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## On Maximum-Likelihood Detection and the Search for the Closest Lattice Point (2003)

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Venue: | IEEE TRANS. INFORM. THEORY |

Citations: | 272 - 9 self |

### Citations

2919 | Capacity of multi-antenna gaussian channels
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Citation Context ...he work. 1) Linear-Dispersion Encoded Quadrature Amplitude Modulation (QAM) Over a Frequency-Flat MIMO Channel: Consider an -transmit, -receive antennas system with frequency-flat quasi-static fading =-=[9]-=-, [10]. The baseband complex received signal after matched filtering and symbol rate sampling is given by SNR (2) (3) (4) (5)s2390 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 200... |

2425 | On limits of wireless communications in a fading environment when using multiple antennas
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Citation Context ...rk. 1) Linear-Dispersion Encoded Quadrature Amplitude Modulation (QAM) Over a Frequency-Flat MIMO Channel: Consider an -transmit, -receive antennas system with frequency-flat quasi-static fading [9], =-=[10]-=-. The baseband complex received signal after matched filtering and symbol rate sampling is given by SNR (2) (3) (4) (5)s2390 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003 wher... |

2096 |
Information Theory and Reliable Communication
- Gallager
- 1968
(Show Context)
Citation Context ...of efficient and powerful decoding techniques able to perform close to ML decoding, without suffering the complexity of exact ML decoding, for coding rates not too close to the channel capacity [32], =-=[34]-=-. Next, we interpret Algorithm I as a chain of sequential decoders. To facilitate this interpretation, we make use of the stack sequential decoding algorithm, briefly summarized as follows. Consider a... |

1464 | Multiuser Detection
- Verdu
- 1998
(Show Context)
Citation Context ...se at the th OFDM subcarrier. 3) Multiple-Access Gaussian Waveform Channel (CDMA): The canonical baseband complex model for a -user synchronous code-division multiple access (CDMA) system is given by =-=[16] w-=-here the columns of represent the (discrete-time) users’ signature waveforms, is a diagonal matrix of amplitudes, and is a vector containing the modulation symbols transmitted by the users in the th... |

961 | Factoring polynomials with rational coefficients, Mathematische Annalen 261
- Lenstra, Lenstra, et al.
- 1982
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Citation Context ...st proposed two modifications of the preprocessing stage to further reduce the complexity of the Pohst enumeration for infinite lattices. a) Apply the Lenstra–Lenstra–Lovász (LLL) reduction algor=-=ithm [35]-=- on the upper triangular matrix . The LLL algorithm finds a unimodular 1 matrix such that and has reduced lattice vectors and almost orthogonal columns. b) Order the columns of according to their Eucl... |

422 | High-rate codes that are linear in space and time,”
- Hassibi, Hochwald
- 2002
(Show Context)
Citation Context ...1]. Assume that the transmitter maps blocks with independent and uniformly distributed entries (information symbols) onto transmit arrays of the form where : are the linear dispersion code generators =-=[11]��-=-�[13]. Then, the array is transmitted column by column in channel uses. Due to the linearity of the channel (5) and of the encoder (6), one can easily see that there exists a matrix such that the ML d... |

414 |
A universal lattice code decoder for fading channels,”
- Viterbo, Boutros
- 1999
(Show Context)
Citation Context ...fficiency in many useful scenarios (see [6] for a comprehensive review of related works). The Pohst enumeration strategy was first introduced in digital communications by Viterbo and Biglieri [4]. In =-=[5]-=-, Viterbo and Boutros applied it to the ML detection of multidimensional constellations transmitted over single antenna fading channels, and gave a flowchart of a specific implementation. More recentl... |

401 | Achieving near-capacity on a multiple-antenna channel - Hochwald, Brink |

356 | Improved methods for calculating vectors of short length in a lattice, including a complexity analysis - Fincke, Pohst |

333 | Closest point search in lattices
- Agrell, Eriksson, et al.
- 2002
(Show Context)
Citation Context ...l vector, (2) takes on the normalized form where the components of the noise have a common variance equal to . In this paper, we consider a class of algorithms, generally known as sphere decoders [2]�=-=��[6], [8], t-=-hat compute or approximate (4) with a polynomial expected complexity within a wide range of system parameters. Throughout the paper, the terms “decoding” and “detection” are used interchangeab... |

