On the distribution of SINR for the MMSE MIMO receiver and performance analysis
TL;DR: A Gamma distribution and a generalized Gamma distribution are proposed as approximations to the finite sample distribution of T and simulations suggest that these approximate distributions can be used to estimate accurately the probability of errors even for very small dimensions.
Abstract: This correspondence studies the statistical distribution of the signal-to-interference-plus-noise ratio (SINR) for the minimum mean-square error (MMSE) receiver in multiple-input multiple-output (MIMO) wireless communications. The channel model is assumed to be (transmit) correlated Rayleigh flat-fading with unequal powers. The SINR can be decomposed into two independent random variables: SINR=SINR/sup ZF/+T, where SINR/sup ZF/ corresponds to the SINR for a zero-forcing (ZF) receiver and has an exact Gamma distribution. This correspondence focuses on characterizing the statistical properties of T using the results from random matrix theory. First three asymptotic moments of T are derived for uncorrelated channels and channels with equicorrelations. For general correlated channels, some limiting upper bounds for the first three moments are also provided. For uncorrelated channels and correlated channels satisfying certain conditions, it is proved that T converges to a Normal random variable. A Gamma distribution and a generalized Gamma distribution are proposed as approximations to the finite sample distribution of T. Simulations suggest that these approximate distributions can be used to estimate accurately the probability of errors even for very small dimensions (e.g., two transmit antennas).
Citations
More filters
TL;DR: In this paper, the tradeoff between the energy efficiency and spectral efficiency of a single-antenna system is quantified for a channel model that includes small-scale fading but not large scale fading, and it is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single antenna system.
Abstract: A multiplicity of autonomous terminals simultaneously transmits data streams to a compact array of antennas. The array uses imperfect channel-state information derived from transmitted pilots to extract the individual data streams. The power radiated by the terminals can be made inversely proportional to the square-root of the number of base station antennas with no reduction in performance. In contrast if perfect channel-state information were available the power could be made inversely proportional to the number of antennas. Lower capacity bounds for maximum-ratio combining (MRC), zero-forcing (ZF) and minimum mean-square error (MMSE) detection are derived. An MRC receiver normally performs worse than ZF and MMSE. However as power levels are reduced, the cross-talk introduced by the inferior maximum-ratio receiver eventually falls below the noise level and this simple receiver becomes a viable option. The tradeoff between the energy efficiency (as measured in bits/J) and spectral efficiency (as measured in bits/channel use/terminal) is quantified for a channel model that includes small-scale fading but not large-scale fading. It is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single-antenna system.
2,770 citations
Posted Content•
TL;DR: It is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single-antenna system.
Abstract: A multiplicity of autonomous terminals simultaneously transmits data streams to a compact array of antennas. The array uses imperfect channel-state information derived from transmitted pilots to extract the individual data streams. The power radiated by the terminals can be made inversely proportional to the square-root of the number of base station antennas with no reduction in performance. In contrast if perfect channel-state information were available the power could be made inversely proportional to the number of antennas. Lower capacity bounds for maximum-ratio combining (MRC), zero-forcing (ZF) and minimum mean-square error (MMSE) detection are derived. A MRC receiver normally performs worse than ZF and MMSE. However as power levels are reduced, the cross-talk introduced by the inferior maximum-ratio receiver eventually falls below the noise level and this simple receiver becomes a viable option. The tradeoff between the energy efficiency (as measured in bits/J) and spectral efficiency (as measured in bits/channel use/terminal) is quantified. It is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single-antenna system.
2,421 citations
Cites methods from "On the distribution of SINR for the..."
...More specifically, the PDF of γk can be approximated by a Gamma distribution as follows [20]:...
[...]
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
TL;DR: An in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multiinput multioutput (MIMO) systems with no fewer receive than transmit antennas reveals several new and surprising analytical results.
Abstract: This paper presents an in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multiinput multioutput (MIMO) systems with no fewer receive than transmit antennas. In spite of much prior work on this subject, we reveal several new and surprising analytical results in terms of output signal-to-noise ratio (SNR), uncoded error and outage probabilities, diversity-multiplexing (D-M) gain tradeoff and coding gain. Contrary to the common perception that ZF and MMSE are asymptotically equivalent at high SNR, we show that the output SNR of the MMSE equalizer (conditioned on the channel realization) is ρmmse = ρzf+η\ssrsnr, where ρzf is the output SNR of the ZF equalizer and that the gap η\ssrsnr is statistically independent of ρzf and is a nondecreasing function of input SNR. Furthermore, as \ssr snr\ura ∞, η\ssrsnr converges with probability one to a scaled F random variable. It is also shown that at the output of the MMSE equalizer, the interference-to-noise ratio (INR) is tightly upper bounded by [(η\ssrsnr)/(ρzf)]. Using the decomposition of the output SNR of MMSE, we can approximate its uncoded error, as well as outage probabilities through a numerical integral which accurately reflects the respective SNR gains of the MMSE equalizer relative to its ZF counterpart. The e-outage capacities of the two equalizers, however, coincide in the asymptotically high SNR regime. We also provide the solution to a long-standing open problem: applying optimal detection ordering does not improve the D-M tradeoff of the vertical Bell Labs layered Space-Time (V-BLAST) architecture. It is shown that optimal ordering yields a SNR gain of 10log10N dB in the ZF-V-BLAST architecture (where N is the number of transmit antennas) whereas for the MMSE-V-BLAST architecture, the SNR gain due to ordered detection is even better and significantly so.
