Does Vector Gaussian Approximation After LMMSE Filtering Improve the LLR Quality?

In this letter, we investigate the extrinsic log-likelihood ratio (LLR) computation of a soft-input soft-output equalizer used in a turbo equalization system. The optimum LLRs are obtained by a maximum a posteriori -based equalizer, which may be computationally expensive. Thus, several reduced-complexity equalizers have been proposed. The most promising approach first applies linear minimum mean square error filtering to the channel output and then computes the LLRs based on a scalar Gaussian approximation of the filter output. The resulting LLRs can be viewed as an approximation of the optimum LLRs. In order to improve the approximation, we investigate the computation of the LLRs based on a vector Gaussian approximation of the filter output, which incorporates the correlation between the estimated symbols after filtering. Surprisingly, it turns out that both approaches, although their derivation is different, give the same LLRs. We verify this remarkable result through an analytical proof and bit error ratio simulations.