323 | Lattice basis reduction: improved practical algorithms and solving subset sum problems - Schnorr, Euchner - 1994 |

256 | Simplified Processing for High Spectral Efficiency Wireless Communication Employing Multi-Element Arrays,
- Foschini, Golden, et al.
- 1999
(Show Context)
Citation Context ...re never occurs. It is also easy to see that the first point found with corresponds to the Babai point [6]. In the communication theory parlance this point is referred to as the nulling and canceling =-=[25]-=- or zero-forcing decision-feedback equalization (ZF-DFE) point. Explicitly, this is given by the back-substitution with integer quantization (slicing) (15) for . As we shall see in Section VI, the dra... |

234 | Lattice codes decoder for space-time codes
- Damen, Chkeif, et al.
- 2000
(Show Context)
Citation Context ...tor, (2) takes on the normalized form where the components of the noise have a common variance equal to . In this paper, we consider a class of algorithms, generally known as sphere decoders [2]–[6]=-=, [8], that c-=-ompute or approximate (4) with a polynomial expected complexity within a wide range of system parameters. Throughout the paper, the terms “decoding” and “detection” are used interchangeably to... |

188 | Signal space diversity: a power- and bandwidth-efficient diversity technique for the rayleigh fading channel
- Boutros, Viterbo
- 1998
(Show Context)
Citation Context ...larly simple case corresponds to the transmission of uncoded QAM symbols. In this case, , , , and is given explicitly by SNR 2) Block Transmission Over Time-Selective or FrequencySelective Fading: In =-=[14]-=-, linearly block-coded transmission for time-selective fading was considered. The complex baseband channel is given by , where now , is the sequence of scalar fading coefficients, and is a suitable ro... |

143 | Universal space-time coding," - Gamal, Damen - 2003 |

140 | Delayed decision-feedback sequence estimation - Duel-Hallen, Heegard - 1989 |

118 | An ecient square-root algorithm for blast. Submitted to IEEE Transactions on Signal Processing. Available for download from http://mars.belllabs.com
- Hassibi
- 1999
(Show Context)
Citation Context ... the observation is given by (20) A very efficient greedy algorithm for computing the permutation that maximizes and, at the same time, providing the corresponding MMSE-DFE filter vectors is given in =-=[37]-=-. 2 Letting denote the MMSE-DFE filter bank for the optimal permutation and the proposed preprocessing and ordering approach now consists of the following elements. a) Computing , , and by using the m... |

82 | Low-complexity near-maximumlikelihood detection and precoding for MIMO systems using lattice reduction
- Windpassinger, Fischer
- 2003
(Show Context)
Citation Context ...nd the basic sphere decoder, known as Pohst enumeration (see details later). However, the same fact has been observed in more general cases by computer simulations in several recent works (e.g., [22]�=-=��[24]-=-). While the exact average complexity analysis of the basic sphere decoder for general linear-dispersion codes and, a fortiori, for improved sphere decoding algorithms as those proposed in this work, ... |

80 |
On the computation of lattice vectors of minimal length, successive minima and reduced basis with applications
- Pohst
- 1981
(Show Context)
Citation Context ...ignal vector, (2) takes on the normalized form where the components of the noise have a common variance equal to . In this paper, we consider a class of algorithms, generally known as sphere decoders =-=[2]–[6], [8-=-], that compute or approximate (4) with a polynomial expected complexity within a wide range of system parameters. Throughout the paper, the terms “decoding” and “detection” are used interchan... |

69 |
On the expected complexity of sphere decoding,”
- Hassibi, Vikalo
- 2002
(Show Context)
Citation Context ...ransmitted point by the additive noise . Therefore, it can be expected that as the SNR increases, the average complexity decreases, for fixed and . This fact has been shown by theoretical analysis in =-=[21]-=- in the case of uncoded QAM over an frequency-flat MIMO channel with , having i.i.d. entries , and the basic sphere decoder, known as Pohst enumeration (see details later). However, the same fact has ... |

56 | Iterative Decoding for MIMO Channels Via Modified Sphere Decoding”, - Vikalo, Hassibi, et al. - 2004 |

