Interference Alignment and Degrees of Freedom of the $K$ -User Interference Channel
TL;DR: For the fully connected K user wireless interference channel where the channel coefficients are time-varying and are drawn from a continuous distribution, the sum capacity is characterized as C(SNR)=K/2log (SNR)+o(log( SNR), which almost surely has K/2 degrees of freedom.
Abstract: For the fully connected K user wireless interference channel where the channel coefficients are time-varying and are drawn from a continuous distribution, the sum capacity is characterized as C(SNR)=K/2log(SNR)+o(log(SNR)) . Thus, the K user time-varying interference channel almost surely has K/2 degrees of freedom. Achievability is based on the idea of interference alignment. Examples are also provided of fully connected K user interference channels with constant (not time-varying) coefficients where the capacity is exactly achieved by interference alignment at all SNR values.
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TL;DR: This paper discusses all of these topics, identifying key challenges for future research and preliminary 5G standardization activities, while providing a comprehensive overview of the current literature, and in particular of the papers appearing in this special issue.
Abstract: What will 5G be? What it will not be is an incremental advance on 4G. The previous four generations of cellular technology have each been a major paradigm shift that has broken backward compatibility. Indeed, 5G will need to be a paradigm shift that includes very high carrier frequencies with massive bandwidths, extreme base station and device densities, and unprecedented numbers of antennas. However, unlike the previous four generations, it will also be highly integrative: tying any new 5G air interface and spectrum together with LTE and WiFi to provide universal high-rate coverage and a seamless user experience. To support this, the core network will also have to reach unprecedented levels of flexibility and intelligence, spectrum regulation will need to be rethought and improved, and energy and cost efficiencies will become even more critical considerations. This paper discusses all of these topics, identifying key challenges for future research and preliminary 5G standardization activities, while providing a comprehensive overview of the current literature, and in particular of the papers appearing in this special issue.
7,139 citations
TL;DR: This paper studies a multiple-input multiple-output (MIMO) wireless broadcast system consisting of three nodes, where one receiver harvests energy and another receiver decodes information separately from the signals sent by a common transmitter, and all the transmitter and receivers may be equipped with multiple antennas.
Abstract: Wireless power transfer (WPT) is a promising new solution to provide convenient and perpetual energy supplies to wireless networks. In practice, WPT is implementable by various technologies such as inductive coupling, magnetic resonate coupling, and electromagnetic (EM) radiation, for short-/mid-/long-range applications, respectively. In this paper, we consider the EM or radio signal enabled WPT in particular. Since radio signals can carry energy as well as information at the same time, a unified study on simultaneous wireless information and power transfer (SWIPT) is pursued. Specifically, this paper studies a multiple-input multiple-output (MIMO) wireless broadcast system consisting of three nodes, where one receiver harvests energy and another receiver decodes information separately from the signals sent by a common transmitter, and all the transmitter and receivers may be equipped with multiple antennas. Two scenarios are examined, in which the information receiver and energy receiver are separated and see different MIMO channels from the transmitter, or co-located and see the identical MIMO channel from the transmitter. For the case of separated receivers, we derive the optimal transmission strategy to achieve different tradeoffs for maximal information rate versus energy transfer, which are characterized by the boundary of a so-called rate-energy (R-E) region. For the case of co-located receivers, we show an outer bound for the achievable R-E region due to the potential limitation that practical energy harvesting receivers are not yet able to decode information directly. Under this constraint, we investigate two practical designs for the co-located receiver case, namely time switching and power splitting, and characterize their achievable R-E regions in comparison to the outer bound.
2,595 citations
24 Apr 2009
TL;DR: This information-theoretic survey provides guidelines for the spectral efficiency gains possible through cognitive radios, as well as practical design ideas to mitigate the coexistence challenges in today's crowded spectrum.
Abstract: Cognitive radios hold tremendous promise for increasing spectral efficiency in wireless systems. This paper surveys the fundamental capacity limits and associated transmission techniques for different wireless network design paradigms based on this promising technology. These paradigms are unified by the definition of a cognitive radio as an intelligent wireless communication device that exploits side information about its environment to improve spectrum utilization. This side information typically comprises knowledge about the activity, channels, codebooks, and/or messages of other nodes with which the cognitive node shares the spectrum. Based on the nature of the available side information as well as a priori rules about spectrum usage, cognitive radio systems seek to underlay, overlay, or interweave the cognitive radios' signals with the transmissions of noncognitive nodes. We provide a comprehensive summary of the known capacity characterizations in terms of upper and lower bounds for each of these three approaches. The increase in system degrees of freedom obtained through cognitive radios is also illuminated. This information-theoretic survey provides guidelines for the spectral efficiency gains possible through cognitive radios, as well as practical design ideas to mitigate the coexistence challenges in today's crowded spectrum.
