# A Minimum Mean-Square-Error (MMSE) Decoder for Quasi-Orthogonal Space-Time Block Code
## 1 Introduction  
Transmit diversity has been applied for mitigating the effect of multi-path fading, as it can improve the performance without adding additional bandwidth and increasing transmit power. A space-time block code (STBC) is most widely used to provide transmit diversity [1; 2; 3; 4; 5; 6; 7; 8; 9]. The orthogonal STBC (O-STBC) was introduced to achieve full diversity and a full transmission rate for two transmit antennas [1]. The O-STBC has computational complexity for two transmit antennas, but that has degraded performance compared to space-time trellis codes. The STBC from orthogonal design (STBC-O) was developed from the theory of orthogonal designs [2]. The STBC-O is a generalization of the O-STBC for multiple antennas systems. However, the STBC-O is not capable of obtaining the full transmission rate for more than two transmit antennas systems. To overcome this drawback, several quasi orthogonal-STBC (QO-STBC) algorithms have been introduced [3; 4; 5] for the four antennas system. In [3], the columns of the transmission matrix were separated into groups according to a quasi-orthogonal design, in which the columns in each group are non-orthogonal and different groups are orthogonal. The algorithm was derived by minimizing the non-orthogonality that arises from increasing the transmission rate [4]. In [3; 4], the QO-STBC algorithms have the full transmission rate for the four antennas system, but the detection matrix of these algorithms has interference terms, which are raised from neighboring channels. This phenomenon increases a computational complexity and decreases the bit-error-rate (BER). To solve these problems, the QO-STBC was developed using the transformation matrix in [5]. However, the BER result of this algorithm is higher than that of the conventional algorithms [10]. In addition, the QO-STBC faces a rank-deficient problem when the detection matrix is singular.  
In this paper, a decoding method for the QO-STBC is proposed to improve the performance for systems that have four antennas, and to address the rank-deficient problem. The proposed algorithm is based on the minimum mean-square-error (MMSE) decoder. This decoder has a lower BER thanthat of the zero forcing (ZF) decoder. However, a limitation is that the MMSE decoder is not used in practical applications, as the noise variance is unknown in real-time applications [11]. To overcome this drawback, the proposed algorithm is derived by estimating the noise variance. Moreover, to reduce computational complexity, elementary matrices are used in the proposed algorithm without matrix inversion. In short, the proposed algorithm achieves a lower BER than other algorithms, while having low computational complexity without matrix inversion. Furthermore, the proposed algorithm is able to overcome a rank-deficient problem because a regularization term, which is an estimated noise variance, is added to the detection matrix. The simulation results illustrate that the BER of the proposed algorithm is smaller than that of the conventional algorithms in various environments.  
In this paper, bold symbols are used for column vector (lower-case) and matrix (upper-case). The notations are used as follows;  
\[\begin{array}{ll}(\cdot)^{T}&\text{transpose of a vector or matrix;}\\ (\cdot)^{*}&\text{conjugate of a complex number;}\\ (\cdot)^{H}&\text{Hermitian transpose of a vector or matrix.}\end{array}\]