In this paper, we propose a novel, energy-efficient directional modulation scheme to enhance the physical-layer security of wireless communication systems. The proposed analytical method designs a uniform linear array (ULA) in the spatial frequency domain, allowing for arbitrary distortion of the constellation pattern. In the secure directions, the design ensures low bit error rates (BERs), while in unsecure angles, it enforces a threshold BER to disrupt potential eavesdroppers. The proposed scheme is not only simple to design but also provides flexible trade-off between the mainlobe and sidelobes, ensuring efficient energy consumption. Furthermore, we show that the new scheme has low computational complexity when optimizing antenna weightings, and while we only consider 4- and 16-QAM in this paper, the proposed method can be generalized to higher order QAM modulation schemes. Numerical results validate the effectiveness of the design in achieving the desired BER thresholds and energy efficiency, demonstrating its suitability for real-world applications.

In this work, we propose a joint security approach based on physical layer security (PhySec) and noisy ciphertext to improve the secrecy rate of a relay network with cooperative jamming via two trusted relays. While the secrecy capacity has been considered as the maximum limit for a perfect secrecy at the physical layer, we demonstrate that the consideration of error- prone ciphertext in encryption and its interaction with PhySec allows us to transmit above the PhySec capacity without any leakage. By formulating and solving a non-convex encryption- aided secrecy rate maximization problem, it is shown that beyond PhySec capacity performances can be achieved even in a simple jamming scenario in which the two jammers just send independent jamming signals. The optimal encryption-aware power allocation and the secrecy rate maximization solution are also established in the case of common jamming signals to further increase the secrecy rate. Numerical results are finally provided to demonstrate the superiority of the proposed joint security framework over traditional PhySec.

This paper addresses the optimal signaling scheme and capacity of an additive Gaussian mixture (GM) noise channel using 1-bit output quantization. The considered GM distribution is a weighted sum Gaussian component densities with arbitrary means, and it can be used to represent any non-Gaussian channel of engineering interest. By first establishing a necessary and sufficient Kuhn-Tucker condition (KTC) for an input signal to be optimal, we demonstrate that the maximum number of mass points in the capacity-achieving signal is four. Our proof relies on novel bounds on the product of Q functions and Dubin’s theorem. By considering a special case of GM with zero mean Gaussian components, which is a realistic accurate model for co-channel interference in heterogeneous wireless networks and impulsive interference, it is shown that the optimal input is $\pi$/2 circularly symmetric. As a result, in this case, the capacity-achieving signal has exactly four mass points forming a square centered at the origin. By further checking the first and second derivatives of the modified KTC, it is then shown that the phase of the optimal mass point located in the first quadrant is $\pi$/4. Thus, with zero-mean GM, the capacity-achieving input signal is QPSK, and the channel capacity can be established in closed-form.

In traditional physical layer security paradigm, no leakage of confidential information to the eavesdropper is tolerated regardless of whether the message is encrypted or not. This will result in an achievable secure transmission rate that is significantly smaller than the channel capacity. This article presents a novel approach that allows treatment of physical layer security in conjunction with encryption to achieve a flexible trade-off of system resources. We propose a novel framework to model the interplay between secured transmission rate and error probability in error prone ciphertexts. To this end, we use the concept of rate-equivocation region to establish such a connection. To clearly describe the application of our framework, we consider the case of 4-node Gaussian wiretap channel. For such a channel, we characterize the rate-equivocation region in different scenarios, and then use it to study the achievable rate of encryption-aware physical layer security. The obtained results show that, for a fixed transmission power, the prior knowledge of encryption can significantly increase the secured transmission rate. In addition, encryption-aware physical layer security can achieve a target transmission rate at a reduced transmission power compared to the conventional encryption-agnostic physical layer security.

The use of low-resolution analog-to-digital converters (ADCs) is considered to be an effective technique to reduce the power consumption and hardware complexity of wireless transceivers. However, in systems with low-resolution ADCs, obtaining channel state information (CSI) is difficult due to significant distortions in the received signals. The primary motivation of this paper is to show that learning techniques can mitigate the impact of CSI unavailability. We study the blind detection problem in multiple-input-multiple-output (MIMO) systems with low-resolution ADCs using learning approaches. Two methods, which employ a sequence of pilot symbol vectors as the initial training data, are proposed. The first method exploits the use of a cyclic redundancy check (CRC) to obtain more training data, which helps improve the detection accuracy. The second method is based on the perspective that the to-be-decoded data can itself assist the learning process, so no further training information is required except the pilot sequence. For the case of 1-bit ADCs, we provide a performance analysis of the vector error rate for the proposed methods. Based on the analytical results, a criterion for designing transmitted signals is also presented. Simulation results show that the proposed methods outperform existing techniques and are also more robust.

This paper investigates the capacity region and detailed characterizations of capacity-achieving signaling schemes of a multiple access channel (MAC) with two users communicating to a base station (BS) equipped with 1-bit quantizers. We consider Rayleigh fading channels where channel state information (CSI) is known only at BS. Towards this end, we first study the weighted sum-rate maximization problem over the set of input distributions of one user for a fixed input signal at another user. By examining a necessary and sufficient Kuhn-Tucker condition (KTC) for an input to be optimal, it is first shown that the power constraint is active, i.e., the equality in the power constraint is achieved. By further exploiting novel bounds on the output distribution functions, the optimal distribution is shown to have a bounded amplitude. In the next step, we prove that if a fixed input with bounded amplitude is used at one user, the other user also needs to use a bounded amplitude signal to maximize the weighted sum-rate. To effectively analyze the KTC, our approach is to divide the domain of fading into two disjoint regions and examine the region with non-zero measure. It is then concluded that any boundary point in the capacity region is achieved by using bounded amplitude signals, and they are π/2 circularly symmetric. Building upon these results, we turn our focus to the sum-capacity segment, and demonstrate that any π/2 circularly symmetric input distribution having a constant amplitude is sum-capacity achieving. The sum-capacity can then be established.

In this paper, we analyze the energy efficiency (EE) of a non-orthogonal multiple access channel with successive interference cancellation for the uplink under Gaussian-mixture aggregate interference, which is a realistic noise plus interference model to capture the asynchronism in a heterogeneous cellular network. The considered EE is measured via the minimum energy per bit Eb/N0 minfor reliable communication and the wideband slope of the spectral efficiency as a function of energy per bit Eb/N0. For completeness, both Gaussian signaling schemes and practical finite alphabet inputs are examined. To this end, our approach is to calculate Eb/N0 minand the wideband slope region via the Kullback-Leibler divergence. Due to the presence of mixture of multiple Gaussian distributions, the Kullback-Leibler divergence cannot be expressed in closed-form. As an alternative, we exploit upper bounds on the divergence, and we show that the bounds are achievable at the limit points, i.e., when the signal-to-noise-ratio (SNR) approaches zero. As a result, Eb/N0 minof each user can be established in closed-form. Next, we apply Taylor series expressions and exploit the achievability in the previous step to evaluate the second derivative of the rates when SNR goes to zero. It is then shown that the wideband slope region can be found effectively. The proposed method can serve as an important tool for making more accurate throughput and EE evaluations of important wireless underlay networks in future cellular systems.

This work addresses the sum-capacity-achieving signaling schemes and the sum-capacity of a multiple access channel (MAC) with two mobile users communicating to a base station equipped with 1-bit quantizers. We consider Rayleigh fading channels between the users and the base station where channel state information (CSI) is known only at the base station. Towards this end, we first establish a necessary and sufficient condition refereed to as Kuhn-Tucker condition (KTC) on the input distribution of one user for a given input signal used at another user so that the input/output mutual information (MI) is maximized. By relaxing the power constraint and establishing upper bounds on the MI, we demonstrate that the power constraint of the first user is active. Then using Fubini-Tonelli theorem to exchange the order of integrations between fading and input distributions, and exploiting novel bounds on the output distribution and a related relative entropy, it is shown that the optimal input distribution has a bounded amplitude. Due to the symmetry of the problem, it is then concluded that the sum-capacity-achieving amplitude distributions are bounded, and both users must use full power to achieve the sum-capacity. Next, we exploit the independence of the amplitude and phase of fading gains to show that the optimal inputs are π/2 circular symmetric. Building upon the results on the amplitude and phase, it is then demonstrated that any π/2 circular symmetric input distribution having a constant amplitude is sum-capacity achieving. The sum-capacity is finally obtained in a precise form.

This paper investigates the optimal signaling schemes and capacities of non-coherent correlated multiple-input single-output (MISO) channels in fast Rayleigh fading. We consider both channels under per-antenna power constraints as well as channels under joint per-antenna and sum power constraints. For per-antenna power constraint channels, we first establish the convex and compact properties of the feasible sets, and demonstrate the existence of optimal input distribution and the uniqueness of optimal effective magnitude input distribution. By exploiting the solutions of a quadratic optimization problem, we show that the Kuhn-Tucker condition on the optimal inputs can be simplified to a single dimension. As a result, we can apply the Identity Theorem to show the discrete and finite nature of the optimal effective magnitude distribution, with a mass point located at the origin. By using this distribution, we then construct a finite and discrete optimal input vector distribution. The use of this input allows us to determine the capacity gain of MISO over SISO via the phase solutions of a constrained quadratic optimization problem on a sphere, which can be obtained using a proposed penalized optimization algorithm. We also extend the results to MISO channels subject to the joint per-antenna and sum power constraints. Under this consideration, it is shown that not all per-antenna constraints are active. While the finiteness and discreteness of the optimal effective magnitude and the optimal input vector distributions still hold, the optimal phases and the optimal power allocation among the transmit antennas need to be determined simultaneously via a quadratic optimization problem under inequality constraints. These solutions can finally be used to obtain the MISO capacity gain.