51 | Soft-input softoutput lattice sphere decoder for linear channels,” in - Boutros, Gresset, et al. - 2003 |

50 | Iterative detection of MIMO transmission using a list-sequential (LISS) detector,” in
- Baro, Hagenauer, et al.
- 2003
(Show Context)
Citation Context ...soft output, in the form of approximated posterior marginal probabilities . The resulting soft-output detector forms the core of some iterative decoders based on Belief–Propagation (see, for example=-=, [27]–[-=-31]). The main contribution of this paper is a fresh look at the class of sphere-decoding algorithms. We start by reviewing the basic Pohst and Schnorr–Euchner enumeration strategies for infinite la... |

50 |
The intersymbol interference channel: Lower bounds on capacity and channel precoding loss
- Shamai, Laroia
- 1996
(Show Context)
Citation Context ...icant. It is also interesting to notice that Algorithm II with the MMSE-DFE preprocessing stage is akin to the standard approach of reduced state sequence estimation by delayed decision feedback (see =-=[38]��-=-�[40] and references therein). A Remark on the Complexity: The impact of the preprocessing approaches on the average complexity depends very much on how often preprocessing is performed. In fact, prep... |

31 |
Biglieri: “A universal lattice decoder,”
- Viterbo, E
- 1993
(Show Context)
Citation Context ...to its efficiency in many useful scenarios (see [6] for a comprehensive review of related works). The Pohst enumeration strategy was first introduced in digital communications by Viterbo and Biglieri =-=[4]-=-. In [5], Viterbo and Boutros applied it to the ML detection of multidimensional constellations transmitted over single antenna fading channels, and gave a flowchart of a specific implementation. More... |

30 | Generalized sphere decoder for asymmetrical space-time communication architecture.
- Damen
- 2000
(Show Context)
Citation Context ...so integer components or, equivalently, if it has integer components and a determinant equal to 6I. We note that this column ordering has not been reported in previous papers on the subject [5], [8], =-=[36]-=-, [22], [6]. We have found that this column ordering also decreases the average complexity of Algorithm II. 2) V-BLAST ZF-DFE Preprocessing and Ordering: As a second preprocessing and ordering approac... |

26 | Linear Threaded Algebraic Space-Time Constellations
- Damen, Gamal, et al.
- 2003
(Show Context)
Citation Context ...ssume that the transmitter maps blocks with independent and uniformly distributed entries (information symbols) onto transmit arrays of the form where : are the linear dispersion code generators [11]�=-=��[13]-=-. Then, the array is transmitted column by column in channel uses. Due to the linearity of the channel (5) and of the encoder (6), one can easily see that there exists a matrix such that the ML detect... |

17 |
Design of low-density parity-check codes for multi-antenna modulation and detection. Submitted to
- Brink, Kramer, et al.
- 2002
(Show Context)
Citation Context ...output, in the form of approximated posterior marginal probabilities . The resulting soft-output detector forms the core of some iterative decoders based on Belief–Propagation (see, for example, [27=-=]–[31]).-=- The main contribution of this paper is a fresh look at the class of sphere-decoding algorithms. We start by reviewing the basic Pohst and Schnorr–Euchner enumeration strategies for infinite lattice... |

15 |
Further results on the sphere decoder
- Damen, A-Meraim, et al.
(Show Context)
Citation Context ...s , and the basic sphere decoder, known as Pohst enumeration (see details later). However, the same fact has been observed in more general cases by computer simulations in several recent works (e.g., =-=[22]��-=-�[24]). While the exact average complexity analysis of the basic sphere decoder for general linear-dispersion codes and, a fortiori, for improved sphere decoding algorithms as those proposed in this w... |

13 |
The shortest vector problem is NP-hard for randomized reductions
- Ajtai
- 1998
(Show Context)
Citation Context ...table translation and scaling, the ML joint detection of the user symbols can be put in the form (4) [17], [18]. It is well known that the minimization (4) for arbitrary and is NP-hard (see [3], [6], =-=[19]-=-, [20]). Nevertheless, it has been shown recently that in many relevant cases, for a certain range of system parameters such as SNR, and , the average complexity of some algorithms implementing or app... |

