Optimal limited feedback technique for small size and low rate MIMO systems
03 Jul 2006-pp 677-682
TL;DR: The proposed antenna selection method performs better than unitary precoding schemes (even optimal precoding) for systems with small number of transmitter and receiver antennas, although precoding techniques exploit more bits of feedback information and computational complexity than the proposed method.
Abstract: In recent investigations, several methods for limited feedback multiple-input multiple-output (MIMO) systems have been proposed. Antenna selection at the transmitter and(or) receiver side is one of the approaches to minimize the average probability of error by using a limited bits of feedback information. In this paper, by using a novel approach, we calculate the optimal signal-to-noise ratio (SNR) for each received symbol for a general space-time block code. We propose an antenna selection method at the transmitter to maximize the average SNR for each symbol. Since we propose the optimal selection, our antenna selection method outperforms antenna selection methods available in the literature for space-time codes with rate⪕ 1 symbol per channel use, derived either based on SNR or capacity maximization. The proposed selection performs better than unitary precoding schemes (even optimal precoding) for systems with small number of transmitter and receiver antennas (particularly for mobile systems), although precoding techniques exploit more bits of feedback information and computational complexity than the proposed method.
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TL;DR: A codebook design method using the genetic algorithm is proposed, which reduces the design complexity and achieves large minimum-distance codebooks and the approximate BER of limited feedback beamforming is comparatively tight even for small size codebooks.
Abstract: Multiple-input multiple-output (MIMO) systems achieve significant diversity and array gains by using transmit beamforming. When complete channel state information (CSI) is not available at the transmitter, a common set of beamformers (codebook) is used by both the transmitter and the receiver. For each channel realization, the best beamformer is selected at the receiver and its index is sent back to the transmitter via a limited feedback channel. In this paper, a codebook design method using the genetic algorithm is proposed, which reduces the design complexity and achieves large minimum-distance codebooks. Exploiting the specific structure of these beamformers, an order and bound algorithm is proposed to reduce the beamformer selection complexity at the receiver side. The exact bit error rate (BER) of the optimal beamforming in finite-series expression is used to facilitate the BER analysis of limited feedback beamforming. By employing a geometrical approach, an approximate BER of limited feedback beamforming is derived when the codebook size is relatively large (high resolution analysis). The simulation results show that the approximate BER is comparatively tight even for small size codebooks.
12 citations
Cites methods from "Optimal limited feedback technique ..."
...CSI at the transmitter may be exploited in two ways: antenna subset selection [12]– [14] and precoding....
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24 Jun 2007
TL;DR: This paper finds the optimal precoder for general space-time codes with rate⩽1 symbol per channel use by maximizing the signal-to-noise ratio (SNR) per received symbol, and proposes a general structure matrix rather than a unitary one for precoding.
Abstract: Despite primary space-time coding where the channel state information (CSI) is available at the receiver only, the capacity and performance of multiple-input multiple-output (MIMO) systems can be increased significantly when a complete or partial CSI is available at the transmitter. Recently, limited feedback methods including antenna subset selection and unitary preceding have been proposed for orthogonal space-time codes where a partial knowledge of the channel is available at the transmitter via an error-free, zero-delay feedback channel. In this paper, we propose a general structure matrix rather than a unitary one for precoding. By maximizing the signal-to-noise ratio (SNR) per received symbol, we find the optimal precoder for general space-time codes with rate⩽1 symbol per channel use. The performance of the optimal scheme is analytically evaluated. Next, we extend the result for limited feedback systems. Simulation results show that the proposed precoder outperforms the previous work.
11 citations
Cites background from "Optimal limited feedback technique ..."
...It can be shown [7] that for any rate 1 STBC, the optimum performance is obtained when the code is orthogonal, i....
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01 Jan 2007
TL;DR: By maximizing the signal-to-noise ratio (SNR) per received symbol, the optimal precoder for general space-time codes with rate1 symbol per channel use is found by relaxing the precoder matrix from being unitary matrix to a general structure matrix.
