Sphere Decoding Complexity Exponent for Decoding Full-Rate Codes Over the Quasi-Static MIMO Channel
TL;DR: The recently introduced threaded cyclic-division-algebra-based codes are shown to take a particularly concise form as a non-monotonic function of the multiplexing gain, which describes the minimum known complexity of any decoder that can provably achieve a gap to maximum likelihood performance that vanishes in the high SNR limit.
Abstract: In the setting of quasi-static multiple-input multiple-output channels, we consider the high signal-to-noise ratio (SNR) asymptotic complexity required by the sphere decoding (SD) algorithm for decoding a large class of full-rate linear space-time codes. With SD complexity having random fluctuations induced by the random channel, noise, and codeword realizations, the introduced SD complexity exponent manages to concisely describe the computational reserves required by the SD algorithm to achieve arbitrarily close to optimal decoding performance. Bounds and exact expressions for the SD complexity exponent are obtained for the decoding of large families of codes with arbitrary performance characteristics. For the particular example of decoding the recently introduced threaded cyclic-division-algebra-based codes—the only currently known explicit designs that are uniformly optimal with respect to the diversity multiplexing tradeoff—the SD complexity exponent is shown to take a particularly concise form as a non-monotonic function of the multiplexing gain. To date, the SD complexity exponent also describes the minimum known complexity of any decoder that can provably achieve a gap to maximum likelihood performance that vanishes in the high SNR limit.
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TL;DR: In this article, the authors provide a recital on the historic heritages and novel challenges facing massive/large-scale multiple-input multiple-output (LS-MIMO) systems from a detection perspective.
Abstract: The emerging massive/large-scale multiple-input multiple-output (LS-MIMO) systems that rely on very large antenna arrays have become a hot topic of wireless communications. Compared to multi-antenna aided systems being built at the time of this writing, such as the long-term evolution (LTE) based fourth generation (4G) mobile communication system which allows for up to eight antenna elements at the base station (BS), the LS-MIMO system entails an unprecedented number of antennas, say 100 or more, at the BS. The huge leap in the number of BS antennas opens the door to a new research field in communication theory, propagation and electronics, where random matrix theory begins to play a dominant role. Interestingly, LS-MIMOs also constitute a perfect example of one of the key philosophical principles of the Hegelian Dialectics, namely, that “quantitative change leads to qualitative change.” In this treatise, we provide a recital on the historic heritages and novel challenges facing LS-MIMOs from a detection perspective. First, we highlight the fundamentals of MIMO detection, including the nature of co-channel interference (CCI), the generality of the MIMO detection problem, the received signal models of both linear memoryless MIMO channels and dispersive MIMO channels exhibiting memory, as well as the complex-valued versus real-valued MIMO system models. Then, an extensive review of the representative MIMO detection methods conceived during the past 50 years (1965–2015) is presented, and relevant insights as well as lessons are inferred for the sake of designing complexity-scalable MIMO detection algorithms that are potentially applicable to LS-MIMO systems. Furthermore, we divide the LS-MIMO systems into two types, and elaborate on the distinct detection strategies suitable for each of them. The type-I LS-MIMO corresponds to the case where the number of active users is much smaller than the number of BS antennas, which is currently the mainstream definition of LS-MIMO. The type-II LS-MIMO corresponds to the case where the number of active users is comparable to the number of BS antennas. Finally, we discuss the applicability of existing MIMO detection algorithms in LS-MIMO systems, and review some of the recent advances in LS-MIMO detection.
626 citations
Posted Content•
TL;DR: Performance of reliable communication over a coherent slow-fading multiple-input multiple-output (MIMO) channel at high signal-to-noise ratio (SNR) is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains.
Abstract: Performance of reliable communication over a coherent slow fading channel at high SNR is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains. We study the problem of designing codes that optimally tradeoff the diversity and multiplexing gains. Our main contribution is a precise characterization of codes that are universally tradeoff-optimal, i.e., they optimally tradeoff the diversity and multiplexing gains for every statistical characterization of the fading channel. We denote this characterization as one of approximate universality where the approximation is in the connection between error probability and outage capacity with diversity and multiplexing gains, respectively. The characterization of approximate universality is then used to construct new coding schemes as well as to show optimality of several schemes proposed in the space-time coding literature.
237 citations
TL;DR: A new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer-valued entries that achieves the optimal diversity-multiplexing tradeoff for the standard multiple-input multiple-output (MIMO) channel with no coding across transmit antennas.