10 | On the Complexity of ML Lattice Decoders for Decoding Linear Full-Rate Space-Time Codes - Rekaya, Belfiore - 2003 |

9 |
On space-time coding for quasi-static multipleantenna channels
- Caire, Colavolpe
- 2001
(Show Context)
Citation Context .... It is also interesting to notice that Algorithm II with the MMSE-DFE preprocessing stage is akin to the standard approach of reduced state sequence estimation by delayed decision feedback (see [38]�=-=��[40]-=- and references therein). A Remark on the Complexity: The impact of the preprocessing approaches on the average complexity depends very much on how often preprocessing is performed. In fact, preproces... |

9 |
Coset codes-Part I: Introduction and geometrical classification
- Jr
- 1988
(Show Context)
Citation Context ...per-rectangles carved from the integer cubic lattice). Consider a lattice with generator matrix , i.e., . A lattice code (or constellation) is the set of lattice points , where is some shaping region =-=[41]-=-, [42]. Typically, the shaping region is chosen to be the -dimensional sphere of given squared radius . In order to handle this case, we first restrict to belong to the hyper-rectangular region The di... |

8 |
Euclidean space lattice decoding for joint detection in CDMA systems
- Brutel, Boutros
- 1999
(Show Context)
Citation Context ...CDMA systems, the user symbols take on values in the (6) (7) (8) 4-QAM signal set. Again, after suitable translation and scaling, the ML joint detection of the user symbols can be put in the form (4) =-=[17]-=-, [18]. It is well known that the minimization (4) for arbitrary and is NP-hard (see [3], [6], [19], [20]). Nevertheless, it has been shown recently that in many relevant cases, for a certain range of... |

8 |
Codes-Part II: Binary Lattices and Related Codes
- “Coset
- 1988
(Show Context)
Citation Context ...ctangles carved from the integer cubic lattice). Consider a lattice with generator matrix , i.e., . A lattice code (or constellation) is the set of lattice points , where is some shaping region [41], =-=[42]-=-. Typically, the shaping region is chosen to be the -dimensional sphere of given squared radius . In order to handle this case, we first restrict to belong to the hyper-rectangular region The differen... |

7 | Iterative QR Detection for BLAST
- Damen, Meraim, et al.
- 2000
(Show Context)
Citation Context ...because of their applications to multiple-antenna systems [8]. This interest is due to the significant performance gain achieved by sphere decoders compared to other suboptimal detection schemes [24]�=-=��[26]-=-, and to their average polynomial complexity (experimentally demonstrated) at medium-to-high SNR. Moreover, the ML detector can be easily augmented to provide symbol-by-symbol soft output, in the form... |

5 | A Safavi, On CDMA with space-time codes over multi-path fading channels
- Damen, Abed-Meraim
- 2003
(Show Context)
Citation Context ...ystems, the user symbols take on values in the (6) (7) (8) 4-QAM signal set. Again, after suitable translation and scaling, the ML joint detection of the user symbols can be put in the form (4) [17], =-=[18]-=-. It is well known that the minimization (4) for arbitrary and is NP-hard (see [3], [6], [19], [20]). Nevertheless, it has been shown recently that in many relevant cases, for a certain range of syste... |

1 |
Linear precoded OFDM transmissions with MMSE equalization: Facts and results
- Debbah, Loubaton, et al.
- 2002
(Show Context)
Citation Context ...ion matrix. Similarly, for frequency-selective slow fading we can work in the frequency domain to obtain linearly precoded orthogonal frequency-division multiplexing (OFDM) schemes (see, for example, =-=[15]-=- and references therein) in the same form as above, where and denotes the fading channel frequency response at the th OFDM subcarrier. 3) Multiple-Access Gaussian Waveform Channel (CDMA): The canonica... |

1 |
The shortest vector in a lattice is hard to approximate to within some constant
- Marcciancio
- 1998
(Show Context)
Citation Context ...translation and scaling, the ML joint detection of the user symbols can be put in the form (4) [17], [18]. It is well known that the minimization (4) for arbitrary and is NP-hard (see [3], [6], [19], =-=[20]-=-). Nevertheless, it has been shown recently that in many relevant cases, for a certain range of system parameters such as SNR, and , the average complexity of some algorithms implementing or approxima... |