Minimum Mean Square Error Multiple-input Multiple-output Research Articles

BER analysis of an optimum MIMO linear receiver in optical SDM systems with mode-dependent loss.

Multiple-input multiple-output (MIMO) receivers designed to optimize the minimum mean square error (MMSE) are a common choice in coherent optical communication systems based on spatial division multiplexing (SDM). This kind of receivers naturally integrate both MIMO equalization and matched filtering functions. However, when the optical channel exhibits significant mode-dependent loss (MDL) and/or mode-dependent gain (MDG), the impact of inter-symbol interference (ISI) and crosstalk that arise, even using an ideal MIMO MMSE linear receiver, is barely analyzed. Moreover, due to the random nature of the MDL/MDG model, the resulting ISI, crosstalk, and bit error rate (BER) also become random variables and, hence the system performance is more unpredictable. In this paper, we first evaluate the residual distortion (ISI and crosstalk) after the MIMO receiver and then we study the validity of assuming it as an additional Gaussian noise term independent of the channel noise. Next, the probability density distribution (PDF) of the BER is analyzed, from both an analytical perspective and numerical simulations. For the latter, we use a single-carrier 2-PAM (pulse amplitude modulation) system, with pulse shaping, and the MIMO MMSE receiver implementation by means of a MIMO fractionally-spaced equalizer (FSE). We carry out simulations of the system under different conditions of MDL/MDG level and signal to noise ratio (SNR), measured at the receiver input. Additionally, we address possible fits of the BER PDF to known closed-form distributions, among which the Generalized Extreme Value (GEV) family of distributions is selected, and polynomial functions are proposed that relate the system parameters with the GEV PDF parameters. Finally, we present contour maps of BER according to a giving target of system outage probability (OP) that depend on the MDL/MDG and SNR conditions.

MMSE Indicator Error Rate Performance on MIMO When Increasing Modulation Order

Research in estimation theory has given evidence that the rich scattering of wireless channels is capable of great theoretical capacity if the multipath is efficiently utilized. Spatial multiplexing using multiple-input multiple-output (MIMO) systems makes efficient use of a limited frequency spectrum and increases the transmission throughput in wireless communications systems. This paper reviews the performance of the minimum mean squared error estimator for MIMO. In this context, it shows that the minimum mean square error (MMSE) for MIMO receiver exhibits an intricate error behavior that depends on channel condition, the antenna configuration, and the degree of modulation order. To this purpose, an explicit step-by-step mathematical model is derived for the receiver estimator and the error covariance based on the MMSE MIMO. Analysis, simulation, and experiment based on open source and programmable hardware and software results show that the MMSE estimator receiver efficiency is also based on the modulation order.

Mode-Dependent Loss and Gain Estimation in SDM Transmission Based on MMSE Equalizers

The capacity in space division multiplexing (SDM) systems with coupled channels is fundamentally limited by mode-dependent loss (MDL) and mode-dependent gain (MDG) generated in components and amplifiers. In these systems, MDL/MDG must be accurately estimated for performance analysis and troubleshooting. Most recent demonstrations of SDM with coupled channels perform MDL/MDG estimation by digital signal processing (DSP) techniques based on the coefficients of multiple-input multiple-output (MIMO) adaptive equalizers. Although these methods provide a valid indication of the order of magnitude of the accumulated MDL/MDG over the link, MIMO equalizers are usually updated according to the minimum mean square error (MMSE) criterion, which is known to depend on the channel signal-to-noise ratio (SNR). Therefore, MDL/MDG estimation techniques based on the adaptive filter coefficients are also impaired by noise. In this paper, we model analytically the influence of the SNR on DSP-based MDL/MDG estimation, and show that the technique is prone to errors. Based on the transfer function of MIMO MMSE equalizers, and assuming a known SNR, we calculate a correction factor that improves the estimation process in moderate levels of MDL/MDG and SNR. The correction factor is validated by simulation of a 6-mode long-haul transmission link, and experimentally using a 3-mode transmission link. The results confirm the limitations of the standard estimation method in scenarios of high additive noise and MDL/MDG, and indicate the correction factor as a possible solution in practical SDM scenarios.