Low-resolution analog-to-digital converter (ADC) has been considered as a promising solution to save power and cost in communication systems using high bandwidth and/or multiple RF chains. The goal of this work is to address the design of optimal signaling schemes and establish the capacity limit of Rician fading channels with low-resolution output quantization. This fading channel can be used to accurately model a wide range of wireless channels with the line-of-sight (LOS) components, including emerging mm-wave communications. The focus is on non-coherent fast fading channels where neither the transmitter nor the receiver knows the channel state information (CSI). By examining the continuity of the input-output mutual information, the existence of the optimal input signal is first validated. Then, considering the case of 1-bit ADC, we show that the optimal input is$\pi /2$circularly symmetric. A necessary and sufficient condition for an input signal to be optimal, which is referred to as the Kuhn-Tucker condition (KTC), and Lagrangian optimization problem are then established. By exploiting the novel log-quadratic bounds on the Gaussian$Q$-function, it is then demonstrated that for a given mass point’s amplitude, the corresponding rotated mass points through the phase of LOS component must form a square grid centered at zero. Furthermore, the amplitude of the mass points in the optimal distribution can take on only one value. As a result, the capacity-achieving input with 1-bit ADC is a rotated quadrature phase-shift keying (QPSK) constellation, and the rotation angle depends on the Rician factor. The characterization of the optimal input has also been extended to the case of multi-bit ADCs. Specifically, it is shown that for a$K$-bit ADC, the optimal input is discrete having atmost$2^{2K}$mass points. In both the cases of 1-bit and$K$-bit ADCs, the channel capacities are established in closed-form.

This paper examines the problem of blind detection in multiple-input-multiple-output (MIMO) systems with low-resolution analog-to-digital converters (ADCs) using learning approaches. Recently, the use of low-resolution ADCs has been considered an effective technique to mitigate the issue of power consumption in millimeter-wave transceivers. One serious problem caused by the low-resolution ADCs is the significant distortion of received signals, resulting in difficulty of obtaining Channel State Information (CSI) at both transmitter and receiver sides. The primary motivation of our work is that learning the input-output relation can help mitigate the impact of CSI unavailability. In both supervised and semi- supervised methods that we propose, a sequence of pilot symbol vectors is used as the initial training data for the learning task. The idea of the supervised learning method is in typical communications systems, cyclic redundancy check (CRC) is used, and thus correctly decoded symbols confirmed by CRC can be exploited as supplementary training data to improve the detection accuracy. In the semi-supervised learning method, the to-be- decoded data is exploited to help the learning process, and so no further training information is required except the pilot symbol vectors. Simulation results show that the two proposed learning methods outperform existing detection techniques.

This paper generalizes and proves the discrete and finite nature of the optimal signaling schemes for general classes of non-Gaussian aggregate interference point-to-point and multiple access channels (MACs) under peak power constraints. Specifically, we first investigate the detailed characteristics of optimal inputs for a single-user channel that is impaired by two types of noise: a Gaussian mixture (GM) noise consisting of Gaussian elements with arbitrary means, and the interference U with an arbitrary distribution. The only very mild condition imposed on U is that its second moment is finite. To this end, we establish the Kuhn-Tucker condition (KTC) on an optimal input and prove the analyticity of the KTC using Fubini-Tonelli's and Morera's theorems. It is then shown that an optimal input is continuous if only if the KTC is zero on the entire real line. However, by examining an upper bound on the output PDF, it is demonstrated that the KTC must be bounded away from zero. As such, any optimal input must be discrete with a finite number of mass points. Finally, we exploit U having an arbitrary distribution to show that the optimal input distributions that achieve the sumcapacity of an M-user MAC under GM noise are discrete and finite. Furthermore, there exist at least two distinct points that achieve the sum capacity on the rate region.

This paper utilizes the tools of stochastic geometry to derive closed-form approximations of coverage probabilities for both cellular and D2D links of a realistic underlaid full-duplex (FD) cellular D2D network. In the considered model, D2D users operate in FD mode under the adverse effect of realistic residual self-interference, and their locations are modeled by a homogeneous spatial Poison point process (PPP). The coverage probabilities involve multiple integrals due to the average over the distributions of transmit power, wireless fading, and link distances. Therefore, our solution is to first apply Laplace transforms and novel approximations that accurately approximate the expected values of fractional and exponential functions of random variables to obtain the distributions of signal to interference and noise ratios (SINRs) at the base station (BS) and D2D users. By further taking the average over the distributions of cellular and D2D link distances, we then arrive at the closed-form approximations of cellular and D2D coverage probabilities. Furthermore, based on the approximation of D2D link coverage probability, an analytical expression of the D2D link sum-rate is obtained, which can be effectively calculated. Our results show that the integration of FD in D2D provides significant sum-rate improvement over the half-duplex D2D counterpart.

Low resolution analog-to-digital converter (ADC) has been considered as a promising solution to save power and cost in communication systems using high bandwidth and/or multiple RF chains. The goal of this work is to address the design of optimal signaling schemes and establish the capacity limit of Rician fading channels with 1- bit output quantization. This fading channel can be used to accurately model a wide range of wireless channels with LOS components, including emerging mmWave communications. The focus is on non-coherent fast fading channels where neither the transmitter nor the receiver knows the channel state information (CSI). By first examining the continuity of the input-output mutual information, the existence of the optimal input signal is validated. Moreover, the optimal input is shown to be π/2 circularly symmetric. A necessary and sufficient condition for an input signal to be optimal, which is referred to as the Kuhn-Tucker condition (KTC), and Lagrangian optimization problem are then established. By exploiting novel log-quadratic bounds on the Gaussian Q-function, it is then demonstrated that for a given mass point's amplitude, the corresponding rotated mass points through the phase of LOS component must form a square grid centered at zero. Furthermore, by establishing an upper bound on the KTC coefficient, we show that there are exact four mass points in the optimal distribution. As a result, the capacity-achieving input is a rotated QPSK constellation, and the rotation angle is the phase of the LOS coefficient. By using this input, the channel capacity is finally established in closed-form.

This paper investigates the secrecy rates and optimal power allocation schemes of a decode-and-forward (DF) wiretap relay channel where a relay operates in a full-duplex (FD) mode. A practical self-interference model is adopted to take into account the effect of residual self-interference. At first, we demonstrate that while the optimal power allocation schemes between the source and the relay are non-convex, closed-form solutions can still be established under different power constraints. An asymptotic behavior of the solutions is then provided to shed important insights on the derived power allocation schemes. Specifically, by using the method of dominant balance, we show that the relay should use full power when its power budget is sufficiently small as compared to power budget at the source. Otherwise, the optimal power consumed at the relay approaches a constant to effectively handle the residual self-interference. The analysis is also helpful to demonstrate that the secrecy capacity of the considered full-duplex relay system is twice as much as that of the half-duplex system. In addition, numerical results reveal that DF relaying provides significantly higher secrecy rate over Amplify-and-Forward (AF) relaying.

This paper details novel methods to accurately estimate the achievable rates of channels with additive Gaussian mixture (GM) noise. Attention is paid to a Gaussian input and discrete inputs. Such discrete inputs represent a wide range of signaling strategies and include the capacity-achieving input as a special case. At first, we propose a simple technique to calculate the GM noise entropy. Specifically, when the noise level is high, a lower bound on the integrand of the noise entropy is established and the noise entropy can be estimated in closed-form. In the low noise region, the piecewise-linear curve fitting (PWLCF) method is applied to calculate the noise entropy. It is then demonstrated this can be estimated in both regions with a predetermined accuracy. We then extend this result to calculate the output entropy and the achievable rate when the input is Gaussian distributed, which is shown to be asymptotically optimal. Next, we propose a simple PWLCF-based method to estimate the output entropy for a given discrete input. In particular, the output entropy is evaluated by examining the output in high and low regions of amplitude using a lower bound on the integrand of the output entropy and PWLCF, respectively. It is demonstrated that the output entropy, and consequently, the achievable rates, can be computed to achieve any desired accuracy level.

This paper investigates the energy efficiency of communication channels subject to additive Gaussian mixture interference in the low-power regime. This channel is widely used to capture the asynchronism in heterogeneous cellular networks. In particular, we characterize the low-signal-to-noise-ratio (low-SNR) metrics of minimum energy per bit or capacity per unit cost and the wideband slope of the spectral-efficiency curve. Instead of directly maximizing the mutual information or determining the optimal input, we make use of the characterization that the first derivative of the capacity with respect to SNR at SNR= 0 can be determined from the Kullback-Leibler divergence without identifying the optimal input. We also identify the wideband slope of spectral efficiency in closed-form by relying on the discreteness in amplitude of the capacity-achieving input. The characterization of these fundamental energy efficiency metrics allows us to find the signaling strategies that are optimally efficient in the low-SNR regime and obtain an analytical confirmation of the earlier information-theoretic findings.

This paper studies the capacity and energy efficiency of non‐coherent Rayleigh fading channels with Gaussian mixture noise where neither the transmitter nor the receiver has the knowledge of channel state information. The channel under consideration is suited for cellular networks having multi‐tier heterogeneous architectures in which the channel conditions change rapidly. In the first part of the paper, we characterize the structure of a capacity‐achieving input signal. Specifically, we establish an integrable upper bound on the integrand in the output entropy and demonstrate that there exists a unique optimal input. By formulating the Kuhn‐Tucker condition and establishing a diverging lower bound on it, we show that the optimal input is discrete having a finite number of mass points. Using this result, we investigate the capacity and energy efficiency of the considered channel in the second part of the paper. In particular, we first develop a numerical method to evaluate the optimal input and compute the capacity. The energy efficiency, which is related to the capacity and optimal input in low‐power regimes, is examined by calculating the minimum bit energy and wideband slope of the spectral‐efficiency curve. We also analytically show the optimality of an on‐off signal in this regime.

