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

We present a novel approach to blind symbol detection in multiple-input multiple-output (MIMO) systems that use orthogonal space-time block codes. Our approach is based on a relaxed maximum likelihood (ML) joint channel and symbol estimation. The proposed technique has a simple closed-form solution and is free of most of the shortcomings of the previously presented blind receivers. In particular, there is no signal-to-noise ratio (SNR) penalty, as in differential encoding techniques, provided that the coherence time of the channel is large enough. Moreover, our approach does not use any pilot symbols as some other (semi-)blind techniques do.

In statistical signal processing, the sequential Monte Carlo (SMC) method is powerful and can approach the theoretical optima. However, its computational complexity is usually very high, especially in multiple-input multiple-output (MIMO) systems. This paper presents a new low-complexity SMC (LC-SMC) algorithm for blind detection in MIMO systems, the main idea of which is to shrink the sampling space via channel estimation which is initialized using the first differentially modulated symbol and then updated using the Monte Carlo samples. Since the a posteriori probability of the transmitted symbols can be calculated separately by each transmit antenna, the proposed LC-SMC algorithm is not only computationally efficient, as compared to the original SMC whose complexity grows exponentially with the number of transmit antennas, but also makes blind turbo receiver more feasible for multilevel/phase modulations. Simulation results are presented to demonstrate the effectiveness of the LC-SMC algorithm

For multiple-input multiple-output (MIMO) spatial multiplexing systems the complexity of the maximum likelihood detector (MLD) can be prohibitively large when the number of transmitting antennas and constellation points is high. To simplify the MLD many linear and nonlinear techniques have been proposed. In this letter the principle of reduced state sequence detection, based on mapping by set partitioning, is applied to perform detection in spatial multiplexing MIMO systems based on non-binary constellations. Simulation results are presented to show the benefits of our proposal. An upper bound to the symbol error probability is also derived.

Despite significant research efforts, closed-form bit error rate (BER) expressions for multiple-input multiple-output (MIMO) systems employing transmit antenna selection and orthogonal space-time block codes (OSTBCs) are not available. We thus derive exact closed-form expressions for the BER of Gray-coded M-ary one and two-dimensional amplitude modulations when an OSTBC is employed and N transmit antennas out of total Lt antennas are selected for transmission. We also derive tight closed-form approximate BER for M-PSK constellations. Our BER expressions are valid for a frequency-flat Rayleigh fading MIMO channel and can be evaluated without numerical integration methods.

In this article, we propose an adaptive multiple stage K-best successive interference cancellation (AMS-KSIC) algorithm for symbol vector detection in multiple-input multiple-output systems. The proposed algorithm employs multiple successive interference cancellation (SIC) stages in parallel, where the number of stages depends on the number of positions at which the minimum mean squared error (MMSE) estimate of the received vector and the SIC solution differ, and each stage is initialized with the partial MMSE estimate of the received vector. In every stage, K-best solutions are generated by using the minimum Euclidean distance criteria. Furthermore, to reduce error propagation, we use two different ordering strategies namely, signal to noise ratio and log-likelihood ratio based orderings. The best solution among all the generated solutions is selected by using maximum likelihood (ML) cost metric. Multiple stages along with K-best solutions in every stage achieves a higher detection diversity, and hence, yield a better performance in terms of bit error rate (BER). From simulations, we observe that the proposed AMS-KSIC algorithm performs better than the MMSE and the SIC based detection schemes, and achieves a near ML performance. Further, the BER performance of the proposed algorithm improves with increase in the number of antennas and shifts towards single-input single-output additive white Gaussian noise performance. In addition, we also check and validate robustness of the proposed algorithm by simulating the BER performance under channel estimation errors.

The bit error rate (BER) performance of orthogonal space-time block codes (STBC) in correlated multiple input multiple output (MIMO) channels is studied. We derive closed-form exact BER equations for pulse amplitude modulation (PAM), quadrature amplitude modulation (QAM), and phase shift keying (PSK) constellations in a correlated Rayleigh MIMO channel. The BER expressions can be easily evaluated without any numerical methods.

