This letter proposes the use of a new near-An-lattice sphere decoding (LSD) technique for optimum detection in block-data transmission systems instead of the traditional LSD technique. The proposed channel matrix mimics the generator matrix of an An lattice. The condition number (λ) of the proposed channel matrix is less than those of traditional LSD matrices and the initial radius (d) is deterministic, thus the proposed technique achieves significant improvement in performance and complexity reduction.

The conventional Equal Gain Transmission and Maximum Ratio Combining (EGT/MRC) requires nonlinear optimization to find the optimal beamforming vector at the receiver. This study shows that the optimal beamforming vector can be easily formed by the geometrical concepts. Accordingly, a novel transmission/reception scheme, called the Scalar Equal Gain Transmission and Generalized Maximum Ratio Combining (SEGT/GMRC), is presented and examined. The Monte-Carlo simulations validate the theory and it is shown that the optimal beamforming vector formed by SEGT is the same as the one determined by the nonlinear optimizer. The closed-form analytical error performance of the SEGT/GMRC scheme is also derived for multiple input single output (MISO) communications. This study also introduces the new limited-feedback geometrical codebooks, called the Quantized Equal gain (QE) codebooks, which can be easily installed as symbol mappers. These codebooks are based on quantized SEGT/GMRC, which eliminates the need for any iterative searching scheme, such as exhaustive search at the receiver. The minimum amount of feedback bits depends on the modulation scheme, where a general M-PSK modulation requires at least log2M bits per quantized phase angle. It is also shown that BPSK modulation requires at least 2 bits per quantized phase angle for near-optimal performance.

In this study, a new method for Decode-Distributed Beamforming (D-DB) relaying is proposed. Each relay node decodes the source symbol by maximum likelihood detection. The detected symbol is entered into the stored Quantized Equal-gain (QE) codebook, where the label of the phase region is provided by a feedback link from the destination node. Therefore, the proposed relay network forms a Decode-Distributed QE (D-DQE) relay network. The performances of the D-DQE codebooks are examined by Monte-Carlo simulations, in which the feedback links and channel estimations are assumed to be error-free. The simulation results reveal that the symbol error rates of the D-DQE relay system improve the error performance of the QE codebooks when relay nodes are close to the source node. When error-free feedback bits are provided, the performance of the proposed D-DQE is better than that of Alamouti's Decode-Distributed Space-Time Coding (D-DSTC) relay network. The weakest relays are rejected to improve the performance of the D-DQE codebooks and reduce the number of feedback bits. This relay network is called Decode-Relay Rejection for Distributed Beamforming (D-RRDB) relay networks.