This paper studies energy-efficient power allocation schemes for secondary users in sensing-based spectrum sharing cognitive radio systems. It is assumed that secondary users first perform channel sensing possibly with errors and then initiate data transmission with different power levels based on sensing decisions. In this setting, the optimization problem is to maximize energy efficiency (EE) subject to peak/average transmission power constraints and peak/average interference constraints. By exploiting the quasi-concave property of the EE maximization problem, the original problem is transformed into an equivalent parameterized concave problem, and an iterative power allocation algorithm based on Dinkelbach's method is proposed. The optimal power levels are identified in the presence of different levels of channel side information (CSI) regarding the transmission and interference links at the secondary transmitter, namely, perfect CSI of both transmission and interference links, perfect CSI of the transmission link, imperfect CSI of the interference link, imperfect CSI of both links, or only statistical CSI of both links. Through numerical results, the impact of sensing performance, different types of CSI availability, and transmit and interference power constraints on the EE of the secondary users is analyzed.

In this paper, we detail effective methods to approximate the achievable rates of channels with additive Gaussian mixture (GM) noise for both real and complex channels to achieve any desired level of accuracy. Attention is paid to a Gaussian input, a discrete real input, and a complex input with discrete amplitude and independent uniform phase. Such discrete inputs represent a wide range of input distributions and they include the capacity-achieving inputs as special cases. At first, we propose a simple technique to accurately calculate the noise entropy. Specifically, when the noise level is high, a lower bound on the integrand of the entropy is established and the noise entropy can be estimated using a closed-form solution. In the low noise region, the piecewise-linear curve fitting (PWLCF) method is applied. We then extend this result to calculate the achievable rate when the input is Gaussian distributed, which is shown to be asymptotically optimal. Next, we propose a simple PWLCF-based method to approximate the output entropy for a real GM channel when the input is discrete, and for a complex GM channel when the input is discrete in amplitude with independent uniform phase. In particular, for the real channel, the output entropy is evaluated by examining the output in high and low regions of amplitude using a lower bound on the integrand of the output entropy and PWLCF, respectively. For the complex channel, the output entropy is approximated a similar manner but using polar coordinates and the Kernel function. It is demonstrated that the output entropy, and consequently, the achievable rates, can be computed to achieve any given accuracy level.

Abstract We propose two novel clustered linear precoding schemes applicable to network multi‐input multi‐output systems using only partial channel state information to enhance the sum‐rate of the system. Using a channel model that decomposes a multi‐input multi‐output channel matrix into transmit and receive steering vectors and assuming that only transmit steering vectors are available at the base transmit stations, we, first, propose a regularized channel inversion precoding scheme to enhance the sum‐rate assuming only single‐antenna users are available in the system. Next, because of the limitation of regularized channel inversion to handle users with multiple receive antennas, a novel block diagonalization method is proposed. We construct the precoding matrices that jointly eliminate inter‐cell interference and maximize the sum‐rate for a given input covariance matrix. Assuming total power constraint and per‐base‐station power constraints, optimal power allocation schemes are further developed to optimize the sum‐rate. We analytically show that the sum‐rate increases linearly with the number of users when only single‐antenna users are present in the system. Numerical results show that at low signal‐to‐noise ratios, the block diagonalization precoding outperforms the regularized channel inversion in terms of the bit error rate; while at high signal‐to‐noise ratios, the regularized channel inversion provides a better performance. Copyright © 2016 John Wiley & Sons, Ltd.

This paper studies the characterization of the optimal input for impulsive ambient noise channels under average power constraint. Our focus is on the two-term Gaussian mixture complex noise model, which has been widely used to model impulsive noise arising in various communication channels. We first demonstrate that there exists a unique input distribution that achieves the channel capacity and the capacity-achieving input distribution has a uniformly distributed phase. By examining the Kuhn-Tucker conditions (KTC), we further show that if the optimal amplitude input distribution contains an infinite number of mass points on a bounded interval, the channel output must be Gaussian distributed. However, by using Bernstein's theorem to examine the completely monotonic condition, it is shown that the assumption of a Gaussian distributed output is not valid. As a result, there is always a finite number of mass points on any bounded interval in the optimal amplitude distribution. In addition, by applying a novel bounding technique on the KTC and using the Envelop Theorem, we demonstrate that the optimal amplitude distribution cannot have an infinite number of mass points. That gives us a unique solution of the optimal input having discrete amplitude with a finite number of mass points. Given such interesting results, we also develop an efficient way to compute the discrete optimal input and the corresponding capacity.

In this paper, we propose an effective method to calculate the average achievable rate and outage probability of a practical cognitive radio (CR) link with finite-alphabet inputs under imperfect spectrum sensing in fast and slow Rayleigh fading, respectively. In the considered CR system, the secondary user (SU) senses and dynamically exploits the spectrum pool via dynamic frequency hopping. Since spectrum sensing is not perfect, miss-detection occurs. Under this event, the interference emerged from collisions due to the simultaneous spectrum access of both primary and cognitive users leads to a non-Gaussian CR link. This makes it very challenging to evaluate the information theoretical limits, especially when finite-alphabet inputs are used. To overcome such challenge, we first introduce a simple method to calculate the instantaneous differential entropy of the channel output for a given fading gain using Laguerre-Gauss quadrature formulas. Using this result, we propose a piece-wise linear curve fitting (PLCF)-based method to calculate the average output entropy and outage probability, respectively. It is then demonstrated that the average achievable rate in fast fading and the outage probability in slow fading of the considered CR channel can be calculated effectively to achieve any predetermined accuracy level for a given finite-alphabet input.

This paper studies a wire-tap channel in which a source node wants to communicate securely to a destination node in the presence of an eavesdropper and under the aid of an amplify-and-forward (AF) relay operating in full-duplex (FD) mode. The residual self-interference due to FD transmission is explicitly taken into account. The secrecy capacity and the respective optimal power allocation schemes for this system are examined under both individual and joint power constraints. At first, the related optimization problems are shown to be quasi-concave. As such, the globally optimal solution exists and is unique. Due to the non-linearity of the derivative, we apply a simple bisection method for root finding and obtain a simple expression for the optimal power allocation scheme. To further provide some insight on the solutions, we apply the method of dominant balance to analyze the capacity and power allocations in different high power regions. It is then demonstrated that full relay power is only needed when the power at the relay is sufficiently small compared to the power at the source. Comparisons with half-duplex (HD) relaying also revealed that FD can achieve a significantly higher secrecy capacity. Finally, numerical results are presented to confirm the optimality of the solutions.

In this study, the authors propose simple methods to evaluate the achievable rates and outage probability of a cognitive radio (CR) link that takes into account the imperfectness of spectrum sensing. In the considered system, the CR transmitter and receiver correlatively sense and dynamically exploit the spectrum pool via dynamic frequency hopping. Under imperfect spectrum sensing, false‐alarm and miss‐detection occur which cause impulsive interference emerged from collisions due to the simultaneous spectrum access of primary and cognitive users. That makes it very challenging to evaluate the achievable rates. By first examining the static link where the channel is assumed to be constant over time, they show that the achievable rate using a Gaussian input can be calculated accurately through a simple series representation. In the second part of this study, they extend the calculation of the achievable rate to wireless fading environments. To take into account the effect of fading, they introduce a piece‐wise linear curve fitting‐based method to approximate the instantaneous achievable rate curve as a combination of linear segments. It is then demonstrated that the ergodic achievable rate in fast fading and the outage probability in slow fading can be calculated to achieve any given accuracy level.

This paper studies the characterization of the optimal input and the computation of the capacity of additive quadrature Gaussian mixture (GM) noise channels under an average power constraint. The considered model can be used to represent a wide variety of channels with impulsive interference, such as the well-known Bernoulli-Gaussian and Middleton class-A impulsive noise channels, as well as multiple-access interference channels and cognitive radio channels under imperfect sensing. At first, we demonstrate that there exists a unique input distribution that achieves the channel capacity, and the capacity-achieving input distribution has a uniformly distributed phase. By examining the Kuhn-Tucker alignment conditions (KTCs), we further show that, if the optimal input amplitude distribution contains an infinite number of mass points on a bounded interval, the channel output must be Gaussian-distributed. However, by using Bernstein's theorem to examine the completely monotonic condition, it is shown that the assumption of a Gaussian-distributed output is not valid. As a result, there are always a finite number of mass points on any bounded interval in the optimal amplitude distribution. In addition, by applying a novel bounding technique on the KTC and using the envelop theorem, we demonstrate that the optimal amplitude distribution cannot have an infinite number of mass points. This gives us the unique solution of the optimal input having discrete amplitude with a finite number of mass points. Given this discrete nature of the optimal input, we then develop a simple method to compute the discrete optimal input and the corresponding capacity. Our numerical examples show that, in many cases, the capacity-achieving distribution consists of only one or two mass points.

In this paper, we propose physical layer security for cooperative cognitive radio networks (CCRNs) with relay selection in the presence of multiple primary users and multiple eavesdroppers. To be specific, we propose three relay selection schemes, namely, opportunistic relay selection (ORS), suboptimal relay selection (SoRS), and partial relay selection (PRS) for secured CCRNs, which are based on the availability of channel state information (CSI) at the receivers. For each approach, we derive exact and asymptotic expressions for the secrecy outage probability. Results show that under the assumption of perfect CSI, ORS outperforms both SoRS and PRS.