Multiple-input multiple-output (MIMO) systems employing both maximal-ratio transmission and receive antenna selection (MRT&RAS) are studied in this paper. The examination has been done for independent and identically distributed (i.i.d.) flat Nakagami-m fading channels. Exact bit error rate (BER) expressions for binary phase shift keying (BPSK), differential binary phase shift keying (DBPSK) and binary frequency shift keying (BFSK) signals are derived by using the moment generating function (MGF)-based analysis method. By deriving upper bounds for exact BER expressions, it is shown that the investigated systems achieve full diversity orders at high signal-to-noise ratios (SNRs). Also, the ergodic system capacity and outage probability of MRT&RAS scheme have been analyzed. Exact BER results for different system and channel conditions are validated by Monte Carlo simulations.

Consider a multiple input multiple output (MIMO) system with pulse amplitude modulation (PAM), where each transmit antenna is allowed to choose a different transmission delay. (We refer to this system as a MIMO system with asynchronous PAM.) Such MIMO systems have been studied before for increasing diversity and avoiding unintentional beamforming. We quantify the capacity/outage capacity improvement due to asynchronism and the corresponding spectral leakage (if any) for MIMO point-to-point and MIMO broadcast channels. We show that under commonly used transmitter spectral mask constraints, the channel has unbounded degrees of freedom. Synchronous PAM exploits only a finite number of these degrees of freedom, and asynchronism helps to improve capacity. For most cases of interest, two distinct shifts attain most of the capacity improvement. These gains are robust to interference from adjacent carriers. Analysis of the spectrum of the asynchronous PAM suggests that optimal water-filling based schemes are useful for improving capacity only in directional links due to possible spectral leakage, while uniform power allocation leads to capacity improvement without any extra spectral leakage.

In this paper, we consider the design of a full-rate linear space-time block code for coherent multiple-input multiple-output (MIMO) communication systems under a quasi-static Rayleigh flat-fading environment. Our design targets specifically at the use of a linear minimum mean-square error (MMSE) receiver that minimizes the asymptotic average bit error rate (BER) when the transmitted signal is selected from a 4-QAM constellation. This optimization problem is solved in two main stages: 1) a lower bound on the BER is first minimized, and 2)how this minimized lower bound can be achieved is then shown. By exploiting a rigorous convex optimization technique without any assumption on the code, we prove that individual unitary and trace-orthogonal structures are the necessary and sufficient conditions to assure the minimum asymptotic average BER with an MMSE detector. An algorithm is provided for an efficient generation of our codes, and simulation results confirm that our optimally designed codes are indeed superior in performance compared to some other commonly used codes

A combined minimum mean square error (MMSE) and maximum likelihood (ML) multiuser detector with reduced-complexity is proposed for uplink space-time block coded (STBC) orthogonal frequency division multiplexing (OFDM) systems. In this combined detector, received signals from co-channels are first detected by an MMSE multiuser detector. Then an ML-based post-detection processor (PDP) is designed to detect estimated outputs from the MMSE detector with the highest probability of errors and correct them to reduce the system bit error rate (BER). An efficient algorithm is proposed for the PDP which requires only a small additional complexity but can improve the system BER performance significantly compared with the conventional MMSE multiuser detector.

We first develop a reduced-rank minimum mean square error (MMSE) detector for direct-sequence (DS) code division multiple access (CDMA) by forcing the linear MMSE detector to lie in a signal subspace of a reduced dimension. While a reduced-rank MMSE detector has lower complexity, it cannot outperform the full-rank MMSE detector. We then concentrate on the blind reduced-rank MMSE detector which is obtained from an estimated covariance matrix. Our analysis and simulation results show that when the desired user's signal is in a low-dimensional subspace, there exists an optimal subspace so that the blind reduced-rank MMSE detector lying in this subspace has the best performance. By properly choosing a subspace, we guarantee that the optimal blind reduced-rank MMSE detector is obtained. An adaptive blind reduced-rank MMSE detector, based on a subspace tracking algorithm, is developed. The adaptive blind reduced-rank MMSE detector exhibits superior steady-state performance and fast convergence speed.