Low-Complexity MIMO MMSE Receiver with Performance Enhancement via Coordinate Interleaving

Multiple-input multiple-output (MIMO) Minimum mean-square error (MMSE) receivers are widely adopted in the latest communication standards and reducing the complexity of these receivers while preserving the error performance is highly desirable. In this work, we study the error performance and implementation complexity of MIMO MMSE receivers when combined with a coordinate interleaved signal space diversity (SSD) technique. Contrary to the well-known trade-off between the error performance and implementation complexity, the proposed system leads to a considerably simplified MIMO MMSE receiver with significant performance gains when compared to the original MIMO MMSE receiver. Unlike the standard MIMO MMSE receiver, the proposed coordinate interleaved technique induces a block diagonal transmit correlation matrix providing both performance enhancement and complexity reduction. The results show that the error performance can be improved more than 10[Formula: see text]dB with up to 71% computational complexity reduction. The complexity comparison between the original and proposed approaches is also verified by means of field-programmable gate array (FPGA) implementation.

On the Distribution of SINR for MMSE MIMO Systems

While the performance analysis of linear zero-forcing (ZF) and minimum mean-square error (MMSE) receivers has received much attention, there has been no exact closed-form distribution of signal-to-interference-plus-noise ratio (SINR) for MMSE multiple-input multiple-output (MIMO) systems in the arbitrary fading environments for over the past 20 years. In this paper, an exact and general distribution of SINR for MMSE MIMO systems is presented under uncorrelated Rayleigh fading environments. We start our analysis by finding regular patterns from the probability density function (PDF) of SINR for small dimensions, and using the regular patterns, we extend the PDF to arbitrary dimensions. From the exact and general PDF of SINR for MMSE MIMO systems, the cumulative distribution function and the moment-generating function are derived in terms of the incomplete gamma function and the confluent hypergeometric function of the second kind, respectively. As applications, the error probability, outage probability, and achievable sum rate are investigated by using the distribution of SINR for MMSE MIMO systems. Most notably, the achievable sum rates for ZF and MMSE MIMO systems are defined by using the Meijer $G$ -function, which are the first closed-form expressions in the literature.

SIC-MMSE Method based Wireless Precoding Technique for Millimetre-Wave MIMO System

Objectives: A communication method is proposed using Minimum Mean Square Error (MMSE) precoding and Successive Interference Cancellation (SIC) technique for millimetre-wave multiple-input multiple-output (mm-Wave MIMO) based wireless communication system. Background: The mm-Wave MIMO technology for wireless communication system is the base potential technology for its high data transfer rate followed by data instruction and low power consumption compared to Long-Term Evolution (LTE). The mm-Wave system is already available in indoor hotspot and Wi-Fi backhaul for its high bandwidth availability and potential lead to rate of numerous Gbps/user. But, in mobile wireless communication system this technique is lagging because the channel faces relative orthogonal coordination and multiple node detection problems while rapid movement of nodes (transmitter and receiver) occur. Methods/Improvement: To improve the conventional mm-wave MIMO nodal detection and coordination performance, the system processes data using symbolized error vector technique for linearization. Then the MMSE precoding detection technique improves the link strength by constantly fitting the channel coefficients based on number of independent service antennas (M), Signal to Noise Ratio (SNR), Channel Matrix (CM) and mean square errors (MSE). To maintain sequentially encoded user data connectivity and to overcome data loss, SIC method is used in combination with MMSE. Improvements: MATLAB was used to validate proposed system performance. Simulation analysis has shown that, with the increase number of antenna use, the spectral efficiency also increased and higher then millimetre-wave MIMO or Single MMSE system. This research observed that, hybrid controller or combined control method have the better efficiency then single method, where SIC-MMSE based hybrid controller is a good example. Keywords: Channel Matrix, Long-term Evolution, Millimetre-wave, Minimum Mean Square Error, Quantized System, Successive Interference-Cancellation, Wireless Communication

A New Turbo Equalizer Conditioned on Estimated Channel for MIMO MMSE Receiver

In this letter, we propose a new equalization scheme based on minimum mean-square error (MMSE) criteria with a channel estimation error. Different from the existing work on linear MMSE receivers, we consider both the a priori information and channel estimation error in the equalizer design, and re-derive the soft output extrinsic information used for the channel decoder. Conditioned on the estimated channel, we developed the new autocorrelation and covariance matrix of the received vector, which are used to derive the equalizer coefficients. One of our major contributions of this letter is a general expression of the MMSE-based turbo equalization to account for both the channel estimation error and the a priori information, which is not presented in the literature. Simulation results show that significant performance improvement can be achieved with the proposed turbo equalizer.

