  In this paper, the issue of designing an iterative-detection-and-decoding
(IDD)-aided receiver, relying on the low-complexity probabilistic data
association (PDA) method, is addressed for turbo-coded
multiple-input-multiple-output (MIMO) systems using general M-ary modulations.
We demonstrate that the classic candidate-search-aided bit-based extrinsic
log-likelihood ratio (LLR) calculation method is not applicable to the family
of PDA-based detectors. Additionally, we reveal that, in contrast to the
interpretation in the existing literature, the output symbol probabilities of
existing PDA algorithms are not the true a posteriori probabilities (APPs) but,
rather, the normalized symbol likelihoods. Therefore, the classic relationship,
where the extrinsic LLRs are given by subtracting the a priori LLRs from the a
posteriori LLRs, does not hold for the existing PDA-based detectors. Motivated
by these revelations, we conceive a new approximate Bayesian-theorem-based
logarithmic-domain PDA (AB-Log-PDA) method and unveil the technique of
calculating bit-based extrinsic LLRs for the AB-Log-PDA, which facilitates the
employment of the AB-Log-PDA in a simplified IDD receiver structure.
Additionally, we demonstrate that we may dispense with inner iterations within
the AB-Log-PDA in the context of IDD receivers. Our complexity analysis and
numerical results recorded for Nakagami-m fading channels demonstrate that the
proposed AB-Log-PDA-based IDD scheme is capable of achieving a performance
comparable with that of the optimal maximum a posteriori (MAP)-detector-based
IDD receiver, while imposing significantly lower computational complexity in
the scenarios considered.