This paper investigates the error performance of a full-duplex (FD) amplify-and-forward (AF) single-relay system under the effect of residual self-interference whose variance is proportional to the λ-th power of the transmitted power (0≤λ ≤1). Our focus is on the cooperative FD linear relaying (LR) protocol that makes use of direct source-destination link. At first, a closed-form expression of the pairwise error probability (PEP) is derived for the considered system. This expression allows us to analyze the diversity behavior in high transmission power regions. Thanks to the use of the direct link, it is shown that the FD LR system can attain the same diversity function as its half-duplex (HD) counterpart as long as a suitable precoder is applied. Different from previous works that either ignore the direct link or treat it as a source of interference, a non-zero diversity order is thus achieved and the error floor behavior can be eliminated despite the existence of self-interference in FD. More interestingly, it is then demonstrated that transmitting a superposition of all symbols in HD mode maximizes the asymptotic coding gain. Although HD relaying is hence asymptotically optimal, simulations results reveal that FD is advantageous at practical bit error rate (BER) levels when λ is sufficiently small.

This paper presents simple methods to tightly estimate the information rate achieved by a Gaussian input and the constrained capacity of a finite-alphabet input of a Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading. Specifically, under the assumption of a Gaussian input, we propose a novel approach to calculate the achievable rate by examining the instantaneous output entropy in two regions of channel gains. In the high-gain region, the rate is evaluated via an upper bound obtained under the Gaussian output assumption. In the other region, we apply the piecewise-linear curve fitting (PWLCF) method to estimate the rate. It is then demonstrated that the information rate achieved by Gaussian inputs can be effectively calculated with a pre-determined accuracy. For a finite-alphabet input, we detail a PWLCF-based method to estimate the constrained capacity. In particular, we first propose a numerical technique to calculate the instantaneous output entropy using 2-dimensional Gauss-Hermite quadrature formulas. The average output entropy is then obtained using PWLCF. Combined with the closed-form expression the entropy of the BG impulse noise, an accurate estimation of the constrained capacity is finally established.

This paper investigates linear precoding designs for a cooperative amplify-and-forward (AF) network with a multi-antenna relay having complete channel state information (CSI). The focus is on both orthogonal AF (OAF) and non-orthogonal AF (NAF) protocols. The precoders at the relay are derived based on the maximum ratio combining (MRC) scheme, followed by an optimal power amplification factor to maximize the end-to-end achievable rate. For OAF, it is a concave optimization problem and the closed-form solution can be obtained using Karush-Kuhn-Tucker (KKT) conditions. However, the optimization problem for NAF is non-convex and getting globally optimal solution in closed-form is more challenging. Our approach is to investigate the achievable rate in different sub-domains of the channel matrix to upper-bound the original problem by a convex optimization problem. It is then shown that the optimal solution to the power amplification factor of the original optimization problem can be obtained in closed-form. The optimal MRC-based relay precoding vector is then established. Numerical results reveal that the proposed system achieves significant end-to-end rate gains over the conventional dual-hop AF multi-antenna as well as cooperative AF single-antenna systems.

This paper investigates the error and diversity performances of full-duplex (FD) amplify-and-forward (AF) singlerelay systems under the effect of residual self-interference. The variance of this interference is assumed to be proportional to the λ-th power of the transmitted power (0 ≤ λ ≤ 1). The study considers the cooperative linear relaying protocol with direct source-destination link and the dual-hop scheme without direct link, both under uncoded and coded frameworks. At first, closed-form pairwise error probability expressions are derived for the uncoded systems, which are then used to obtain tight bounds to the bit error rate (BER) of the coded systems. To shed an insight on the diversity behavior, asymptotic expressions at high transmission powers are also presented. Different from previous works that treat the direct link as interference, this paper shows that FD linear relaying systems with a suitable precoder can attain the same diversity function as their half-duplex (HD) counterparts. However, further analysis shows that HD orthogonal AF using a superposition constellation is asymptotically optimal in terms of maximum coding gain. In addition, it is shown that the diversity of FD dual-hop systems is a decreasing function of λ and is equal to zero when λ = 1. Although HD relaying is asymptotically optimal under the considered protocols and interference model, illustrative results show that FD relaying is advantageous at practical BER levels when λ is sufficiently small.

Spatial modulation (SM) is a new transmission technique for multi-antenna systems in which the transmit antennas are used to modulate the signal. In this paper, the symbol error rate (SER) performance of SM is investigated. In SM, a signal domain modulation (i.e. amplitude-phase modulation) and an antenna domain modulation (i.e. space shift keying) are combined together to achieve a certain transmission rate while exploiting the properties of two independent modulation domains. A key question is the fine balance between the constellation sizes in the two domains when a constant rate is targeted. For a fixed rate, there are many ways to assign constellation vectors to the spatial and signal domains. In this paper, we investigate optimal constellation breakdown between space and signal domains. The analysis is based on the union bound of the error probability of SM with two typical APM schemes, i.e. phase-shift keying (PSK) and square quadrature amplitude modulation (S-QAM). It is shown that, at any transmission rate, there exists an optimal APM dimension in which the SER is minimized. Furthermore, a trade-off between the number of transmit antennas and the transmit power is introduced.

This paper investigates the achievable rate and ergodic capacity of a non-orthogonal amplify-and-forward (NAF) half-duplex multi-relay network where multiple relays exploit channel state information (CSI) to cooperate with a pair of source and destination. In the first step, for a given input covariance matrix at the source, we derive an optimal power allocation scheme among the relays via optimal instantaneous power amplification coefficients to maximize the achievable rate. Given the nature of broadcasting and receiving collisions in NAF, the considered problem in this step is non-convex. To overcome this drawback, we propose a novel method by evaluating the achievable rate in different sub-domains of the vector channels. It is then demonstrated that the globally optimal solution can be derived in closed-form. In the next step, we establish the ergodic channel capacity by jointly optimizing the input covariance matrix at the source and the power allocation among the relays. We show that this is a bi-level non-convex problem and solve it using Tammer decomposition method. This approach allows us to transform the original optimization problem into an equivalent master problem and a set of sub-problems having closed-form solutions derived in the first step. The channel capacity is then obtained using an iterative water-filling-based algorithm. Finally, we analyze the capacity-achieving input covariance matrix at the source in high and low signal-to-noise ratio (SNR) regimes. At sufficiently high SNRs, it is shown that the transmit power at the source should be equally distributed in all broadcasting and cooperative phases. On the other hand, in low SNR regions, the source should spend all its power in the broadcasting phase associated with a relay having the strongest cascaded source-relay and relay-destination channels.

This paper presents a novel approach to tightly estimate the ergodic Shannon and constrained capacities of an additive Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading environments where channel gains are known at the receiver, but not at the transmitter. We first show that the differential entropy of the BG impulsive noise can be established in closed-form using Gaussian hypergeometric function2F1(1, 1; ·; ·). The Shannon capacity is then calculated via upper and lower bounds. Specifically, we derive in closed-form two upper bounds on the Shannon capacity using the assumption of a Gaussian output and using full knowledge of noise state, respectively. Under the assumption of a Gaussian input, we propose a novel approach to calculate a lower bound by examining the instantaneous output entropy in two regions of channel gains. In the high-gain region, the lower bound is evaluated via the upper bound obtained under the Gaussian output assumption. In the other region, we apply the piecewise-linear curve fitting (PWLCF) method to estimate the lower bound. It is then demonstrated that the lower bound can be calculated with a predetermined accuracy. By establishing the difference between the lower bound and the two upper bounds, we show that the lower bound can be used to effectively estimate the Shannon capacity. Finally, we detail a PWLCF-based method to estimate the constrained capacity for a finite-alphabet constellation. To this end, we first propose a numerical technique to calculate the instantaneous entropy of the output using 2-dimensional (2-D) Gauss-Hermite quadrature formulas. The average output entropy is then obtained using the PWLCF method. Combined with the closed-form expression of the entropy of the BG impulsive noise, the constrained capacity can be effectively estimated.

In this paper, we analyze the capacity of the static half-duplex single-relay amplify-and-forward (AF) system under both per-node and joint power constraints. Different from multiple-input multiple-output systems, the channel matrix of the cooperative AF system is a function of the parameters to be optimized and hence water-filling over the square of the singular values is no longer optimal. Furthermore, given that the mutual information of the AF system is not a concave function, conventional optimization methods cannot be applied. Instead, by deriving and comparing all local solutions, we characterize the optimal input covariance matrix at the source and the optimal power allocation scheme at the relay that maximize the achievable rate. First, for the individual power constraint scenario, it is shown that the capacity of the AF system is achieved by either a direct transmission (DT) scheme, a non-orthogonal AF (NAF) beamforming (BF) protocol with a unit-rank covariance matrix, or a NAF system using a specific full-rank covariance matrix. Then, for the global power constraint scenario, it is shown that only a DT or a NAF-BF protocol can achieve the capacity. In both cases, orthogonal transmission is strictly suboptimal. The capacity of the AF system is finally analyzed for some concrete examples, such as under asymptotically high and low transmission powers and for several network models.

