A Closed-Form Power Allocation and Signal Alignment for a Diagonalized MIMO Two-Way Relay Channel With Linear Receivers
TL;DR: Simulation results demonstrate that the proposed GSVD-based relaying scheme, with the signal alignment and closed-form power allocation, significantly improves the ASR while retaining the diagonalized channel structure.
Abstract: A novel channel diagonalization scheme for an amplify-and-forward, multiple-input multiple-output (MIMO), two-way relay channel (TWRC) is proposed using generalized singular value decomposition (GSVD). Diagonalization of the MIMO TWRC is a sub-optimal approach that achieves two main purposes: reducing the computational complexity for optimizing the linear precoders at each node; and reducing the detection complexity at the source nodes by separating the multiple data streams. For the given diagonalized structure, we first align the entries of the diagonalized channels using a permutation to maximize a lower bound of average achievable sum rate (ASR), and a joint source-relay power allocation is then performed to maximize the ASR of the aligned TWRC; the overall problem is divided into two convex subproblems, the solutions to which are provided in closed-form. Our analysis for the proposed scheme underlines the benefits of acquiring channel state information. Simulation results demonstrate that the proposed GSVD-based relaying scheme, with the signal alignment and closed-form power allocation, significantly improves the ASR while retaining the diagonalized channel structure. In addition, the proposed scheme achieves the same level of ASR with much less computational complexity as compared to the iterative schemes.
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TL;DR: Simulation results show that the proposed overhearing protocol for two-way cooperative multiantenna relaying systems shows not only lower mean squared error but also higher achievable sum rate than existing cooperative relaying schemes.
Abstract: In this paper, an overhearing protocol is proposed for two-way cooperative multiantenna relaying systems, where the relays equipped with multiple antennas collaborate to relay signals between the base station (BS) and two user equipment units (UEs). In the proposed overhearing protocol, the UE in the uplink transmission phase transmits only in the first time slot, i.e., it remains silent in the second time slot, whereas the previous overhearing protocol assumes that the UE transmits also in the second time slot. Therefore, the proposed overhearing protocol is more power efficient. The precoding matrix at each cooperative relay is optimized in the sense of minimizing the weighted mean squared error (WMSE). Simulation results show that the proposed scheme shows not only lower mean squared error but also higher achievable sum rate than existing cooperative relaying schemes.
118 citations
TL;DR: A general two-way relay (TWR) transmission method is proposed under per-antenna power constraints by combining space-time line codes (STLCs) with transmit power allocation for two source nodes and an optimal encoder structure with power allocation is proposed in terms of maximizing the minimum SINRs.
Abstract: In this paper, a general two-way relay (TWR) transmission method is proposed under per-antenna power constraints by combining space-time line codes (STLCs) with transmit power allocation for two source nodes. We introduce a general STLC-based encoding scheme for a decode-and-forward TWR and derive the detection signal-to-interference-plus-noise ratio (SINR) values at two source nodes. An optimal encoder structure with power allocation is proposed in terms of maximizing the minimum SINRs, and it is verified that the optimal encoder is identical to the superposition of two conventional STLCs. An iterative method is proposed to find the optimal power allocation for detection at two source nodes. Moreover, a low-complexity suboptimal encoder is proposed for practical implementation. Numerical simulations present that the proposed STLC-based transmission method outperforms a conventional eigen-beamforming scheme with nulling and an STLC-based scheme with equal power allocation, in terms of the average bit error rate of source nodes, regardless of the distribution of source nodes, the number of TWR antennas, and the transmit power.
28 citations
Cites methods from "A Closed-Form Power Allocation and ..."
...2905992 have been designed based on the minimum mean square error (MMSE) criterion [8]–[11] and the maximum achievable rate criterion [12], [13]....
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TL;DR: PNC coupled with SA (PNC-SA) has the potential of fully exploiting the precoding space at the senders, and can better utilize the spatial diversity of a MIMO network for higher system degrees-of-freedom (DoF).
