Multiple beamforming is realized by singular value decomposition of the channel matrix which is assumed to be known to both the transmitter and the receiver. Bit-Interleaved Coded Multiple Beamforming (BICMB) can achieve full diversity as long as the code rate Rcand the number of employed subchannels S satisfy the condition RcS ≤ 1. Bit-Interleaved Coded Multiple Beamforming with Constellation Precoding (BICMB-CP), on the other hand, can achieve full diversity without the condition RcS ≤ 1. However, the decoding complexity of BICMB-CP is much higher than BICMB. In this paper, a reduced complexity decoding technique, which is based on Sphere Decoding (SD), is proposed to reduce the complexity of Maximum Likelihood (ML) decoding for BICMB-CP. The decreased complexity decoding achieves several orders of magnitude reduction, in terms of the average number of real multiplications needed to acquire one precoded bit metric, not only with respect to conventional ML decoding, but also, with respect to conventional SD.

Beamforming techniques that employ Singular Value Decomposition (SVD) are commonly used in Multi-Input Multi-Output (MIMO) wireless communication systems. In the absence of channel coding, when a single symbol is transmitted, these systems achieve the full diversity order provided by the channel; whereas when multiple symbols are simultaneously transmitted, this property is lost. When channel coding is employed, full diversity order can be achieved. For example, when Bit-Interleaved Coded Modulation (BICM) is combined with this technique, full diversity order of NM in an M x N MIMO channel transmitting S parallel streams is possible, provided a condition on S and the BICM convolutional code rate is satisfied. In this paper, we present constellation precoded multiple beamforming which can achieve the full diversity order both with BICM-coded and uncoded SVD systems. We provide an analytical proof of this property. To reduce the computational complexity of Maximum Likelihood (ML) decoding in this system, we employ Sphere Decoding (SD). We report an SD technique that reduces the computational complexity beyond commonly used approaches to SD. This technique achieves several orders of magnitude reduction in computational complexity not only with respect to conventional ML decoding but also, with respect to conventional SD.

Diversity analysis of a number of multiple-input multiple-output (MIMO) applications requires the calculation of the expectation of a function whose variables are the ordered multiple eigenvalues of the Wishart matrices. To solve this, we need the marginal pdf of an arbitrary subset of the ordered eigenvalues. The marginal pdf shown in the literature is useful in numerical analysis, but not beneficial to diversity analysis. In this paper, we derive an upper bound to the marginal pdf of the eigenvalues. The derivation is based on the multiple integration of the well-known joint pdf, which is very complicated due to the exponential factors of the joint pdf. We suggest an alternative function that provides simpler calculation of the multiple integration. As a result, the marginal pdf is shown to consist of a multivariate polynomial with a given degree. By applying the marginal pdf to the calculation of the expectation, the diversity order for a number of MIMO systems can be calculated. Simulation results that support the analysis are presented.

In this paper, we present the diversity order analysis of bit-interleaved coded multiple beamforming (BICMB) combined with the constellation precoding scheme. Multiple beamforming is realized by singular value decomposition of the channel matrix which is assumed to be perfectly known to the transmitter as well as the receiver. Previously, BICMB is known to have a diversity order bound related with the product of the code rate and the number of parallel subchannels, losing the full diversity order in some cases. In this paper, we show that BICMB combined with the constellation precoder and maximum likelihood detection achieves the full diversity order. We also provide simulation results that match the analysis.

In this paper, diversity analysis of bit-interleaved coded multiple beamforming (BICMB) is extended to the case of general spatial interleavers, removing a condition on their previously known design criteria. We provide a method to get diversity order, simplifying the calculation of pairwise error probability (PEP). By using the Singleton bound, we also show the maximum achievable diversity for given code rate and the number of subchannels.

We present and analyze the performance of constellation precoded beamforming. This multi-input multi-output transmission technique is based on the singular value decomposition of a channel matrix. In this work, the beamformer is precoded to improve its diversity performance. It was shown previously that while single beamforming achieves full diversity without channel coding, multiple beamforming results in diversity loss. In this paper, we show that a properly designed constellation precoder makes uncoded multiple beamforming achieve full diversity order. We also show that partially precoded multiple beamforming gets better diversity order than multiple beamforming without constellation precoder if the subchannels to be precoded are properly chosen. We propose several criteria to design the constellation precoder. Simulation results match the analysis, and show that precoded multiple beamforming actually outperforms single beamforming without precoding at the same system data rate while achieving full diversity order.

Cooperative relaying has been proposed as a way to create a virtual antenna array, hence providing spatial diversity in slow fading wireless environment. From the practical implementation perspective, one needs to design a relaying system that can achieve the maximum diversity order with a low system complexity. One promising candidate is to incorporate equal gain combining (EGC) technique into relaying systems and it is the main focus of this paper. In particular, the techniques of EGC and smart relaying are combined in the decode-and-forward (DF) relay processing method. It is shown that for a system with one relay and M phase-shift-keying (M-PSK) modulation, a maximum diversity order of 2m is achieved in Nakagami-m fading environment. With K relays, numerical results suggest that the diversity order is m(K + 1).

This paper addresses the performance of bit-interleaved coded multiple beamforming (BICMB) [1], [2] with imperfect knowledge of beamforming vectors. Most studies for limited-rate channel state information at the transmitter (CSIT) assume that the precoding matrix has an invariance property under an arbitrary unitary transform. In BICMB, this property does not hold. On the other hand, the optimum precoder and detector for BICMB are invariant under a diagonal unitary transform. In order to design a limited-rate CSIT system for BICMB, we propose a new distortion measure optimum under this invariance. Based on this new distortion measure, we introduce a new set of centroids and employ the generalized Lloyd algorithm for codebook design. We provide simulation results demonstrating the performance improvement achieved with the proposed distortion measure and the codebook design for various receivers with linear detectors.We show that although these receivers have the same performance for perfect CSIT, their performance varies under imperfect CSIT.