In this paper, achievable rates and power allocation (PA) schemes are studied for a two-way (TW) amplify-and-forward (AF) relaying system over Rayleigh fading channels in high and low signal-to-noise ratio (SNR) regimes. We consider three different AF techniques: i) the channel inversion (CI) scheme where the relay has full channel side information (CSI) of the two incoming links; ii) the fixed-gain scheme where the relay has only channel distribution information (CDI); and iii) a mixed (MX) scheme where the relay has CSI of one link but only CDI of the other. First, focusing on high SNR regimes, approximations to the achievable rates of the three systems are derived. The approximations are shown to be tight and can be used to analyze the systems under consideration. In particular, it is demonstrated that the CI system provides the best sum rate performance, followed by the MX and CDI techniques. A suboptimal yet effective PA scheme to maximize the sum rate at high SNR is then proposed. In low SNR regimes, we first derive rate approximations for the considered AF systems. It is then shown that the three systems achieve a similar sum rate. An asymptotically optimal PA is then proposed to maximize the sum rate at low SNRs. Finally, the sum rate of the direct transmission (DT) scheme is compared to that of the TWAF system. Although the TWAF is proved to be inferior at low SNRs, it outperforms the DT at high SNRs in various network configurations.

This paper addresses the problem of optimal power allocation among relaying nodes for a non-orthogonal amplify-and-forward (NAF) half-duplex relay network where multiple relays exploit channel state information (CSI) to cooperate with a pair of source and destination to maximize the source-destination mutual information (MI). In particular, assuming that the relays have complete CSI of all source-relay, source-destination, and relay-destination links, we investigate an optimal power sharing scheme via optimal power amplification coefficients at the relays. Given the nature of broadcasting and receiving collisions in NAF, the considered problem is non-convex. To overcome this drawback, we propose a novel method by evaluating the MI in different sub-domains of the vector channels. It is then demonstrated that the globally optimal solution can be obtained. In particular, the optimal solutions are derived in closed-form for the system under the total average power constraint (TAPC) and for the system under both TAPC and individual average power constraint (IAPC) at each relay. Numerical results are provided to quantify the significant gains offered by the proposed power sharing schemes over conventional schemes using either channel distribution information (CDI) or channel inversion (CI).

This paper investigates the precoder design for a non-orthogonal amplify-and-forward (NAF) half-duplex single-relay channel using mutual information (MI) as the main performance metric. Different from precoder design methods using pairwise error probability (PEP) analysis which are valid only at high signal-to-noise ratios (SNR), our precoder design can apply to any SNR region, which is of more interest from both information-theoretic and practical points of view. We develop a MI-based criterion for an arbitrary cooperative length of 2T , which corresponds to the case of using a 2T ×2T precoder. The design criterion is established in a closed-form, which can be helpful in finding an optimal precoder. Then by focusing on the 2×2 precoder design, we analytically show that a good precoder should have all entries that are equal in magnitude, which is different from the optimal precoders obtained thus far using the conventional PEP criterion. Simulation results indicate that the proposed class of precoder outperforms the existing precoders in terms of the mutual information performance.

In this paper, we investigate the channel capacity of an additive Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading via lower and upper bounds. To this end, we first show that the differential entropy of the BG impulse noise can be established in closed-form using Gaussian hypergeometric function2F1(1, 1; .; .). This closed-form expression allows us to derive a lower bound on the capacity limit obtained by a Gaussian input using the Gauss-Hermite quadrature formula. We also derive in closed-form two upper bounds on the channel capacity. The first upper bound is obtained under the assumption of full knowledge of noise state, while the second upper bound is developed using a Gaussian distributed output. At high power regions, the lower bound achieved by Gaussian inputs and the upper bound generated by Gaussian inputs are indistinguishable. These two bounds can therefore be used as an accurate estimation for the channel capacity. When the channel input power is small compared to the power of the impulsive noise component, the lower bound obtained by using a Gaussian input and the upper bound under the perfect knowledge of impulse noise state are almost identical, which are useful to predict the capacity. The establishment of the lower bound and the two upper bounds in closed-form helps us to confirm the near-optimality of the Gaussian input in a wide range of input power levels over BG impulsive noise channels in Rayleigh fading.

In this paper, we investigate the channel regularization and vector perturbation techniques at the source and linear precoding at the relay to enhance the performance of a dual-hop amplify and-forward (AF) relay downlink system. At first, an optimal regularized channel inversion scheme via regularization parameter is proposed. The joint channel regularization and linear precoder design at the relay is then investigated using the mean-square-error (MSE) criterion. Since dealing directly with such a criterion is quite involved, we propose a simple iterative solution. Our simulation results show that the proposed solution significantly outperforms the conventional channel inversion method. To further improve the performance of the regularization scheme, we further incorporate vector perturbation into channel regularization using a sphere encoder at the source. Numerical results show that the bit-error-rate (BER) performance of the regularized perturbation technique outperforms other well-known precoder designs in dual-hop AF relay downlink systems.

In this paper, we establish the achievable sum-rate of a half-duplex single-relay two-way amplify-and-forward network in which the relay uses channel knowledge of the two incoming links to cooperate with the two sources. Specifically, by assuming that the relay can acquire full channel knowledge and that Gaussian codebooks are used at the source nodes, the optimal power adaptation scheme at the relay that maximizes the achievable sum-rate under a long-term average power constraint is derived. While the maximization of the sum-rate is shown to be a convex optimization problem, obtaining optimal solutions are challenging. By using the Karush-Kuhn-Tucker conditions, we first show that finding the optimal relay adaptation scheme is equivalent to finding the root of a quartic polynomial. The closed-form optimal solutions are then obtained. Important insights on the proposed adaptation scheme are also presented and discussed. Numerical results reveal that the derived relay adaptation technique outperforms the conventional fixed-gain and variable-gain amplification coefficients at low signal-to-noise ratio regimes, thanks to the benefit of dynamic power allocation.

This paper investigates the achievable rate and ergodic capacity of an orthogonal amplify-and-forward (OAF) half-duplex multiple-relay network with direct link where multiple relays use channel state information (CSI) to cooperate with the source and destination. The relays are subject to two types of power constraint: the total average power constraint (TAPC) and the individual average power constraint (IAPC). In the first step, by assuming a fixed input covariance matrix at the source, we derive an optimal power allocation (OPA) scheme among the relays via optimal instantaneous power amplification coefficients to maximize the achievable rate. The closed-form optimal solutions are obtained for the considered system under either the TAPC or both the TAPC and IAPC. Next, we derive the ergodic capacity by jointly optimizing the input covariance matrix and the power allocation at the relays. We show that this is a bi-level non-convex problem and solve this using Tammer decomposition method. This approach allows us to convert the original optimization problem to a master problem and a set of sub-problems that have closed-form solutions as obtained in the first step. The ergodic capacity is then obtained using an iterative water-filling-based algorithm.

In this paper, we rigorously analyze the optimal input distribution and capacity of an additive Bernoulli-Gaussian (BG) impulsive noise (IN) channel in high and low input power regimes. First, we obtain an input distribution for which the channel output is Gaussian distributed. This distribution, if valid, shall result in the capacity of the channel. At an asymptotically high input power level, we then show that the derived input is always valid and in fact, it resembles a Gaussian distribution. As such, the Gaussian channel input is considered approximately optimal. Using the monotonicity property of the characteristic function (CF), we then develop a necessary condition for the existence of the derived optimal input for a finite level of input power. The condition indicates that a sufficiently high input power is usually required. Then focusing on the low power region, we first derive an upper bound on the channel capacity assuming full knowledge of noise state. A closed-form expression of the mutual information (MI) achieved by Gaussian inputs, which is considered as a lower bound on the channel capacity, is then developed. By comparing these two bounds, it is shown that a Gaussian input asymptotically results in the capacity. Interestingly, it is also demonstrated that such a capacity is the same as the capacity of an erasure channel in low power regimes.

In this paper, we develop optimal power adaptation schemes by means of power amplification coefficients at the relay for the half-duplex single-relay non-orthogonal amplify-and-forward (NAF) system with channel side information (CSI) available at the relay. In particular, assuming that the relay has full knowledge of the channel gains, optimal power adaptation schemes are established in closed-form using both the mutual information (MI) and pairwise error probability (PEP) criteria with Gaussian inputs at the source. The proposed solutions can be understood as a multi-dimensional (multi-D) distributed water-filling in time and space. Numerical results show that the proposed power amplification methods provide a significant improvement over conventional schemes using either channel distribution information (CDI) or channel inversion (CI). The gain is observed with regard to both the MI with Gaussian inputs and the bit error rate (BER) performance using finite constellations such as quadrature amplitude modulation (QAM).

In this paper, we establish in closed-form the capacity and characterize the optimal input covariance matrix at the source and the optimal power allocation scheme between source and relay for a half-duplex single-relay amplify-and-forward (AF) system with static channel gains. Different from multiple-input multiple-output (MIMO) systems, the channel matrix of the AF system is a function of the parameters to be optimized and hence water filling over the square of the singular values of this matrix is no longer optimal. Furthermore, given that the mutual information of the AF system is not a convex function, conventional optimization methods cannot be applied to find the optimal input covariance, the power allocation, and the capacity. Instead, by analyzing all local maximizers, it is shown that the capacity of the AF system is achieved by either the direct transmission (DT) scheme, the orthogonal AF (OAF) protocol, or the non-orthogonal AF (NAF) protocol using a non-diagonal covariance matrix. The choice of protocol depends on the signal-to-noise ratio (SNR) and network configuration. By further analyzing the asymptotic mutual information, it is shown that the DT scheme is dominant in low and high SNR regimes. The capacity of the AF system is also provided for comparison among several network models. Specifically, it is demonstrated that in a general linear network model, the NAF protocol can provide significant gains over the other schemes at medium SNRs.