Abstract: We apply signal alignment (SA), a wireless communication technique that enables physical layer network coding (PNC) in multi-input multi-output (MIMO) wireless networks. Through calculated precoding, SA contracts the perceived signal space at a node to match its receive capability, and hence facilitates the demodulation of linearly combined data packets. PNC coupled with SA (PNC-SA) has the potential of fully exploiting the precoding space at the senders, and can better utilize the spatial diversity of a MIMO network for higher system degrees-of-freedom (DoF). PNC-SA adopts the idea of `demodulating a linear combination' from PNC. The design of PNC-SA is also inspired by recent advances in IA, though SA aligns signals not interferences. We study the optimal precoding and power allocation problem of PNC-SA, for SNR (singal-to-noise-ratio) maximization at the receiver. The mapping from SNR to BER is then analyzed, revealing that the DoF gain of PNC-SA does not come with a sacrifice in BER. We then design a general PNC-SA algorithm in larger systems, and demonstrate general applications of PNC-SA, and show via network level simulations that it can substantially increase the throughput of unicast and multicast sessions, by opening previously unexplored solution spaces in multi-hop MIMO routing.
23 citations
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...VIII concludes the paper....
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TL;DR: Under the premise that all nodes have local channel state information at the receiver only and operate in half-duplex mode, it is shown that a total K/2 DoF is achievable when M=K-1, and the key to showing this result is a novel encoding and decoding scheme.
Abstract: This paper studies a network information flow problem for a multiple-input multiple-output (MIMO) Gaussian wireless network with K users and a single intermediate relay having M antennas. In this network, each user sends a multicast message to all other users while receiving K-1 independent messages from the other users via an intermediate relay. This network information flow is termed a MIMO Gaussian K-way relay channel. For this channel, it is shown that the optimal sum degrees of freedom (sum-DoF) is KM/K-1, assuming that all nodes have global channel knowledge and operate in full-duplex. A converse argument is derived by cut-set bounds. The achievability is shown by a repetition coding scheme with random beamforming in encoding and a zero-forcing method combined with self-interference cancelation in decoding. Furthermore, under the premise that all nodes have local channel state information at the receiver only and operate in half-duplex mode, it is shown that a total K/2 DoF is achievable when M=K-1. The key to showing this result is a novel encoding and decoding scheme, which creates a set of network code messages with a chain structure during the multiple access phase and performs successive interference cancelation using side-information for the broadcast phase. One major implication of the derived results is that efficient exploitation of the transmit message as side-information leads to an increase in the sum-DoF gain in a multiway relay channel with multicast messages.
17 citations
TL;DR: This paper proposes to employ singular value decomposition (SVD) for both the multiple-access phase and the broadcast phase to parallelize the two-way relay channels, which is optimal in the sense of exploiting full diversity gain.
Abstract: Considering simultaneous wireless information and power transfer (SWIPT), we investigate joint two sources and the relay beamforming design problem for orthogonal-space–time-block-code (OSTBC)-based amplify-and-forward (AF) multiple-input–multiple-output (MIMO) two-way relay networks in this paper. It is found that to exploit full diversity gain, maximum ratio combining should be employed at the relay, which requires the received signals from two sources at the relay to be aligned and proportional. Thus, unlike conventional relay networks without SWIPT, we propose to employ singular value decomposition (SVD) for both the multiple-access phase and the broadcast phase to parallelize the two-way relay channels, which is optimal in the sense of exploiting full diversity gain. Our objective is to maximize the achievable sum rate under a sum transmit power constraint and an energy harvesting constraint. We propose to solve the formulated optimization problem by alternating optimization combined with the line search method and obtain a locally optimal solution. We also propose a near-optimal solution that simplifies the optimization problem by replacing the objective function with its approximation at a high signal-to-noise ratio (SNR). Simulation results demonstrate that the proposed two-way relay network has significant performance improvement over the one-way relay network.
17 citations
Cites methods from "A Closed-Form Power Allocation and ..."
...For conventional AF-MIMO two-way relay networks, it is well known that generalized singular value decomposition (GSVD) for the MAC phase and singular value decomposition (SVD) for the BC phase parallelize the two-way relay channels, which results in near-optimal performance [21], [22]....
