# Low-Complexity QRD-\(M\) with Path Eliminations in MIMO-OFDM Systems
## 1 Introduction  
Recently, multiple input multiple output (MIMO) systems have been used to increase spectral efficiency and channel capacity without high transmit power or wide bandwidth in a rich scattering wireless channel [1; 2]. A MIMO system has been used in combination with orthogonal frequency division multiplexing (OFDM, a well-known multi-carrier transmission system), and is called MIMO-OFDM [3]. The MIMO-OFDM systems have high frequency efficiency and strong frequency-selective fading. However, since the received signal is composed of many distorted transmit signals by the fading channel, a complex signal detector is required for reliable communication. So, huge MIMO-OFDM systems which use high modulation order and have many antennas require a complex signal detector for accurate estimation of the transmit symbols. The optimal detection in MIMO-OFDM systems is maximum likelihood (ML) [4; 5]. The ML calculates squared Euclidean distance (SED) between received symbols and all reference symbols for optimal error performance. However, the main disadvantage of the ML is its extremely high complexity. The number of metrics in the ML increases exponentially with respect to the number of transmit antennas and the modulation order. The QR decomposition-\(M\) algorithm (QRD-\(M\)) is the simplified ML detection based on tree search. The performance between QRD-\(M\) and ML detectors is not the same, but is similar. Unlike the ML, the QRD-\(M\) calculates accumulated squared Euclidean distance (ASED) at each layer, and selects only \(M\) small distances. These \(M\) small distances are usually called survival paths. To approach the error performance of the ML, \(M\) must be large, but it requires a very high complexity. The main reason for the high complexity of the QRD-\(M\) is that it contains unnecessary survival paths among the \(M\) survival paths. Several detections were developed to reduce the complexity of the conventional QRD-\(M\) in [6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. Among several detections, an adaptive QRD-\(M\) was developed in [10], a modified tree structure QRD-\(M\) was developed in [11],and a lattice reduction (LR)-aided adaptive QRD-\(M\) was developed in [12]. In [10; 11; 12], adaptive path eliminations were used at each layer for low complexity. However, these proposed detections still have high complexity since the average number of unnecessary survival paths is high despite the path eliminations. Specifically, modified tree structure QRD-\(M\) has high complexity when the number of transmit antennas is high because the conventional QRD-\(M\) is applied from the \(\lceil N_{t}/O\rceil\)-th layer to the first layer, where \(\lceil\cdot\rceil\) denotes rounding function which towards positive infinity and \(O\) denotes modulation order. Therefore, this paper proposes very low complexity QRD-\(M\) which has the same error performance as ML. At first, the proposed QRD-\(M\) calculates the threshold at each layer. Then, ASEDs are compared with the threshold and paths which are larger than the threshold are eliminated. The path eliminations continue until the first layer for very low complexity. For the efficient signal detection, the threshold is calculated adaptively at each layer, depending on the modulation order. The complexity of the proposed QRD-\(M\) is much lower than that of other QRD-\(M\) detection schemes in [10; 11; 12] because of the efficient threshold value considering the modulation order.