390 citations
Cites background or methods from "On the distribution of SINR for the..."
...In the independent work [12], the authors show that such a decomposition is possible even if the columns of are correlated (but the rows need to be independent)....
[...]
...In the independent work of [12], the authors show that and are independent even if the columns of are correlated but with the rows of being independent....
[...]
...The approach of [12] is to approximate the first three asymptotic moments of as and then approximate it by a Gamma (or generalized Gamma) random variable....
[...]
References
More filters
Book•
01 Jan 2004
TL;DR: The book gives many numerical illustrations expressed in large collections of system performance curves, allowing the researchers or system designers to perform trade-off studies of the average bit error rate and symbol error rate.
Abstract: noncoherent communication systems, as well as a large variety of fading channel models typical of communication links often found in the real world, including single- and multichannel reception with a large variety of types. The book gives many numerical illustrations expressed in large collections of system performance curves, allowing the researchers or system designers to perform trade-off studies of the average bit error rate and symbol error rate. This book is a very good reference book for researchers and communication engineers and may also be a source for supplementary material of a graduate course on communication or signal processing. Nowadays, many new books attach a CD-ROM for more supplementary material. With the many numerical examples in this book, it appears that an attached CD-ROM would be ideal for this book. It would be even better to present the computer program in order to be interactive so that the readers can plug in their arbitrary parameters for the performance evaluation. —H. Hsu
6,469 citations
6,036 citations
Book•
01 Jan 1990TL;DR: In this paper, a comprehensive introduction to probability theory covering laws of large numbers, central limit theorem, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion is presented.
Abstract: This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.
5,168 citations
Book•
[...]
01 Aug 1998
TL;DR: This self-contained and comprehensive book sets out the basic details of multiuser detection, starting with simple examples and progressing to state-of-the-art applications.
Abstract: From the Publisher:
The development of multiuser detection techniques is one of the most important recent advances in communications technology. This self-contained and comprehensive book sets out the basic details of multiuser detection, starting with simple examples and progressing to state-of-the-art applications. The only prerequisites assumed are undergraduate-level probability, linear algebra, and digital communications. The book contains over 240 exercises and will be a suitable textbook for electrical engineering students. It will also be an ideal self-study guide for practicing engineers, as well as a valuable reference volume for researchers in communications, information theory, and signal processing.
5,048 citations
"On the distribution of SINR for the..." refers background or methods or result in this paper
...Based on the asymptotic Normality results, Poor and Verdu [2] (also in [4], [8]) proposed using the limiting BER (denoted by BER∞), which is a single Q-function, BER∞ = Q ( √...
[...]
...When p ≥ 32 (or p ≥ 64), the computations become intractable [4], [8]....
[...]
...For binary inputs, Verdu [4] (6....
[...]
..., [2]–[4], [6], [8]–[11]) has received a lot of attention....
[...]
...[3], [4] (6....
[...]
Book•
29 May 2003
TL;DR: This book is an accessible introduction to every fundamental aspect of space-time wireless communications and a powerful tool for improving system performance that already features in the UMTS and CDMA2000 mobile standards.
Abstract: Wireless networks are under constant pressure to provide ever-higher data rates to increasing numbers of users with greater reliability. This book is an accessible introduction to every fundamental aspect of space-time wireless communications. Space-time processing technology is a powerful tool for improving system performance that already features in the UMTS and CDMA2000 mobile standards. The ideal volume for graduate students and professionals, it features homework problems and other supporting material on a companion website.
3,798 citations
"On the distribution of SINR for the..." refers background or methods in this paper
...Rt ∈ Cp×p is the transmitter correlation matrix....
[...]
...We consider the popular linear minimum mean square error (MMSE) receiver....
[...]
...Conditional on the channel matrix H, the signal-to-interference-plus-noise ratio (SINR) on the k th spatial stream can be expressed as (e.g., [1]–[6]) SINRk = 1 MMSEk − 1 = 1[( Ip + 1mH †H )−1] kk − 1, (2) where Ip is a p × p identity matrix, and H† is the Hermitian transpose of H. Note that (2),…...
[...]