Abstract: Despite primary space-time coding where the chan- nel state information (CSI) is available at the receiver only, the capacity and performance of multiple-input multiple-output (MIMO) systems can be increased significantly when a complete or partial CSI is available at the transmitter. Recently, limited feedback methods including antenna subset selection and unitary precoding have been proposed for orthogonal space-time codes where a partial knowledge of the channel is available at the transmitter via an error-free, zero-delay feedback channel. In this paper, we propose a general structure matrix rather than a unitary one for precoding. By maximizing the signal-to-noise ratio (SNR) per received symbol, we find the optimal precoder for general space-time codes with rate1 symbol per channel use. The performance of the optimal scheme is analytically evaluated. Next, we extend the result for limited feedback systems. Simulation results show that the proposed precoder outperforms the previous work. I. INTRODUCTION Multiple-input multiple-output (MIMO) wireless channels, created by deploying antenna arrays at both the transmitter and receiver, promise high capacity and high-quality wireless communication links (1), (2). To fully exploit the benefits of MIMO channels, space-time modulation and receiver algo- rithms are required to provide a sensible performance and complexity tradeoff. Space-time block codes (STBC) with rate1 symbol per channel use are of interest when the number of receive antennas may be one or more, particularly in the downlink of mobile systems. Orthogonal STBCs (OSTBCs) (3) are a class of STBCs that guarantee full diversity and simple decoding. Primary schemes proposed for exploiting multiple antennas at the transmitter and/or receiver commonly assume that by applying pilots or training sequences, the receiver can estimate the channel gains accurately, but this information is not avail- able at the transmitter. However, in communication systems that experience a slow fading environment, complete or partial knowledge of the channel may be available at the transmitter. Channel state information (CSI) at the transmitter may be exploited in two ways: antenna subset selection and precoding. The optimum precoder matrix can be obtained based on the eigen structure of the channel matrix (4). Due to the bandwidth limits on feedback channel, however, full CSI is not always available at the transmitter. Therefore, precoding techniques using limited feedback are of interest (5). In (5), the authors propose a codebook of unitary precoders derived from Grassmannian subspace packing for limited feed- back systems. The codebook is known to both the transmitter and receiver and for each channel realization, only the index of the appropriate matrix (precoder) is sent back to the trans- mitter. The precoder structure is originally proposed in (6) for differential unitary space-time modulation (DUSTM) which consists of a diagonal matrix and a rectangular sub-matrix of the Discrete Fourier Transform (DFT) matrix. The diagonal terms are some points on the unit circle in the complex plain where their angles are defined by some integers that should be optimized. In this paper, when CSI is available at the transmitter, we relax the precoder matrix from being unitary matrix to a general structure matrix. We extend the precoder design for all rate1 STBCs. Considering the power constraint at the transmitter, we maximize the received signal-to-noise ratio (SNR) for each transmitted symbol to find the optimal precoder. We show that any precoding for STBCs with rate1 symbol per channel use (e.g. (5)) is not optimal. In fact, we show that the optimal precoding for any STBCs with rate1 symbol per channel use is reduced to transmit beamforming of the transmit signals individually, by the weighting vector equal to the corresponding right singular vector of the largest singular value of the channel matrix. Due to the use of a general matrix, the proposed precoding method outperforms the unitary precoding proposed in (5) for OSTBCs. To show this, we analytically derive the exact bit error rate (BER) of the system and compare it with the performance of the previous work. Finally we extend the results for limited feedback systems. Simulation results show that our proposed precoding method outperforms the previous limited feedback precoder for OSTBCs (5).
1 citations
References
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Book•
01 Jan 1983
25,017 citations
"Optimal limited feedback technique ..." refers background in this paper
...Since each entry inH has CN (0, 1) distribution, the square norm of each row has a central Chi-square χ(2)(Mr, 1) distribution with the following probability density function (f(y)) and cumulative density function (F (y)) [14]:...
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Book•
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TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.
23,986 citations
AT&T1
TL;DR: This paper presents a simple two-branch transmit diversity scheme that provides the same diversity order as maximal-ratio receiver combining (MRRC) with one transmit antenna, and two receive antennas.
Abstract: This paper presents a simple two-branch transmit diversity scheme. Using two transmit antennas and one receive antenna the scheme provides the same diversity order as maximal-ratio receiver combining (MRRC) with one transmit antenna, and two receive antennas. It is also shown that the scheme may easily be generalized to two transmit antennas and M receive antennas to provide a diversity order of 2M. The new scheme does not require any bandwidth expansion or any feedback from the receiver to the transmitter and its computation complexity is similar to MRRC.
13,706 citations
"Optimal limited feedback technique ..." refers background in this paper
...Antenna subset selection have been proposed in [4] to maximize the system capacity, in [5] to suppress the rank deficiency of the channel matrix, in [6] to minimize the average symbol error rate (SER) in spatial multiplexing systems with linear receivers where the number of receiver antennas are greater than (or equal to) the number of selected transmit antennas and in [7] for orthogonal space-time codes, particularly Alamouti’s code [8]....
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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
AT&T1
TL;DR: A generalization of orthogonal designs is shown to provide space-time block codes for both real and complex constellations for any number of transmit antennas and it is shown that many of the codes presented here are optimal in this sense.
Abstract: We introduce space-time block coding, a new paradigm for communication over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space-time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas. The received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise. Maximum-likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection. This uses the orthogonal structure of the space-time block code and gives a maximum-likelihood decoding algorithm which is based only on linear processing at the receiver. Space-time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm. The classical mathematical framework of orthogonal designs is applied to construct space-time block codes. It is shown that space-time block codes constructed in this way only exist for few sporadic values of n. Subsequently, a generalization of orthogonal designs is shown to provide space-time block codes for both real and complex constellations for any number of transmit antennas. These codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as PAM. For an arbitrary complex constellation such as PSK and QAM, space-time block codes are designed that achieve 1/2 of the maximum possible transmission rate for any number of transmit antennas. For the specific cases of two, three, and four transmit antennas, space-time block codes are designed that achieve, respectively, all, 3/4, and 3/4 of maximum possible transmission rate using arbitrary complex constellations. The best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well.
7,348 citations