Various precoders have been recently studied by the wireless community to combat the channel fading effects. Two prominent precoders are implemented with the discrete Fourier transform (DFT) and Walsh-Hadamard transform (WHT). The WHT precoder is implemented with less complexity since it does not need complex multiplications. Also, spreading can be applied sparsely to decrease the transceiver complexity, leading to sparse DFT (SDFT) and sparse Walsh-Hadamard (SWH). Another relevant topic is the design of iterative receivers that deal with inter-symbol-interference (ISI). In particular, many detectors based on expectation propagation (EP) have been proposed recently for channels with high levels of ISI. An alternative is the maximum a-posterior (MAP) detector, although it leads to unfeasible high complexity in many cases. In this paper, we provide a relatively low-complexity \textcolor{black}{computation} of the MAP detector for the SWH. We also propose two \textcolor{black}{feasible methods} based on the Log-MAP and Max-Log-MAP. Additionally, the DFT, SDFT and SWH precoders are compared using an EP-based receiver with one-tap FD equalization. Lastly, SWH-Max-Log-MAP is compared to the (S)DFT with EP-based receiver in terms of performance and complexity. The results show that the proposed SWH-Max-Log-MAP has a better performance and complexity trade-off for QPSK and 16-QAM under highly selective channels, but has unfeasible complexity for higher QAM orders.

翻译：近期无线通信领域研究了多种预编码器以对抗信道衰落效应。其中两种典型预编码器基于离散傅里叶变换（DFT）和沃尔什-哈达玛变换（WHT）实现。由于无需复数乘法，WHT预编码器实现复杂度更低。此外，可采用稀疏扩展技术降低收发复杂度，由此衍生出稀疏DFT（SDFT）和稀疏沃尔什-哈达玛（SWH）方案。另一个相关研究方向是应对符号间干扰（ISI）的迭代接收机设计。近年来，针对高ISI信道提出多种基于期望传播（EP）的检测器。最大后验（MAP）检测器作为替代方案，但在多数场景下计算复杂度过高。本文为SWH提出了一种计算复杂度相对较低的MAP检测器实现方案，同时基于Log-MAP和Max-Log-MAP提出了两种可行方法。此外，采用基于EP接收机配合单抽头频域均衡，比较了DFT、SDFT和SWH预编码器的性能。最后，将SWH-Max-Log-MAP与基于EP接收机的(S)DFT方案进行性能与复杂度对比。结果表明，在高选择性信道条件下，所提出的SWH-Max-Log-MAP在QPSK和16-QAM调制中展现出更优的性能-复杂度折衷，但对高阶QAM调制则因复杂度不可行而受限。