Abstract: Linear receivers are often used to reduce the implementation complexity of multiple-antenna systems. In a traditional linear receiver architecture, the receive antennas are used to separate out the codewords sent by each transmit antenna, which can then be decoded individually. Although easy to implement, this approach can be highly suboptimal when the channel matrix is near singular. This paper develops a new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer-valued entries. Rather than attempting to recover transmitted codewords directly, the decoder recovers integer combinations of the codewords according to the entries of the effective channel matrix. The codewords are all generated using the same linear code, which guarantees that these integer combinations are themselves codewords. Provided that the effective channel is full rank, these integer combinations can then be digitally solved for the original codewords. This paper focuses on the special case where there is no coding across transmit antennas and no channel state information at the transmitter(s), which corresponds either to a multiuser uplink scenario or to single-user V-BLAST encoding. In this setting, the proposed integer-forcing linear receiver significantly outperforms conventional linear architectures such as the zero forcing and linear minimum mean-squared error receiver. In the high signal-to-noise ratio regime, the proposed receiver attains the optimal diversity-multiplexing tradeoff for the standard multiple-input multiple-output (MIMO) channel with no coding across transmit antennas. It is further shown that in an extended MIMO model with interference, the integer-forcing linear receiver achieves the optimal generalized degrees of freedom.
177 citations
13 Jun 2010
TL;DR: A new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer-valued entries that achieves the optimal diversity-multiplexing tradeoff for the standard multiple-input multiple-output (MIMO) channel with no coding across transmit antennas.
Abstract: Linear receivers are often used to reduce the implementation complexity of multiple antenna systems. In a traditional linear receiver architecture, the receive antennas are used to separate out the codewords sent by each transmit antenna, which can then be decoded individually. Although easy to implement, this approach can be highly sub-optimal when the channel matrix is near singular. In this paper, we develop a new linear architecture that uses the receive antennas to create an effective channel matrix with integer-valued entries. Instead of attempting to recover a transmitted codeword directly, each decoder recovers a different integer combination of the codewords according to the effective channel matrix. If the effective channel is full rank, these linear equations can be digitally solved for the original codewords. By allowing the receiver to equalize the channel to any matrix with integer entries, this scheme can outperform traditional linear architectures such as decorrelators and MMSE receivers while maintaining a similar complexity. Furthermore, in the case where each transmit antenna encodes an independent data stream, the proposed receiver attains the optimal diversity multiplexing tradeoff.
170 citations
Cites background from "Sphere Decoding Complexity Exponent..."
...Rather than naively checking all possible codewords, the sphere decoder only examines codewords that lie within a ball around the received vector....
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TL;DR: The performance achieved by the proposed DL-based sphere decoding algorithm is very close to the optimal maximum likelihood decoding (MLD) over a wide range of signal-to-noise ratios (SNRs), while the computational complexity, compared to existing sphere decoding variants, is significantly reduced.
Abstract: In this paper, a deep learning (DL)-based sphere decoding algorithm is proposed, where the radius of the decoding hypersphere is learned by a deep neural network (DNN). The performance achieved by the proposed algorithm is very close to the optimal maximum likelihood decoding (MLD) over a wide range of signal-to-noise ratios (SNRs), while the computational complexity, compared to existing sphere decoding variants, is significantly reduced. This improvement is attributed to the DNN’s ability of intelligently learning the radius of the hypersphere used in decoding. The expected complexity of the proposed DL-based algorithm is analytically derived and compared with existing ones. It is shown that the number of lattice points inside the decoding hypersphere drastically reduces in the DL-based algorithm in both the average and worst-case senses. The effectiveness of the proposed algorithm is shown through the simulation for high-dimensional multiple-input multiple-output (MIMO) systems, using high-order modulations.
64 citations
Cites background from "Sphere Decoding Complexity Exponent..."
...THE problem of optimum maximum likelihood decoding (MLD) in spatial multiplexing multiple-input multipleoutput (MIMO) systems leads to an integer least-squares (LS) problem, which is equivalent to finding the closest lattice point to a given point [1]–[3]....
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TL;DR: A simple characterization of the optimal tradeoff curve is given and used to evaluate the performance of existing multiple antenna schemes for the richly scattered Rayleigh-fading channel.
Abstract: Multiple antennas can be used for increasing the amount of diversity or the number of degrees of freedom in wireless communication systems. We propose the point of view that both types of gains can be simultaneously obtained for a given multiple-antenna channel, but there is a fundamental tradeoff between how much of each any coding scheme can get. For the richly scattered Rayleigh-fading channel, we give a simple characterization of the optimal tradeoff curve and use it to evaluate the performance of existing multiple antenna schemes.
4,422 citations
"Sphere Decoding Complexity Exponent..." refers background in this paper
...Building on this, Theorem 4 establishes that, given any full rate design of arbitrary DMT performance, there is always at least one non-random SD column ordering [3], [4] for which c(r) = c̄(r), i.e., for which the exactc(r) can be explicitly calculated from the result of Theorem 2....
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...We let Z, R, and C, denote the set of integer, real and complex numbers respectively andFn andFm×n the set ofnvectors andm × n-matrices overF ∈ {Z,R,C}....
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