2,516 citations
Cites background from "Interference Alignment and Degrees ..."
...Unfortunately, the regulatory bodies governing spectrum allocation have not shown much appetite for change since their inception in the early 1900s....
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Book•
16 Jan 2012TL;DR: In this article, a comprehensive treatment of network information theory and its applications is provided, which provides the first unified coverage of both classical and recent results, including successive cancellation and superposition coding, MIMO wireless communication, network coding and cooperative relaying.
Abstract: This comprehensive treatment of network information theory and its applications provides the first unified coverage of both classical and recent results. With an approach that balances the introduction of new models and new coding techniques, readers are guided through Shannon's point-to-point information theory, single-hop networks, multihop networks, and extensions to distributed computing, secrecy, wireless communication, and networking. Elementary mathematical tools and techniques are used throughout, requiring only basic knowledge of probability, whilst unified proofs of coding theorems are based on a few simple lemmas, making the text accessible to newcomers. Key topics covered include successive cancellation and superposition coding, MIMO wireless communication, network coding, and cooperative relaying. Also covered are feedback and interactive communication, capacity approximations and scaling laws, and asynchronous and random access channels. This book is ideal for use in the classroom, for self-study, and as a reference for researchers and engineers in industry and academia.
2,442 citations
TL;DR: An overview of the theory and currently known techniques for multi-cell MIMO (multiple input multiple output) cooperation in wireless networks is presented and a few promising and quite fundamental research avenues are also suggested.
Abstract: This paper presents an overview of the theory and currently known techniques for multi-cell MIMO (multiple input multiple output) cooperation in wireless networks. In dense networks where interference emerges as the key capacity-limiting factor, multi-cell cooperation can dramatically improve the system performance. Remarkably, such techniques literally exploit inter-cell interference by allowing the user data to be jointly processed by several interfering base stations, thus mimicking the benefits of a large virtual MIMO array. Multi-cell MIMO cooperation concepts are examined from different perspectives, including an examination of the fundamental information-theoretic limits, a review of the coding and signal processing algorithmic developments, and, going beyond that, consideration of very practical issues related to scalability and system-level integration. A few promising and quite fundamental research avenues are also suggested.
1,911 citations
Cites background from "Interference Alignment and Degrees ..."
...ference alignment, has been shown to achieve significantly improved multiplexing gain for the MIMO interference network, where both the transmitters and the receivers are equipped with multiple antennas [72]....
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..., [72])....
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References
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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
01 Jun 2002
TL;DR: This work shows that true beamforming gains can be achieved when there are sufficient users, even though very limited channel feedback is needed, and proposes the use of multiple transmit antennas to induce large and fast channel fluctuations so that multiuser diversity can still be exploited.
Abstract: Multiuser diversity is a form of diversity inherent in a wireless network, provided by independent time-varying channels across the different users. The diversity benefit is exploited by tracking the channel fluctuations of the users and scheduling transmissions to users when their instantaneous channel quality is near the peak. The diversity gain increases with the dynamic range of the fluctuations and is thus limited in environments with little scattering and/or slow fading. In such environments, we propose the use of multiple transmit antennas to induce large and fast channel fluctuations so that multiuser diversity can still be exploited. The scheme can be interpreted as opportunistic beamforming and we show that true beamforming gains can be achieved when there are sufficient users, even though very limited channel feedback is needed. Furthermore, in a cellular system, the scheme plays an additional role of opportunistic nulling of the interference created on users of adjacent cells. We discuss the design implications of implementing. this scheme in a complete wireless system.
3,041 citations
"Interference Alignment and Degrees ..." refers background in this paper
...beamforming [ 18 ]. While the pre-log term is the principal determinant of capacity as SNR approaches infinity, terms are quite significant at SNR values of practical interest....
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TL;DR: A new achievable rate region for the general interference channel which extends previous results is presented and evaluated and the capacity of a class of Gaussian interference channels is established.
Abstract: A new achievable rate region for the general interference channel which extends previous results is presented and evaluated. The technique used is a generalization of superposition coding to the multivariable case. A detailed computation for the Gaussian channel case clarifies to what extent the new region improves previous ones. The capacity of a class of Gaussian interference channels is also established.