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References
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Book•
01 Jan 1991TL;DR: The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Abstract: Preface to the Second Edition. Preface to the First Edition. Acknowledgments for the Second Edition. Acknowledgments for the First Edition. 1. Introduction and Preview. 1.1 Preview of the Book. 2. Entropy, Relative Entropy, and Mutual Information. 2.1 Entropy. 2.2 Joint Entropy and Conditional Entropy. 2.3 Relative Entropy and Mutual Information. 2.4 Relationship Between Entropy and Mutual Information. 2.5 Chain Rules for Entropy, Relative Entropy, and Mutual Information. 2.6 Jensen's Inequality and Its Consequences. 2.7 Log Sum Inequality and Its Applications. 2.8 Data-Processing Inequality. 2.9 Sufficient Statistics. 2.10 Fano's Inequality. Summary. Problems. Historical Notes. 3. Asymptotic Equipartition Property. 3.1 Asymptotic Equipartition Property Theorem. 3.2 Consequences of the AEP: Data Compression. 3.3 High-Probability Sets and the Typical Set. Summary. Problems. Historical Notes. 4. Entropy Rates of a Stochastic Process. 4.1 Markov Chains. 4.2 Entropy Rate. 4.3 Example: Entropy Rate of a Random Walk on a Weighted Graph. 4.4 Second Law of Thermodynamics. 4.5 Functions of Markov Chains. Summary. Problems. Historical Notes. 5. Data Compression. 5.1 Examples of Codes. 5.2 Kraft Inequality. 5.3 Optimal Codes. 5.4 Bounds on the Optimal Code Length. 5.5 Kraft Inequality for Uniquely Decodable Codes. 5.6 Huffman Codes. 5.7 Some Comments on Huffman Codes. 5.8 Optimality of Huffman Codes. 5.9 Shannon-Fano-Elias Coding. 5.10 Competitive Optimality of the Shannon Code. 5.11 Generation of Discrete Distributions from Fair Coins. Summary. Problems. Historical Notes. 6. Gambling and Data Compression. 6.1 The Horse Race. 6.2 Gambling and Side Information. 6.3 Dependent Horse Races and Entropy Rate. 6.4 The Entropy of English. 6.5 Data Compression and Gambling. 6.6 Gambling Estimate of the Entropy of English. Summary. Problems. Historical Notes. 7. Channel Capacity. 7.1 Examples of Channel Capacity. 7.2 Symmetric Channels. 7.3 Properties of Channel Capacity. 7.4 Preview of the Channel Coding Theorem. 7.5 Definitions. 7.6 Jointly Typical Sequences. 7.7 Channel Coding Theorem. 7.8 Zero-Error Codes. 7.9 Fano's Inequality and the Converse to the Coding Theorem. 7.10 Equality in the Converse to the Channel Coding Theorem. 7.11 Hamming Codes. 7.12 Feedback Capacity. 7.13 Source-Channel Separation Theorem. Summary. Problems. Historical Notes. 8. Differential Entropy. 8.1 Definitions. 8.2 AEP for Continuous Random Variables. 8.3 Relation of Differential Entropy to Discrete Entropy. 8.4 Joint and Conditional Differential Entropy. 8.5 Relative Entropy and Mutual Information. 8.6 Properties of Differential Entropy, Relative Entropy, and Mutual Information. Summary. Problems. Historical Notes. 9. Gaussian Channel. 9.1 Gaussian Channel: Definitions. 9.2 Converse to the Coding Theorem for Gaussian Channels. 9.3 Bandlimited Channels. 9.4 Parallel Gaussian Channels. 9.5 Channels with Colored Gaussian Noise. 9.6 Gaussian Channels with Feedback. Summary. Problems. Historical Notes. 10. Rate Distortion Theory. 10.1 Quantization. 10.2 Definitions. 10.3 Calculation of the Rate Distortion Function. 10.4 Converse to the Rate Distortion Theorem. 10.5 Achievability of the Rate Distortion Function. 