In this paper, we investigate the capacity limit of a single secondary user (SU) communication link in a cognitive radio system. An SU transmitter establishes the communication with its receiver by dynamically hopping in a frequency spectrum pool, and by sensing the spectrum to exploit the temporal communication opportunities. We characterize the performance of sensing by false-alarm and miss-detection probabilities. Firstly, focusing on a high channel input power region, we develop an upper bound on the SU capacity by assuming a Gaussian distributed output. A lower bound on the SU capacity is also derived using a Gaussian input. We then show that the lower bound closely approaches the upper bound at a high channel input power level. This means that the Gaussian input is nearly optimal in this case. Furthermore, we characterize the impact of primary user activities and sensing performance on the SU capacity by developing a closed-form tight approximation. Secondly, paying attention to a low channel input power region, we propose a genie-aided upper bound and a lower bound using the Gaussian input. By comparing these two bounds, a closed-form approximation to the capacity is developed and the near optimality of the Gaussian input is demonstrated. Finally, numerical results are provided to complement the theoretical discussion.

In this paper, we investigate the capacity and optimal power allocation (PA) scheme between the source and relay for a dual-hop amplify-and-forward (AF) system over non-symmetric Rayleigh fading channels with channel information available at the relay. At first, a closed-form expression of the mutual information (MI) between the input and output of the considered channel is obtained. Since only the exponential integral is involved, the derived expression is useful in finding the optimal PA to achieve the capacity. By further considering high and low signal-to-noise ratio (SNR) regimes, we present tight yet simple approximations to this MI, which can be used to show the advantage of knowing channel information at the relay. Then, focusing on the problem of optimal PA, we first derive a closed- form derivative of the MI. A simple bisection method is then proposed to find the optimal PA scheme. While uniform PA is shown to achieve the capacity at any SNR over the symmetric channel, its optimality can only be observed at low SNRs over a non- symmetric channel. In other SNR regimes, numerical results reveal that uniform PA experiences a significant loss. A comparison between the dual-hop and direct transmission scheme is also made, where we show that the dual-hop scheme using the optimal PA can provide impressive rate increases in medium SNR ranges in various network configurations.

In this paper, we investigate the achievable rate and ergodic capacity of a general non-symmetric half-duplex non-orthogonal amplify-and-forward (NAF) single-relay channel in Rayleigh fading environments, assuming that the channel state information is only available at the destination. The considered channel, which captures pathloss and shadowing effects over the transmission links, includes the symmetric one as a special case. At first, for a given power allocation scheme, simple and closed-form expressions of the upper and lower bounds on the achievable rate are derived. As shown by various numerical examples, the gap between the upper and lower bounds is small in the entire range of SNRs, which makes them useful in finding the optimal power allocation solution to achieve the capacity. Focusing on the two extreme cases of low and high SNRs, we then provide relatively tight approximations of the achievable rate. Using these approximations, it is then revealed that at both high and low SNR regimes, the ergodic capacity is achieved when the relay is inactive. Equivalently, NAF relaying does not yield any advantage over direct transmission at low and high SNRs. The results and observations in this paper therefore provide some further important insights on NAF relaying.

This paper studies the mutual information improvement attained by rotated multidimensional (multi-D) constellations via a unitary precoder G in Rayleigh fading. At first, based on the symmetric cut-off rate of the N-D signal space, we develop a design criterion with regard to the precoder G. It is then demonstrated that the use of rotated constellations in only a reasonably low dimensional signal space can significantly increase the mutual information in high-rate regimes. Based on parameterizations of unitary matrices, we then construct good unitary precoder G in 4-D signal space using a simple optimization problem, which involves only four real variables and it is applicable to any modulation scheme. To further illustrate the potential of multi-D constellation and to show the practical use of mutual information improvement, we propose a simple yet powerful bit-interleaved coded modulation (BICM) scheme in which a (multi-D) mapping technique employed in a multi-D rotated constellation is concatenated with a short-memory high-rate convolutional code. By using extrinsic information transfer (EXIT) charts, it is shown that the proposed technique provides an exceptionally good error performance. In particular, both EXIT chart analysis and simulation results indicate that a turbo pinch-off and a bit error rate around 10-6happen at a signal-to-noise ratio that is well below the coded modulation and BICM capacities using traditional signal sets. For example, with code rates ranging from 2/3 to 7/8, the proposed system can operate 0.82 dB-2.93 dB lower than the BICM capacity with QPSK and Gray labeling. The mutual information gain offered by rotated constellations can be therefore utilized to design simple yet near Shannon limit systems in the high-rate regions.

This paper proposes a precoding scheme over multiple cooperative frames to increase the diversity order of a bandwidth efficient Bit Interleaved Coded Modulation (BICM) system over a Non-orthogonal Amplify-and-Forward (NAF) half-duplex single-relay channel. By deriving a union bound on the bit error probability, it is shown that the diversity gain function of the considered system is (Nf· dH)-th power of that of uncoded cooperative systems, where Nfis the number of precoded cooperative frames and dHis the minimum Hamming distance of the outer code. An optimal class of precoders is then derived to optimize the asymptotic coding gain. It is then shown that the source should transmit a superposition of all symbols in the broadcasting phases, while being silent in all cooperative phases for best asymptotic performance. By further analyzing the first iteration performance, a design criterion is then developed to find optimal superposition angles for good convergence behavior. A pragmatic approach is then proposed to find good rotation angles. Analytical and simulation results show that the proposed scheme provides a significantly higher order of time and cooperative diversities and better coding gains than previous precoding schemes.

In this paper, we design a capacity-approaching coded modulation scheme for half-duplex non-orthogonal amplify-and-forward (NAF) single-relay channel. We apply the idea of multi-dimensional (multi-D) mapping employed in precoded multiple cooperative frames, concatenated with a simple outer binary code. Using union bounding techniques, it is first shown that for any unitary and full-diversity rotation G, the optimal multi-D labeling for NAF relaying shall maximize the average Euclidean distance between all pairs in the multi-D rotated constellation whose labels differ in only one bit. The extrinsic information transfer (EXIT) charts are then used to match the outer code, the multi-D mapping, and the precoder for near-capacity performance. It is demonstrated that the proposed scheme is promising for NAF relaying, in the sense that it can operate below the achievable rate obtained by using a conventional modulation scheme, i.e., the constrained capacity. In particular, for various spectral efficiencies, we obtain the bit error rate (BER) of 10-5or lower at a signal-to-noise ratio (SNR) that is 0.45dB-1dB below the traditional achievable rate and within 1.55dB-1.95dB from the ergodic capacity with Gaussian inputs.

This paper studies the mutual information improvement attained by rotated multidimensional (multi-D) constellations via a unitary precoder G ∈ CN×Nin Rayleigh fading. At first, based on the symmetric cut-off rate of the N-D signal space, we develop a design criterion with regard to the precoder G. It is then demonstrated that the use of rotated constellations in only a reasonably low dimensional signal space can significantly increase the mutual information in high-rate regimes. By considering the QPSK system, we then propose a class of good unitary rotation G in 4-D signal space using parameterization approach, which are shown to provide remarkable improvement. To further illustrate the potential of multi-D constellation and to show the practical use of mutual information improvement, we propose a simple yet powerful coded modulation scheme in which a (multi-D) mapping technique employed in a multi-D rotated constellation is concatenated with a short-memory high-rate convolutional code (CC). By using extrinsic information transfer (EXIT) charts, it is demonstrated that the proposed technique provides an exceptionally good error performance. For example, by using the derived 4×4 rotation, together with QPSK constellation, and a simple rate-3/4 outer convolutional code, it is shown that the proposed system can operate 1.39 dB lower than the traditional coded modulation capacity at the bit error rate (BER) level around 10-6.

ABSTRACT This paper proposes the idea of precoding over multiple cooperative frames with multi‐dimensional (multi‐D) mapping for a bit interleaved coded modulation system over an ergodic non‐orthogonal amplify‐and‐forward (NAF) half‐duplex single‐relay channel. The benefits of multiple‐frame precoding and multi‐D labeling are analyzed in two different regions: the error‐floor region to exploit diversity and the turbo pinch‐off region for near‐capacity performance. In the error‐floor area, it is shown that the diversity gain function of the considered system is th power of that of uncoded cooperative systems, where Nf is the number of precoded cooperative frames and dH is the minimum Hamming distance of the outer code. To optimize the asymptotic coding gain, it is then shown that the source and relay must transmit orthogonally for best asymptotic performance. In the turbo pinch‐off region, we demonstrate that the proposed system concatenated with a simple outer binary code can be employed to achieve near‐capacity performance. Using union bounding techniques, we show that the optimal multi‐D labeling for NAF relaying shall maximize the average Euclidean distance between all pairs in the multi‐D rotated constellation whose labels differ in only 1 bit. The extrinsic information transfer charts are then used to match the outer code, the multi‐D mapping, and the precoder. It is demonstrated that the proposed system is also promising for NAF relaying in the turbo pinch‐off region, in the sense that it can operate below the achievable rate achieved by a conventional coded modulation scheme. Copyright © 2011 John Wiley & Sons, Ltd.

This paper considers the precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) over non-orthogonal amplify-and-forward (NAF) single-relay channels with an arbitrary length of cooperative frame 2N. Based on the tight union bound on the bit error probability (BEP), we first derive an asymptotic design criterion with regard to a general 2N × 2N rotation matrix. This expression allows us to develop a class of precoder that not only achieves full cooperative diversity but also optimizes the asymptotic error performance. Interestingly, the developed class of optimal precoder indicates that the source should be kept silent in the cooperative phase. In the broadcasting phase, it is shown that power is distributed equally to 2N information symbols at the source. By further examining the structure of the optimal class of 2N × 2N precoders, we then reveal that precoding over a group of at least 2 information symbols is sufficient to fully exploit diversity and coding advantages. Such precoding technique, which is referred to as symbol grouping, therefore significantly reduces the system complexity without degrading the error performance.