1,872 citations
"Interference Alignment and Degrees ..." refers background in this paper
...Recently, a special case of the Han–Kobayashi scheme [3] is shown in [10] to achieve the capacity of the two-user interference channel within one bit....
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...A combination of Han–Kobayashi [3] type achievable schemes and structured codes is a promising avenue in the quest for the capacity of wireless networks....
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...926344 While less common in practice due to the complexity of multi-user detection, this approach is supported by the capacity results on the “very strong interference” [1], and “strong interference” [3], [4] scenarios in the context...
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TL;DR: The capacity of the two-user Gaussian interference channel has been open for 30 years and the best known achievable region is due to Han and Kobayashi as mentioned in this paper, but its characterization is very complicated.
Abstract: The capacity of the two-user Gaussian interference channel has been open for 30 years. The understanding on this problem has been limited. The best known achievable region is due to Han and Kobayashi but its characterization is very complicated. It is also not known how tight the existing outer bounds are. In this work, we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges, and by deriving new outer bounds, we show that a very simple and explicit Han-Kobayashi type scheme can achieve to within a single bit per second per hertz (bit/s/Hz) of the capacity for all values of the channel parameters. We also show that the scheme is asymptotically optimal at certain high signal-to-noise ratio (SNR) regimes. Using our results, we provide a natural generalization of the point-to-point classical notion of degrees of freedom to interference-limited scenarios.
1,473 citations
TL;DR: The sum capacity of the vector Gaussian broadcast channel is characterized by showing that the existing inner bound of Marton and the existing upper bound of Sato are tight for this channel.
Abstract: We characterize the sum capacity of the vector Gaussian broadcast channel by showing that the existing inner bound of Marton and the existing upper bound of Sato are tight for this channel. We exploit an intimate four-way connection between the vector broadcast channel, the corresponding point-to-point channel (where the receivers can cooperate), the multiple-access channel (MAC) (where the role of transmitters and receivers are reversed), and the corresponding point-to-point channel (where the transmitters can cooperate).
1,369 citations
"Interference Alignment and Degrees ..." refers background or methods in this paper
...],i,j ∈ {1,2,3} have a full rank of M almost surely, the above equations can be equivalently represented as span(V[1]) = span(EV[1]) (50) V[2] = FV[1] (51) V[3] = GV[1] (52) where E = (H[31])−1H[32](H[12])−1H[13](H[23])−1H[21] F = (H[32])−1H[31] G = (H[23])−1H[21] Let e 1,e 2,...eM be the M eigenvectors of E. Then we set V 1 to be V[1] = [ e 1 ... (M/2)] Then V[2] and V[3] are found using equations (5...
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...of the interference has to be less than or equal to M/2. The following three interference alignment equations ensure that the dimension of the interference is equal to M/2 at all the receivers. span(H[12]V[2]) = span(H[13]V[3]) (47) H[21]V[1] = H[23]V[3] (48) H[31]V[1] = H[32]V[2] (49) where span(A) represents the vector space spanned by the column vectors of matrix A We now wish to choose V[i],i = 1,...
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...ced by equations (40), the interference vectors from transmitters 3,4...K are perfectly aligned with the interference from transmitter 2 and therefore, all interference arrives along the nN vectors H¯[12]V¯[2]. In order to prove that there are (n+1)N interference free dimensions it suffices to show that the col umns of the square, Mn ×Mn dimensional matrix h H¯[11]V¯[1] H¯[12]V¯[2] i (46) are linearly ...
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...corresponds to the term αmk = 0,∀(m,k)), the linear independence condition boils down to the condition that 1 = 0 with non-zero probability - an obvious contradiction. Thus the matrix h H¯[11]V¯[1] H¯[12]V¯[2] i can be shown to be non-singular with probability1. Similarly, the desired signal can be chosen to be linearly independent of the interference at all other receivers almost surely. Thus ( ( n+1...
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...mensional received signal vector Y¯[1], the dimension of the interference should be not more than nN. This can be ensured by perfectly aligning the interference from transmitters 2,3...K as follows H¯[12]V¯[2] = H¯[13]V¯[3] = H¯[14]V¯[4] = ... = H¯[1K]V¯[K] (37) At the same time, receiver 2 zero-forces the interference from X¯[i],i 6= 2 . To extract nN interference-free dimensions from a Mn = (n+1)N +...
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