10.6 Strongly Typical Sequences and Rate Distortion. 10.7 Characterization of the Rate Distortion Function. 10.8 Computation of Channel Capacity and the Rate Distortion Function. Summary. Problems. Historical Notes. 11. Information Theory and Statistics. 11.1 Method of Types. 11.2 Law of Large Numbers. 11.3 Universal Source Coding. 11.4 Large Deviation Theory. 11.5 Examples of Sanov's Theorem. 11.6 Conditional Limit Theorem. 11.7 Hypothesis Testing. 11.8 Chernoff-Stein Lemma. 11.9 Chernoff Information. 11.10 Fisher Information and the Cram-er-Rao Inequality. Summary. Problems. Historical Notes. 12. Maximum Entropy. 12.1 Maximum Entropy Distributions. 12.2 Examples. 12.3 Anomalous Maximum Entropy Problem. 12.4 Spectrum Estimation. 12.5 Entropy Rates of a Gaussian Process. 12.6 Burg's Maximum Entropy Theorem. Summary. Problems. Historical Notes. 13. Universal Source Coding. 13.1 Universal Codes and Channel Capacity. 13.2 Universal Coding for Binary Sequences. 13.3 Arithmetic Coding. 13.4 Lempel-Ziv Coding. 13.5 Optimality of Lempel-Ziv Algorithms. Compression. Summary. Problems. Historical Notes. 14. Kolmogorov Complexity. 14.1 Models of Computation. 14.2 Kolmogorov Complexity: Definitions and Examples. 14.3 Kolmogorov Complexity and Entropy. 14.4 Kolmogorov Complexity of Integers. 14.5 Algorithmically Random and Incompressible Sequences. 14.6 Universal Probability. 14.7 Kolmogorov complexity. 14.9 Universal Gambling. 14.10 Occam's Razor. 14.11 Kolmogorov Complexity and Universal Probability. 14.12 Kolmogorov Sufficient Statistic. 14.13 Minimum Description Length Principle. Summary. Problems. Historical Notes. 15. Network Information Theory. 15.1 Gaussian Multiple-User Channels. 15.2 Jointly Typical Sequences. 15.3 Multiple-Access Channel. 15.4 Encoding of Correlated Sources. 15.5 Duality Between Slepian-Wolf Encoding and Multiple-Access Channels. 15.6 Broadcast Channel. 15.7 Relay Channel. 15.8 Source Coding with Side Information. 15.9 Rate Distortion with Side Information. 15.10 General Multiterminal Networks. Summary. Problems. Historical Notes. 16. Information Theory and Portfolio Theory. 16.1 The Stock Market: Some Definitions. 16.2 Kuhn-Tucker Characterization of the Log-Optimal Portfolio. 16.3 Asymptotic Optimality of the Log-Optimal Portfolio. 16.4 Side Information and the Growth Rate. 16.5 Investment in Stationary Markets. 16.6 Competitive Optimality of the Log-Optimal Portfolio. 16.7 Universal Portfolios. 16.8 Shannon-McMillan-Breiman Theorem (General AEP). Summary. Problems. Historical Notes. 17. Inequalities in Information Theory. 17.1 Basic Inequalities of Information Theory. 17.2 Differential Entropy. 17.3 Bounds on Entropy and Relative Entropy. 17.4 Inequalities for Types. 17.5 Combinatorial Bounds on Entropy. 17.6 Entropy Rates of Subsets. 17.7 Entropy and Fisher Information. 17.8 Entropy Power Inequality and Brunn-Minkowski Inequality. 17.9 Inequalities for Determinants. 17.10 Inequalities for Ratios of Determinants. Summary. Problems. Historical Notes. Bibliography. List of Symbols. Index.
45,034 citations
TL;DR: Using distributed antennas, this work develops and analyzes low-complexity cooperative diversity protocols that combat fading induced by multipath propagation in wireless networks and develops performance characterizations in terms of outage events and associated outage probabilities, which measure robustness of the transmissions to fading.