This paper investigates the error performance and precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) system over a non-orthogonal amplify-and-forward (NAF)half-duplex single-relay channel. A tight union bound on the bit error probability (BEP) is first derived for an arbitrary block length of 2N using a 2N × 2N precoder. This bound provides an useful tool to predict the error performance. Attention is then devoted to the system using 2×2 precoder, where a closed-form expression of the bound is obtained. Based on this expression, an optimal 2×2 precoder applicable to any modulation scheme is developed. Different from the optimal precoders designed for uncoded NAF systems, the derived precoder indicates that the source only needs to send the superposition of signals in the first time slot and being silent in the second time slot in order to achieve the best asymptotic performance. Analytical and simulation results show that the proposed precoder not only exploits full cooperative diversity but also offers a significant coding gain over optimal precoders for uncoded NAF systems.

This paper studies the problem of pilot-aided joint carrier frequency offset (CFO) and channel estimation using a Bayesian approach in multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) transmissions over time- and frequency-selective (doubly selective) channels. Unlike the joint CFO and channel impulse response (CIR) estimation over block-fading channels, the joint CFO and time-variant CIR estimation gives rise to the identifiability problem where the number of observations (received samples) is smaller than that of both CFO and time-variant CIR parameters to be estimated. To reduce a large number of the time-variant CIR parameters to be estimated, various basis expansion models (BEMs) are deployed as fitting parametric models for capturing the time variation of the MIMO channels. As the main purpose of using BEMs, the resulting dimension reduction in the time-variant channel representation helps to avoid the identifiability issue in the joint estimation problem. Under Bayesian estimation, CFO and BEM coefficients are treated as random variables to be estimated by the maximum-a-posteriori (MAP) technique. Numerical results demonstrate that the deployment of BEMs is able to alleviate performance degradation in the considered estimation technique using the conventional assumption of block fading over time varying channels.

This paper proposes a capacity-approaching, yet simple scheme for multi-input multiple-output (MIMO) channels. The proposed scheme is based on a concatenation of a mixture of short memory-length convolutional codes or repetition codes and a short, and simple rate-1 linear block code, followed by either 1-dimensional (1-D) anti-Gray or Gray mapping of quadrature phase-shift keying (QPSK) modulation. By interpreting the rate-1 code and the 1-D mapping as a multi-D mapping performed over multiple transmit antennas, the error performance is analyzed in two regions. In the error-floor region, a tight union bound and the corresponding design criterion on the asymptotic performance are derived. The bound provides a useful tool to predict the error performance at relatively low bit error rate (BER) values. Based on the obtained design criterion, an optimal rate-1 code for each 1-D mapping is then constructed to achieve the best asymptotic performance. In the turbo pinch-off region, by using extrinsic information transfer (EXIT) charts, the most suitable mixed codes are selected for both symmetric and asymmetric antenna configurations. It is demonstrated that the simple concatenation scheme can achieve a near-capacity performance over the MIMO channels. Furthermore, its error performance is shown to be comparable to that obtained by using well-designed irregular LDPC and RA codes, and therefore, the proposed scheme significantly outperforms a scheme employing a parallel concatenated turbo code. Simulation results in various cases are provided to verify the analysis.

This paper proposes a capacity-approaching, yet simple scheme over a multiple-input single-output (MISO) wireless fading channel, which is very common in the downlink of a cellular system. The proposed scheme is based on a concatenation of a mixture of short memory-length convolutional codes or repe

The paper introduces a turbo (iterative) receiver design for joint channel estimation, synchronization and soft decoding in convolutional-coded multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems over time- and frequency-selective (doubly selective) channels. Employing the complex-exponential basis expansion model (CE-BEM) for representing doubly selective channels, a maximum likelihood (ML) objective function of carrier frequency offset (CFO) and MIMO time-varying channel responses (BEM coefficients) is formulated to develop a semi-blind ML framework for joint time-variant channel estimation and synchronization. To reduce the overhead of pilot signals without sacrificing estimation accuracy, the soft bit information from a soft-input soft-output (SISO) decoder is exploited in computing soft estimates of data symbols to be functioned as pilots for further enhancing the estimation accuracy after CFO and channel acquisition phase (initial coarse estimation) using pilots. In other words, the resulting semi-blind ML estimation scheme operates in conjunction with soft decoding process in a (iteratively) progressive manner to exploit remarkable gains of turbo processing (iterative extrinsic information exchange). Simulation results show that the proposed turbo joint channel estimation and synchronization scheme offers high estimation accuracy that approaches Cramer-Rao lower bounds (CRLBs) over a wide range of CFO values under low signal-to-noise ratio (SNR) conditions.

This paper proposes a simple yet capacity-approaching concatenation of a mixture of short memory length convolutional codes and simple rate-1 block code followed by either complex 1-dimensional (1-D) anti-Gray or Gray mapping over multiple antenna channels with quadrature phase-shift keying (QPSK). By interpreting rate-1 code together with 1-D mapping as a multi-D mapping employed over multiple transmit antennas, the error performance is analyzed in two regions, the error-floor and turbo pinch-off regions. In the former one, a tight union bound and design criterion on the asymptotic performance are first derived, which provide an useful tool to predict the error performance. Based on the design criterion, an optimal rate-1 code for each 1-D mapping is then constructed to achieve the best asymptotic performance. In the turbo pinch-off area, by using extrinsic information transfer (EXIT) chart, the most suitable mixed codes are selected for both symmetric and asymmetric antenna setups. It is demonstrated that the simple concatenation scheme can achieve near-capacity. Furthermore, its error performance is comparable to that obtained by using well-designed irregular low-density parity-check (LDPC) and repeat accumulate (RA) codes, and thereby, outperforms a scheme employing a parallel concatenated turbo code.

This paper proposes a near-capacity yet simple bit-interleaved coded modulation with iterative decoding (BICM-ID) scheme by employing a multi-dimensional (multi-D) mapping technique in a multi-D constellation carved from a rotated lattice. Using extrinsic information transfer (EXIT) charts, it is shown that the proposed technique fits well with simple convolutional codes in terms of the area property, for which turbo pinch-off can happen at a low Eb/N0value. In particular, both EXIT chart analysis and simulation results indicate that by using just a simple convolutional code together with a 4-D mapping, a turbo pinch-off and a bit error rate (BER) close to 10-6happen at a signal-to-noise ratio (SNR) that is even lower than the BICM constraint capacity limit with a uniform input. The proposed BICM-ID scheme can be considered as an attractive alternative to other bandwidth-efficient coded modulation techniques using powerful turbo-like codes such as turbo or low-density parity-check (LDPC) codes over a Rayleigh fading channel.

This letter generalizes the application of signal space diversity (SSD) over multiplicative fading channels, where fading is represented by the product ofKstatistically independent Nakagami-mrandom variables. The pairwise error probability (PEP) for the systems is first obtained in a closed-form using generalized hypergeometric functions. Based on the obtained PEP expression, it is shown that the error performance over multiplicative fading channels can be significantly improved by using high diversity constellations. Furthermore, by employing SSD with a sufficiently large dimension, it is observed that the adverse effects of multiplicative fading can be practically eliminated without any power nor bandwidth expansion. Simulation results for both uncoded and coded systems are provided to show the agreement with the analysis.

This paper generalizes the application bit-interleaved coded modulation with iterative decoding (BICM-ID) using signal space diversity (SSD) overkeyholeNakagami-mfading channels. The tight union bound on the asymptotic error performance is first analytically derived. The near-optimal rotation matrix with respect to both the asymptotic performance and the convergence behavior is then determined. In particular, it is demonstrated that the suitable rotation matrix is the one that has 1) all entries equal in magnitude, 2) a high diversity order, and 3) a large minimum product of the ratios between squared distances to the powermand log-squared distances to the powermof the rotated constellation scaled by factors of signal-to-noise ratio (SNR) and the parameterm. Various analytical and simulation results show that by employing SSD with a sufficiently large dimension, the error performance can closely approach that over an additive white Gaussian noise (AWGN) channel, even in the worst case of keyhole fading.

The symbol error rate (SER) and the bit error rate (BER) of orthogonal space-time block codes (OSTBCs) with antenna selection over keyhole fading channels are examined. Considered are receive antenna selection, transmit antenna selection, and joint antenna selection at both the transmitter and the receiver. The exact SER of OSTBCs for M-PSK and square M-QAM constellations is obtained using the technique of moment generating function. By applying the Bonferroni-type bounds, tight lower and upper bounds for both the SER and BER are provided in closed-form expressions with finite-range single integrals. The bounds can be applied to arbitrary constellations and mappings. Numerical results show that the bounds can be used to provide practically the exact SER and BER over a wide range of the signal-to-noise ratio.

Bit-interleaved coded modulation with iterative decoding (BICM-ID) using signal space diversity (SSD) is considered for cascaded Rayleigh fading channels. A tight bound on the asymptotic error probability is derived to determine the optimal rotation matrix for SSD design and to identify the key parameters that influence the system performance. It is shown that, for small modulation constellation, a cascaded Rayleigh fading causes a much more severe performance degradation than a conventional Rayleigh fading. However, BICM-ID employing SSD with a sufficiently large constellation can close the performance gap between the conventional and cascaded Rayleigh fading channels, and their performance can closely approach that over an AWGN channel. Illustrative simulation results for various scenarios are in a good agreement with analytical derivations.

