  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.
