Chih-Hao Liu

Generalized Geometric Mean Decomposition and DFE Transceiver Design—Part I: Design and Complexity

This paper considers a new matrix decomposition which decomposes a complex matrix as a product of several sets of semi-unitary matrices and upper triangular matrices in an iterative manner. The inner most triangular matrix has its diagonal elements equal to the geometric mean of the singular values of the target complex matrix. The complexity (defined in terms of the number of floating point operations) of the new decomposition, generalized geometric mean decomposition (GGMD), depends on its parameters, but is always less than or equal to that of geometric mean decomposition (GMD). The optimal parameters which yield the minimal complexity are derived. The paper also shows how to use GGMD to design an optimal decision feedback equalizer (DFE) transceiver for multiple-input multiple-output (MIMO) channels without zero-forcing constraint. A novel iterative receiving detection algorithm for the specific receiver is also proposed. For the application to cyclic prefix systems in which the SVD of the equivalent channel matrix can be easily computed, the proposed GGMD transceiver has K/log2(K) times complexity advantage over the GMD transceiver, where is the number of data symbols per data block and is a power of 2. In a companion paper, performance analyses of the proposed GGMD transceiver in terms of arithmetic mean square error (MSE), symbol error rate (SER) and Gaussian mutual information are performed, and comparisons with well-known transceivers are made. The results show that the proposed transceiver reaches the same optimality that a GMD MMSE transceiver can possibly achieve.

Zero-Forcing DFE Transceiver Design Over Slowly Time-Varying MIMO Channels Using ST-GTD

This paper considers the optimization of transceivers with decision feedback equalizers (DFE) for slowly time-varying memoryless multi-input multi-output (MIMO) channels. The data vectors are grouped into space-time blocks (ST-blocks) for the spatial and temporal precoding to take advantage of the diversity offered by time-varying channels. The space-time generalized triangular decomposition (ST-GTD) is proposed for application in time-varying channels. Under the assumption that the instantaneous channel state information at the transmitter (CSIT) and receiver (CSIR), and the channel prediction are available, we also propose the space-time geometric mean decomposition (ST-GMD) system based on ST-GTD. Under perfect channel prediction, the system minimizes both the arithmetic MSE at the feedback detector, and the average un-coded bit error rate (BER) in moderate high signal to noise ratio (SNR) region. For practical applications, a novel ST-GTD based system which does not require channel prediction but shares the same asymptotic BER performance with the ST-GMD system is also proposed. At the moderate high SNR region, our analysis and numerical results show that all the proposed systems have better BER performance than the conventional GMD-based systems over time-varying channels; the average BERs of the proposed systems are non-increasing functions of the ST-block size.

Generalized Geometric Mean Decomposition and DFE Transceiver Design—Part II: Performance Analysis

In a companion paper, the generalized geometric mean decomposition (GGMD) was proposed and used to design GGMD decision feedback equalizer (DFE) transceivers for arbitrary multiple-input multiple-output (MIMO) channels without zero-forcing constraint. For the application to cyclic prefix (CP) systems, the GGMD DFE transceiver has the most advantage over the GMD DFE MMSE transceiver in terms of design and implementation complexity. This paper presents the performance analysis for the GGMD DFE transceiver implementation proposed in Part I. The arithmetic mean-square error (MSE), symbol error rate (SER) and Gaussian mutual information of the proposed system are investigated. The performance advantages of GGMD DFE transceiver over popular orthogonal-frequency-division multiplexing (OFDM) and single-carrier CP MMSE systems are shown analytically and verified by numerical simulations.

MMSE DFE Transceiver Design Over Slowly Time-Varying MIMO Channels Using ST-GTD

In a companion paper, we have studied the zero-forcing (ZF) transceiver with decision feedback equalizer (DFE) over slowly time-varying narrowband multiinput multioutput (MIMO) channels. The space-time generalized triangular decomposition (ST-GTD) was used for the design of ZF-DFE transceivers. The space-time geometric mean decomposition (ST-GMD) ZF transceiver minimizes both the arithmetic mean square error (MSE) at the feedback detector and the average uncoded bit error rate (BER) in moderate high signal-to-noise ratio (SNR). This paper addresses the design problem of DFE transceiver without zero-forcing constraint. In the first part, a channel independent temporal precoder is superimposed on the conventional block-wise GMD-based minimum mean square error (MMSE) DFE transceiver to take advantage of the temporal diversity. In the second part, ST-GTD is applied for the design of MMSE DFE transceivers. With accurate channel prediction and space-time powerloading, the proposed ST-GMD MMSE transceiver minimizes the arithmetic MSE at the feedack detector, and maximizes Gaussian mutual information. For practical applications, the ST-GTD MMSE transceiver which does not require channel prediction but shares the same asymptotic BER performance with the ST-GMD MMSE system is also developed. In the convex region, our analysis shows that the proposed MMSE transceivers has better BER performance than the conventional GMD-based MMSE transceiver; the average BERs of the proposed systems are nonincreasing functions of the ST-block size. The superior performance of ST-GMD MMSE transceiver over the ST-GMD ZF transceiver is also verified analytically.