Abstract: We develop and analyze low-complexity cooperative diversity protocols that combat fading induced by multipath propagation in wireless networks. The underlying techniques exploit space diversity available through cooperating terminals' relaying signals for one another. We outline several strategies employed by the cooperating radios, including fixed relaying schemes such as amplify-and-forward and decode-and-forward, selection relaying schemes that adapt based upon channel measurements between the cooperating terminals, and incremental relaying schemes that adapt based upon limited feedback from the destination terminal. We develop performance characterizations in terms of outage events and associated outage probabilities, which measure robustness of the transmissions to fading, focusing on the high signal-to-noise ratio (SNR) regime. Except for fixed decode-and-forward, all of our cooperative diversity protocols are efficient in the sense that they achieve full diversity (i.e., second-order diversity in the case of two terminals), and, moreover, are close to optimum (within 1.5 dB) in certain regimes. Thus, using distributed antennas, we can provide the powerful benefits of space diversity without need for physical arrays, though at a loss of spectral efficiency due to half-duplex operation and possibly at the cost of additional receive hardware. Applicable to any wireless setting, including cellular or ad hoc networks-wherever space constraints preclude the use of physical arrays-the performance characterizations reveal that large power or energy savings result from the use of these protocols.
12,761 citations
01 Nov 1999
TL;DR: In this paper, the authors investigate the use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading, and derive formulas for the capacities and error exponents of such channels, and describe computational procedures to evaluate such formulas.
Abstract: We investigate the use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading. We derive formulas for the capacities and error exponents of such channels, and describe computational procedures to evaluate such formulas. We show that the potential gains of such multi-antenna systems over single-antenna systems is rather large under independenceassumptions for the fades and noises at different receiving antennas.
12,542 citations
Book•
01 Nov 19968,608 citations
"A Closed-Form Power Allocation and ..." refers background or methods in this paper
...Proof: From [45], [46], and have generalized singular values which are arranged in increasing order: zero values, ordinary values from the diagonal entries of , and infinite values....
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...From Table II, it is worth emphasizing that increasing the number of antennas as well as the number of alternations and iterations makes the iterative approaches to the AF MIMO TWRC impractical due to the burden of complexity....
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...From this observation, the data streams indexed by at can be divided into three cases and their indices sets are defined by , and in Table I....
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...From Theorem 1, we have and ....
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...From (59), implies that and ....
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TL;DR: In this article, the capacity of the Gaussian relay channel was investigated, and a lower bound of the capacity was established for the general relay channel, where the dependence of the received symbols upon the inputs is given by p(y,y) to both x and y. In particular, the authors proved that if y is a degraded form of y, then C \: = \: \max \!p(x,y,x,2})} \min \,{I(X,y), I(X,Y,Y,X,Y
Abstract: A relay channel consists of an input x_{l} , a relay output y_{1} , a channel output y , and a relay sender x_{2} (whose transmission is allowed to depend on the past symbols y_{1} . The dependence of the received symbols upon the inputs is given by p(y,y_{1}|x_{1},x_{2}) . The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)If y is a degraded form of y_{1} , then C \: = \: \max \!_{p(x_{1},x_{2})} \min \,{I(X_{1},X_{2};Y), I(X_{1}; Y_{1}|X_{2})} . 2)If y_{1} is a degraded form of y , then C \: = \: \max \!_{p(x_{1})} \max_{x_{2}} I(X_{1};Y|x_{2}) . 3)If p(y,y_{1}|x_{1},x_{2}) is an arbitrary relay channel with feedback from (y,y_{1}) to both x_{1} \and x_{2} , then C\: = \: \max_{p(x_{1},x_{2})} \min \,{I(X_{1},X_{2};Y),I \,(X_{1};Y,Y_{1}|X_{2})} . 4)For a general relay channel, C \: \leq \: \max_{p(x_{1},x_{2})} \min \,{I \,(X_{1}, X_{2};Y),I(X_{1};Y,Y_{1}|X_{2}) . Superposition block Markov encoding is used to show achievability of C , and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.
4,311 citations