Rayleigh flat fading channel. Closed-form expressions for the probabilities of correct decision and symbol error for the DTT, Viterbi's ratio threshold test (RTT), and the OTT are derived when square-law combining (SLC) diversity is used. Several erasure tests including the Z-BET, the RTT, the OTT, and the DTT are compared in terms of probability of codeword error. The 2-BET provides almost the same performance as the BET that requires all the sums of energy detector outputs. The 2-BET provides power gains of 0.9 dB and 0.7 dB over the RTT for (7,3) and (15,7) Reed-Solomon (RS) codes, respectively, when the probability of codeword error is 10 3 and the diversity order is one. Higher-order diversity has reduced gains.

Optimum selection combining using the maximum energy (OSC-ME) criterion for M-ary noncoherent frequency shift keying signaling in independent and nonidentically distributed fading channels is examined. A weighted maximum energy selection combining (WMESC) scheme is proposed and found to provide virtually the same symbol error probability as OSC-ME in Rician fading channels. The WMESC always provides power gain over the classical selection combining regardless of diversity order and signal-to-noise ratio.

We present the optimum selection-combining (SC) rule for M-ary signals based on the log-likelihood ratio (LLR) in frequency-nonselective, independent and nonidentically distributed fading channels. The motivation for using the LLR in selecting the diversity branch is that it provides the reliability information of the maximum a posteriori probability decision which minimizes the probability of symbol error. We present coherent and noncoherent optimum SC rules with and without fading information that minimize the probability of symbol error. It is shown that the optimum SC rule provides a significant power gain over the conventional signal-to-noise ratio-based SC rule, and the power gain increases with an increasing number of available diversity branches and decreasing alphabet size. We also present suboptimum SC rules that require less computation, but provide a power gain close to that provided by the optimum SC rule.

We present a generalized selection combining (GSC) of M-ary signals that selects K out of L diversity branches based on the log-likelihood ratio (LLR). The motivation for using the LLR in selecting diversity branches is that the magnitude of LLR provides the reliability of hard decision and the LLR-based hard decision minimizes the bit error probability. It is shown that the proposed LLR-based GSC rule provides a significant power gain over the conventional SNR-based GSC rule, and the power gain increases with decreasing alphabet size and decreasing number of selected branches. We also present suboptimum LLR-based GSC rules that require a less computation, but provide virtually identical performance as the LLR-based GSC rule.

We calculate the capacity for binary phase shift keying (BPSK) signaling in Rayleigh fading channels with generalized selection combining (GSC). In GSC, the K diversity branches with largest fading amplitudes among L available diversity branches are combined. We also calculate the capacity of BPSK signaling in Nakagami-m fading channel when there is no diversity available, i.e. L is equal to one. Based on the capacity, the minimum required signal-to-noise ratio (SNR) per information bit for reliable communication is found as a function of code rate with both hard decision and soft decision. We find that the minimum required SNR per information bit increases monotonically as code rate increases. The channel dispersion induced by fading is found to be more harmful than the information loss by hard decision for large available SNR per bit. Also, we find that the gain provided by increasing diversity order is larger than the loss by hard decision for large available SNR per bit.

We propose truncating the transmission power (allocating no power) for symbols with low channel gain, and tagging erasures on the corresponding symbols at the receiver. The motivation is that symbols with low channel gain are highly likely to be in error and yet, if transmitted, consume the energy resource and generate interference to other users. Truncating the power for those symbols has the effect of reducing the interference to other users and allocating more power on symbols with high channel gain (thereby reducing the error probability). Since block codes can correct twice as many erasures as errors, the coded performance can be improved by properly combining the power control with the error-control coding. In this letter, we analyze the performance of the Reed-Solomon-coded multicarrier direct-sequence code-division multiple-access systems with two power-control schemes. We show that the probability of incorrect decoding can be significantly improved by properly combining the power control with the error control coding.

We propose a generalized selection combining (GSC) scheme for binary signaling in which a subset of diversity branches providing the largest magnitude of log-likelihood ratio (LLR) are selected and combined. It is shown that the bit-error probability with maximum ratio combining (MRC) or square-law combining of L branches is identical to that with LLR-based GSC of L/2 branches. We also propose a simple, but suboptimal, GSC based on a noncoherent envelope detection and discuss its potential advantages over the conventional signal-to-noise-ratio-based GSC and MRC.