The symbol error rate (SER) and the bit error rate (BER) of orthogonal space-time block codes (OSTBCs) with antenna selection over keyhole fading channels are examined. Considered are receive antenna selection, transmit antenna selection, and joint antenna selection at both the transmitter and the receiver. The exact SERs of OSTBCs for M-PSK and square MQAM constellations are obtained using the technique of moment generating function (MGF). By applying the Bonferroni-type bounds, tight lower and upper bounds for both the SER and BER are provided in closed-form expressions with finite-range single integrals. The bounds can be applied to arbitrary constellations and mappings. Numerical results show that the bounds can be used to provide practically the exact SER and BER over a wide range of the signal-to-noise ratio.

Exploiting signal space diversity (SSD) to improve the error performance of communications systems over fading channels has been shown to be a very effective technique. The application of SSD in bit-interleaved coded modulation with iterative decoding (BICM-ID) is considered for cascaded Rayleigh fading channels, which are suitable for mobile-to-mobile communications. A tight bound on the asymptotic error performance is first derived. The bound is then used to find the optimal rotation matrix. It is shown that employing SSD in a sufficiently large constellation can close the performance gap between a conventional Rayleigh fading channel and a cascaded Rayleigh fading channel. In fact, similar to the case of conventional Rayleigh fading, it is demonstrated that the error performance of BICM-ID with SSD over a cascaded Rayleigh fading channel can also closely approach the performance of BICM-ID over an AWGN channel. Various analytical and simulation results are provided to confirm the analysis.

This paper generalizes the application of bit interleaved coded modulation with iterative decoding (BICM-ID) using signal space diversity (SSD) over keyhole Nakagami-m fading channels. The tight union bound on the asymptotic error performance is first analytically derived. The optimal rotation matrix with respect to both the asymptotic performance and the convergence behavior is then determined. In particular, it is demonstrated that the most suitable rotation matrix is the one that has i) all entries equal in magnitude, ii) high diversity order, and iii) large minimum product of the ratios between squared distances and log-squared-distances of the rotated constellation scaled by factors of SNR and the parameter m. Various analytical and simulation results show that by employing SSD with a sufficiently large dimension, the error performance can closely approach that over an AWGN channel, even in the worst case of keyhole fading.

Multi-dimensional mapping over groups of subcarriers, called subcarrier mapping, is proposed and investigated for bit-interleaved coded in OFDM with iterative decoding (BI-COFDM-ID) over correlated frequency-selective Rayleigh fading channels. A tight bound on the asymptotic error performance is first presented and used to establish the design criterion. It is then shown that one could choose an optimal subcarrier grouping scheme and an optimal multi-dimensional mapping independently to achieve the best overall error performance. Analytical and simulation results show that it is not necessary to implement linear constellation precoding (LCP) in the proposed system while still achieving the full diversity offered by frequency-selective fading channels, and at the same time providing significant coding gains compared to the previously studied BI-COFDM-ID. Such coding gains are obtained without any power nor bandwidth expansion and with the same receiver complexity.

The application of Bonferroni-type bounds is investigated for orthogonal space-time block codes over a keyhole Nakagami-m fading channel, which includes a cascaded Rayleigh fading channel as a special case. In particular, upper and lower bounds on the symbol error rate and bit error rate are derived and shown to be very tight at any signal-to-noise ratio. The developed bounds are applicable for arbitrary signal constellations and mappings and can be accurately computed with single finite-range integrals.

This paper studies the bit error probability of coded unitary space-time modulation with iterative decoding where neither the transmitter nor the receiver knows the channel fading coefficients. The tight error bound with respect to the asymptotic performance is first analytically derived for any given unitary constellation and mapping rule. Design criteria regarding the choice of unitary constellation and mapping are then established. Furthermore, using the unitary constellation obtained from orthogonal design with quadrature phase-shift keying (QPSK or 4-PSK) and 8-PSK, two different mapping rules are proposed. The first mapping rule gives the most suitable mapping for systems that do not implement iterative processing, which is similar to a Gray mapping in coherent channels. The second mapping rule yields the best mapping for systems with iterative decoding. In particular, analytical and simulation results show that with the proposed mappings of the unitary constellations obtained from orthogonal designs, the asymptotic error performance of the iterative systems can closely approach a lower bound which is applicable to any unitary constellation and mapping

Employing multi-dimensional constellation and mapping to improve the error performance of the bit-interleaved coded modulation with iterative decoding (BICM-ID) has recently received a lot of attention, both in single-antenna and multiple-antenna systems. To date, except for the cases of BPSK and QPSK constellations, good multi-dimensional mappings have only been found by computer searching techniques. This paper introduces an explicit algorithm to construct a good multidimensional mapping of 8-PSK for improving the asymptotic performance of BICM-ID systems. By comparing the performance of the proposed mapping with an unachievable lower bound, it is conjectured that the proposed mapping is the global optimal mapping. The superiority of the proposed mapping over the best conventional (two-dimensional) mapping and the multidimensional mapping found previously by computer search is also thoroughly demonstrated

This paper presents a performance analysis of bit-interleaved coded-modulation with iterative decoding (BICM-ID) and complex N-dimensional signal space diversity in fading channels to investigate its performance limitation, the choice of the rotation matrix and the design of a low-complexity receiver. The tight error bound is first analytically derived. Based on the design criterion obtained from the error bound, the optimality of the rotation matrix is then established. It is shown that using the class of the optimal rotation matrices, the performance of BICM-ID systems over a Rayleigh fading channel approaches that of the BICM-ID systems over an AWGN channel when the dimension of the signal constellation increases. Furthermore, by exploiting the sigma mapping for any M-ary QAM constellation, a very simple sub-optimal, but yet effective iterative receiver structure suitable for signal constellations with large dimensions is proposed. Simulation results in various cases and conditions indicate that the proposed receiver can achieve the analytical performance bounds with low complexity

Employing multi-dimensional constellation and mapping to improve the error performance of bit-interleaved coded modulation with iterative decoding (BICM-ID) has recently received a lot of attention, both in single-antenna and multiple-antenna systems. To date, except for the cases of BPSK and QPSK constellations, good multi-dimensional mappings have only been found by computer searching techniques. This paper introduces an explicit algorithm to construct a good multi-dimensional mapping of 8-PSK for improving the asymptotic performance of BICM-ID systems. By comparing the performance of the proposed mapping with an unachievable lower bound, it is conjectured that the proposed mapping is the globally optimal mapping. The superiority of the proposed mapping over the best conventional (two-dimensional) mapping and the multi-dimensional mapping found previously by computer search is also demonstrated.

This paper presents a performance analysis of bit-interleaved coded-modulation with iterative decoding (BICM-ID) and complex N-dimensional signal space diversity in fading channels to investigate its performance limitation, the choice of the rotation matrix and the design of a low-complexity receiver. The tight error bound is first analytically derived. Based on the design criterion obtained from the error bound, the optimality of the rotation matrix is then established. It is shown that using the class of the optimal rotation matrices, the performance of BICM-ID systems over a Rayleigh fading channel approaches that of the BICM-ID systems over an AWGN channel when the dimension of the signal constellation increases. Furthermore, by exploiting the sigma mappings for M-QAM constellations, a very simple suboptimal, but yet effective iterative receiver structure suitable for signal constellations with large dimensions is proposed. Simulation results in various cases and conditions indicate that the proposed receiver can achieve the analytical performance bounds with low complexity.

The application of bit-interleaved coded modulation and iterative decoding (BICM-ID) in OFDM systems with modulation diversity over frequency selective Rayleigh fading channels is considered. A tight bound on the asymptotic error performance is first derived for general preceding over all N subcarriers and used to establish the best achievable performance with modulation diversity. It is then shown that preceding over subgroups of at least L subcarriers per group, where L is the number of channel taps, is sufficient to achieve this best performance while keeping the receiver complexity at minimum. The jointly optimum subcarrier grouping and rotation matrix are derived by solving the Vandermonde linear system

This paper examines the bit error probability (BEP) of bit-interleaved coded modulation and iterative decoding (BICM-ID) in OFDM systems with modulation diversity (MD) over correlated frequency selective Rayleigh fading channels. A tight bound on the asymptotic error performance is derived for the general preceding over all the N subcarriers, which includes preceding over subcarrier groups as a special case. A design parameter that characterizes the effects of signal constellation, mapping and modulation diversity on the asymptotic performance is then established. Various analytical and simulation results are provided to confirm the tightness of the derived error bound.

This paper studies the bit error probability (BEP) of coded unitary space-time modulation with iterative decoding where neither the transmitter nor the receiver knows the channel fading coefficients. The tight error bound on the asymptotic performance is first analytically derived for given unitary constellation and mapping rule. Design criterion regarding the choice of unitary constellation and mapping is then established to achieve the best asymptotic performance. Furthermore, using the unitary constellation obtained from orthogonal design and 4-PSK, two mapping rules are proposed. In particular, one mapping is the best mapping for systems with iterative decoding, whereas the other mapping is most suitable for systems that do not implement iteration process. The latter mapping is similar to Gray mapping considered for coherent channels. Analytical and simulation results show that with the proposed mapping of the unitary constellation obtained from orthogonal design, the error performance of the iterative systems can approach very near the performance of the ideal unitary constellation and mapping

This paper introduces new mappings of QPSK symbols, viewed as a multi-dimensional hypercube, to improve the performance of bit-interleaved coded modulation with iterative decoding (BICM-ID). By evaluating the upper bound of the bit error rate performance of BICM-ID, a condition to find the best mapping of a hypercube constellation in terms of the asymptotic performance under different channel models is established. A general and simple algorithm to construct the best mapping of a hypercube is then proposed. Analytical and simulation results show that the use of the proposed mappings together with very simple convolutional codes can offer significant coding gains over the conventional BICM-ID systems for all the channel models considered. Such coding gains are achieved without bandwidth or power expansion and with a very small increase in the system complexity.