SINR Statistics of Correlated MIMO Linear Receivers

Linear receivers offer a low complexity option for multiantenna communication systems. Therefore, understanding the outage behavior of the corresponding SINR is important in a fading mobile environment. In this paper, we introduce a large deviation method, valid nominally for a large number M of antennas, which provides the probability density of the SINR of Gaussian channel MIMO minimum mean square error (MMSE) and zero-forcing (ZF) receivers, with arbitrary transmission power profiles and in the presence of receiver antenna correlations. This approach extends the Gaussian approximation of the SINR, valid for large M asymptotically close to the center of the distribution, to obtain the non-Gaussian tails of the distribution. Our methodology allows us to calculate the SINR distribution to next-to-leading order ( O(1/M)) and showcase the deviations from approximations that have appeared in the literature (e.g., the Gaussian or the generalized Gamma distribution). We also analytically evaluate the outage probability, as well as the uncoded bit-error-rate. We find that our approximation is quite accurate even for the smallest antenna arrays (2 × 2).

Exact and Closed-Form Error Performance Analysis for Hard MMSE-SIC Detection in MIMO Systems

We investigate the exact error performance of hard minimum mean-squared error (MMSE) detection possibly with successive interference cancellation (SIC) in multiple-input multiple-output (MIMO) systems. To facilitate the analysis, we start with an exact bit-error rate (BER) analysis for a general system with decision statistic, z=ax+u, where a>;0, x is the transmitted signal, and u is an arbitrarily distributed noise component which is possibly dependent on the signal component x. For this general system, we derive the exact and closed-form BER expressions for M-ary pulse amplitude modulation (PAM) and arbitrary rectangular quadrature amplitude modulation (QAM), which include the well-known BER result of as a special case. Furthermore, by formulating the MIMO MMSE decision statistics in the same form as z in the general system, we derive the exact and closed-form instantaneous BER and symbol-error rate (SER) expressions for MIMO hard MMSE detection with/without SIC employing PAM and QAM. Finally, the validity of our derived error probability expressions is verified through extensive Monte Carlo simulations and the results reported in the literature.

Power Allocation between Pilot and Data Symbols for MIMO Systems with MMSE Detection under MMSE Channel Estimation

For multiple-input multiple-output (MIMO) wireless communication system with minimum mean square error (MMSE) detection, a new scheme of power allocation between pilot and data symbols is investigated under MMSE channel estimation in this paper. First, we propose a novel soft-output MMSE MIMO detector by taking into consideration the channel estimation error. Then, through the application of randommatrix theorem, we propose an efficient scheme for power allocation between pilot and data symbols which maximizes the lower bound of postprocessing signal-to-interference-and-noise ratio (SINR) for MIMO systems with equal number of transmitter and receiver antennas. We have proven the existence and uniqueness of the proposed optimal power allocation settings. Furthermore, our analysis shows that the proposed power allocation is also valid and applicable for those MIMO systems with unequal number of transmitter and receiver antennas. Finally, our extensive simulation results have validated this novel power allocation scheme.

Bit Error Rate Performance of MIMO MMSE Receivers in Correlated Rayleigh Flat-Fading Channels

This paper analyzes the average bit error rate (BER) of multiple-input–multiple-output (MIMO) systems in transmit-correlated Rayleigh flat-fading channels. The receiver scheme is based on the minimum mean square error (MMSE) criterion, and the input may be precoded to optimize the communication. Accurate closed-form formulations for the average BER are derived.

MMSE Beamforming for SC-FDMA Transmission over MIMO ISI Channels

We consider transmit beamforming for single-carrier frequency-division multiple access (SC-FDMA) transmission over frequency-selective multiple-input multiple-output (MIMO) channels. The beamforming filters are optimized for minimization of the sum of the mean-squared errors (MSEs) of the transmitted data streams after MIMO minimum mean-squared error linear equalization (MMSE-LE), and for minimization of the product of the MSEs after MIMO MMSE decision-feedback equalization (MMSE-DFE), respectively. We prove that for SC-FDMA transmission in both cases eigenbeamforming, diagonalizing the overall channel, together with a nonuniform power distribution is the optimum beamforming strategy. The optimum power allocation derived for MMSE-LE is similar in spirit to classical results for the optimum continuous-time transmit filter for linear modulation formats obtained by Berger/Tufts and Yang/Roy, whereas for MMSE-DFE the capacity achieving waterfilling strategy well known from conventional single-carrier transmission schemes is obtained. Moreover, we present a modification of the beamformer design to mitigate an increase of the peak-to-average power ratio (PAPR) which is in general associated with beamforming. Simulation results demonstrate the high performance of the proposed beamforming algorithms.