We present the difference threshold test (DTT) as a method for generating reliability information for M-ary signaling with coherent detection. The proposed DTT declares an erasure whenever the difference between the shortest Euclidean distance and the second shortest Euclidean distance from the received signal is less than a threshold. We show that the DTT is an approximation of the Bayesian test, and, for binary signaling, it is equivalent to the Bayesian test. We also extend the DTT for L-fold diversity systems. The performance of Reed-Solomon (RS) codes with errors-and-erasures decoding is analyzed when the DTT and the conventional SNR threshold test are applied.

We present the difference threshold test (DTT) for M-ary orthogonal frequency shift keying (FSK) signaling in Rayleigh flat fading channels. The proposed DTT declares an erasure whenever the difference between the largest and the second largest energy detector output does not exceed a fixed threshold. We show that the DTT is an approximation of the Bayesian erasure test, but provides almost the same performance as the Bayesian erasure test. For (7,3) and (31,15) Reed-Solomon (RS)-coded systems, the DTT provides power gains of 0.8 dB and 0.4 dB over the ratio threshold test (RTT), respectively. We prove that the limiting performance of the DTT approaches that of the RTT for very large M.

We propose a generalized selection combining (GSC) scheme for binary signaling in which M diversity branches providing the largest magnitude of log-likelihood ratio (LLR) are selected and combined. The bit error probability provided by LLR-based GSC serves as a lower bound on the bit error probability provided by any GSC techniques. We also propose a suboptimal GSC based on a noncoherent envelope detection. We derive the bit error probability with LLR-based and envelope-based GSC techniques and examine their power gains over the conventional SNR-based GSC technique. We show that the bit error probability with maximum ratio combining or square-law combining of L branches is identical to that with LLR-based GSC of L/2 branches.

We present the optimum selection combining for M-ary signaling based on the log-likelihood ratio that minimizes the probability of symbol error. Then, we present the noncoherent optimum selection combining that does not require knowledge of the signal phases on each branch. Numerical results indicate that a significant gain over the SNR-based selection combining can be provided by the proposed optimum selection combining. Finally, we present a suboptimum selection combining that requires a lower complexity, but provides a power gain close to that of the optimum selection combining.

We propose a new generalized selection combining (GSC) scheme that selects m diversity branches out of L (/spl ges/m) diversity branches based on the magnitude of the log-likelihood ratio (LLR), which, for BPSK signals, is proportional to the product of the fading amplitude and the matched filter output. The proposed GSC is shown to provide a significant power gain over the conventional GSC which selects diversity branches based on signal-to-noise ratio. The proposed GSC scheme for m=3 is found to be only 0.3 dB inferior to the maximal ratio combining scheme (m=L), when L=8, in a Rayleigh fading channel.

We propose a new selection diversity scheme, called |ar|-selection diversity, that selects the branch providing the largest magnitude of log-likelihood ratio (LLR). The LLR for BPSK signals in fading channels is found to be proportional to the product of the fading amplitude and the matched filter output after phase compensation. The proposed |ar|-selection diversity scheme is shown to be optimal in the sense of minimizing the bit error rate (BER), and outperform existing selection diversity schemes. We also propose a suboptimal selection diversity scheme, called |aw|-selection diversity, that does not require a phase compensation in the selection process, thereby significantly reducing implementation complexity. We show that the proposed |ar|-selection and |aw|-selection diversity schemes exhibit significant power gains over existing selection diversity schemes in Rayleigh fading channels.

To estimate the convolutional encoder state from received data, one may use the inverse to the encoder G. However, channel errors make this method unreliable. We propose a method that uses the received data in the following way. We calculate the syndrome, and after a specific number of received syndrome values equal to zero, we expect that the corresponding received data is also error-free. The received data is then used to build the inverse and give an estimate for the encoder state. The method can be used in situations where knowledge of the encoder state helps the decoding process or for synchronization purposes. We analyze the performance of the described method with respect to state estimation error probability and the average time it takes before we can estimate the encoder state with a certain desired reliability.