ZF-DFE transceiver for time-varying MIMO channels with channel-independent temporal precoder

This paper considers the DFE transceiver optimization for time-varying memoryless MIMO channels under zero-forcing (ZF) constraint. For time-varying channels, the uncoded average BER of the conventional geometric mean decomposition (GMD) based systems is not minimized because of the diverse arithmetic MSEs at different block times. To minimize the BER, a new GMD transceiver is proposed, in which the data vectors are grouped into space-time blocks (ST-blocks). A channel independent-temporal precoder is superimposed on the conventional blockwise GMD for the equalization of arithmetic MSEs across different blocks. So, the proposed system can take advantage of both the temporal and spatial diversity offered by time-varying MIMO channels. At moderate high SNR, corresponding to reasonable BER, there exists a class of optimal channel-independent temporal precoders, e.g., DFT and Hadamard matrices, for the minimization of average BER. Furthermore, simulation results show that the average BER performance of the proposed system improves with ST-block size and converges at moderate ST-block size.1

MIMO Broadcast DFE Transceivers With QoS Constraints: Min-Power and Max-Rate Solutions

This paper considers two joint design problems of linear precoder, decision feedback equalizer (DFE) and bit allocation for multi-input multi-output (MIMO) broadcast (BC) channels. The first problem is a power minimization problem (min-power) with a total bitrate constraint and per data stream symbol error rate (SER) specifications. The second problem is a rate maximization problem (max-rate) with a total transmit power constraint and per data stream SER specifications. For a given broadcast DFE transceiver, optimal bit allocation formulas for both problems are derived. A particular class of joint triangularization (JT) is applied to obtain the optimal broadcast DFE transceivers for the min-power and max-rate QoS problems, namely the minimum power JT broadcast DFE transceiver (MPJT) and the maximum rate JT broadcast DFE transceiver (MRJT), respectively. Two suboptimal broadcast DFE transceivers, the power minimized QR broadcast DFE transceiver (PMQR) and the rate maximized QR broadcast DFE transceiver (RMQR), are also proposed for the min-power and max-rate QoS problems, respectively. The proposed suboptimal designs apply QR decompositions instead of the particular class of JT. Moreover, integer bit allocation problems for both QoS problems are addressed. This work also shows the duality of the proposed MPJT and MRJT transceivers. At the end, numerical results are presented to demonstrate the performance of the proposed MPJT, MRJT, PMQR and RMQR transceivers under different QoS constraints, and verify the duality of the proposed MPJT and MRJT transceivers.

MIMO broadcast DFE transceiver design with bit allocation under QoS constraints

This paper considers the joint design problem of linear precoder, decision feedback equalizer (DFE) and bit allocation for multi-input multi-output (MIMO) broadcast (BC) channels. The problem is formulated as the power minimization problem when the target bit rate and per stream symbol error rate (SER) are specified. For the two-user case, a particular class of joint triangularization (JT) can be applied to obtain the optimal design for the joint optimization problem with quality of service (QoS) constraints, namely, the Minimum Power JT broadcast DFE transceiver (MPJT). For arbitrary number of users, a suboptimal design, the Power Minimized QR DFE broadcast transceiver (PMQR), is also proposed. Numerical results are presented to demonstrate the performance of MPJT and PMQR transceivers under different QoS constraints.

Joint MAX-SER-minimized DFE transceiver design with bit allocation for broadcast channels

This paper addresses the joint design problem of bit allocation and decision feedback equalizer (DFE) transceiver for multi-input multi-output (MIMO) broadcast (BC) channels. Channel state information (CSI) is assumed to be available both at the transmitter and receivers. The transceiver is designed by minimizing maximum symbol error rate (MAX-SER) under the total power and bitrate constraints. The bit allocation, precoder, feed-forward matrices and feedback matrices are the optimization variables. For the two-user case, a particular class of joint triangularization (JT) can be applied to obtain the optimal design for the problem, which is the optimal JT minimum Max-SER BC DFE transceiver (JTMMS). For arbitrary number of users, the suboptimal design, the QR BC DFE transceiver with bit allocation, is discussed. In the simulations, the Max-SER and average BER performance of the proposed systems are investigated.

Generalized geometric mean decomposition and DFE MMSE transceiver design for cyclic prefix systems

This paper considers the decomposition of a complex matrix as the product of several sets of semi-unitary matrices and upper triangular matrices in iterative manner. The innermost triangular matrix has its diagonal elements equal to the geometric mean of the singular values of the complex matrix. This decomposition, generalized geometric mean decomposition (GGMD), has one order less complexity than the geometric mean decomposition (GMD) if the target matrix is a diagonal matrix. GGMD can be used to design the optimal decision feedback equalizer (DFE) MMSE transceiver for arbitrary multi-input-multi-output (MIMO) channels. The GGMD transceiver shares the same performance as the transceiver designed by using GMD. For the applications over cyclic prefix system, the GGMD transceiver has K/ log2(K) times lower complexity1 than the GMD transceiver, where K is the number of subchannels and is a power of 2.

ZF-DFE transceiver design for time-varying MIMO channels using space-time generalized triangular decomposition

We consider the design of MIMO transceivers with zeroforcing (ZF) decision feedback detection over time-varying MIMO channels. The data vectors are grouped into spacetime blocks (ST-blocks) for the spatial and temporal precoding to take advantage of the diversity offered by time-varying channels. We extend the generalized triangular decomposition (GTD) for the case of time-varying channels by introducing the space-time GTD (ST-GTD). Based on ST-GTD and the channel prediction, we propose the space-time geometric mean decomposition (ST-GMD) based system which minimizes the arithmetic mean square error (MSE) for every ST-block. We also present the causal ST-GTD based system which does not require channel prediction. The simulations show that this system achieves the same BER performance asymptotically as the ST-GMD based system. In moderate high SNR, the proposed systems have superior BER performance over the conventional GMD-based systems.