Achievable Sum Rate of MIMO MMSE Receivers: A General Analytic Framework

This paper investigates the achievable sum rate of multiple-input multiple-output (MIMO) wireless systems employing linear minimum mean-squared error (MMSE) receivers. We present a new analytic framework which exploits an interesting connection between the achievable sum rate with MMSE receivers and the ergodic mutual information achieved with optimal receivers. This simple but powerful result enables the vast prior literature on ergodic MIMO mutual information to be directly applied to the analysis of MMSE receivers. The framework is particularized to various Rayleigh and Rician channel scenarios to yield new exact closed-form expressions for the achievable sum rate, as well as simplified expressions in the asymptotic regimes of high and low signal-to-noise ratios (SNRs). These expressions lead to the discovery of key insights into the performance of MIMO MMSE receivers under practical channel conditions.

Soft-Output MMSE MIMO Detector Under ML Channel Estimation and Channel Correlation

As channel estimation errors are not taken into account by existing soft-output minimum mean square error (MMSE) multiple-input multiple-output (MIMO) detector, its performance can therefore be degraded under imperfect channel estimation. In this letter, we propose a novel soft-output MMSE MIMO detector under maximum likelihood (ML) channel estimation for a MIMO system with spatially correlated receiver and transmitter antennas. Our proposed detector takes both channel estimation errors and spatial correlation of antennas into account when constructing MMSE filter and computing log-likelihood ratio (LLR) of each coded bit. Simulation results show the proposed novel detector outperforms existing soft-output MMSE MIMO detector with considerable improvement.

MIMO Equalization With State-Space Channel Models

State-space models are proposed to represent multiple-input multiple-output (MIMO) frequency-selective wireless channels with the motivation of better model approximation and more robust channel equalization performance when the order of the channel model is lower that of the true channel. We develop a simple framework under which the equalizers for state-space channel models can be designed using the existing methods for designing equalizers for finite impulse response (FIR) models. In particular, a MIMO minimum mean-squared error decision feedback equalizer is developed for state-space models. When only estimates of the channel are available to the receiver, the equalization performance is affected by the channel estimation accuracy. Because reduced-order state-space models can provide lower <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> channel estimation error than reduced-length FIR models, state-space based equalizers typically exhibit significantly smaller symbol error rate than FIR-based ones. Thus, state-space channel models can be a more robust choice than FIR models in the presence of model order selection error. Numerical simulation also shows that adaptive state-space based receivers have slower but comparable convergence rates compared to FIR-based adaptive equalizers.

A Practical, Hardware Friendly MMSE Detector for MIMO-OFDM-Based Systems

Design and implementation of a highly optimized MIMO (multiple-input multiple-output) detector requires cooptimization of the algorithm with the underlying hardware architecture. Special attention must be paid to application requirements such as throughput, latency, and resource constraints. In this work, we focus on a highly optimized matrix inversion free 4 × 4 MMSE (minimum mean square error) MIMO detector implementation. The work has resulted in a real-time field-programmable gate array-based implementation (FPGA-) on a Xilinx Virtex-2 6000 using only 9003 logic slices, 66 multipliers, and 24 Block RAMs (less than 33% of the overall resources of this part). The design delivers over 420 Mbps sustained throughput with a small 2.77-microsecond latency. The designed 4 × 4 linear MMSE MIMO detector is capable of complying with the proposed IEEE 802.11n standard.

On the distribution of SINR for the MMSE MIMO receiver and performance analysis

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).

Adaptive cochannel interference cancellation in space-time coded communication systems

Space-time coding is a powerful scheme that combines channel coding, modulation, and multiple transmit antennas to achieve higher data rates and combat fading in wireless systems. In this letter, we propose a multiple-input multiple-output minimum mean-square error spatial-filtering-based adaptive antenna arrays method to suppress cochannel interference (CCI) in space-time coded systems. It is shown that the proposed method can effectively suppress CCI while preserving the space-time structure, thereby significantly improving the system's interference suppression ability without significant bit error rate performance degradation.