Minimum mean square error (MMSE) linear detector is able to achieve the near-optimal bit error rate (BER) performance for uplink multi-user massive Multiple-Input Multiple-Output (MIMO) systems due to the increase in the number of base station (BS) antennas. However, this detector involves matrix inversion with high complexity, especially when the number of users is large. In order to reduce the complexity, in this paper, a low-complexity MMSE detector algorithm based on the Damped Jacobi (DJ) method is proposed. Further, a simple approach to determine the optimum and quasi-optimum damped parameter by exploiting the massive MIMO channel property of asymptotic orthogonality is developed. The analysis shows that the proposed algorithm can reduce the complexity of the classical MMSE detector by about one order of magnitude without performance loss. Finally, it is verified through simulation results that the proposed algorithm outperforms the recently proposed linear detector based on the conventional Jacobi method, and achieves the near-optimal performance of the classical MMSE detector with a small number of iterations in Rayleigh fading channels.

In recent years, multiple-input multiple-output (MIMO) systems, which use several transmit and receive antennas, have attracted much attention for high performance radio systems. In MIMO systems, the channel estimation is important to distinguish transmit signals from multiple transmit antennas. While the space-alternating generalized expectation-maximization (SAGE) algorithm is known to offer good channel estimation and data detection. We proposed earlier the minimum mean square error (MMSE)-based SAGE algorithm for MIMO systems where the MMSE estimation is used for channel estimation. We showed that the proposed MMSE-based SAGE algorithm can achieve the better bit error rate (BER) than the maximum likelihood (ML) detection with training symbols. The MMSE channel estimation needs the knowledge of the maximum Doppler frequency F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> for deriving the covariance matrix of the channel and the variance sigma <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> of additive white Gaussian noise (AWGN). Additionally, the computation of the MMSE channel estimation requires O(L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) operations where L is the transmitted frame length. Thus, its computational complexity is high. In this paper, we propose a zero-forcing (ZF)-based SAGE algorithm for channel estimation and data detection in MIMO systems that does not need the knowledge of Fd and sigma <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Since the computation of the proposed ZF-based SAGE algorithm requires O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) operations where N is the number of transmit antennas, its computational complexity is low. We show that the proposed ZF-based tracking SAGE algorithm with less computational complexity can achieve almost the same BER as that of the MMSE-based tracking SAGE algorithm

We consider very general code-division-multiple-access (CDMA) systems, namely multiple-chip-rate CDMA systems, where signals can be transmitted at different chip rates, carrier frequencies, processing gains, and transmitted powers to satisfy the given quality of service (QoS) requirements. Also, non-zero and possibly different carrier frequency offsets are assumed for different users. For these systems, a closed-form bit error rate (BER) expression is derived based on the simplified improved Gaussian approximation. Unlike the conventional simplified improved Gaussian approximation, however, we utilize the correlations between chip waveforms and their integrations rather than using the correlations between chip sequences in order to obtain the closed-form BER expression. Numerical results demonstrate that the proposed method provides much more accurate BER values compared to the standard Gaussian approximation

Enhanced genetic algorithms (GA) applied in space-time block coded (STBC) multiuser detection (MUD) systems in Rayleigh flat-fading channels are reported in this paper. Firstly, an improved objective function, which is designed to help speed up the search for the optimal solution, is introduced. Secondly, a decorrelating detector (DD) and a minimum mean square error (MMSE) detector have been added to the GA STBC MUD receiver to create the seed chromosome in the initial population. This operation has improved the receiver performance further because some signal information has been intentionally embedded in the initial population. Simulation results show that the receiver employing the improved objective function and the DD or MMSE detector can converge faster with the same bit error rate (BER) performance than the receiver with the initial population chosen randomly. The total signal-to-noise ratio (SNR) improvement contributed by these two modifications can reach 4 dB. Hence the proposed GA receiver is a promising solution of the STBC MUD problem.