Closed-Form Bit Error Probability of ZF Detection for OFDM-M-MIMO Systems Using Effective Noise PDF

This paper derives the closed-form bit error probability (BEP) of massive multiple-input, multiple-output (M-MIMO) systems using orthogonal frequency-division multiplexing (OFDM) and zero-forcing (ZF) detection. We improve the BEP accuracy by increasing the Neumann series expansion (NSE) to second order for the system that we previously analyzed by deriving the probability distribution function (PDF) of the effective noise. The proposed PDF is then utilized to evaluate the BEP, the PDF of output signal-to-noise ratio (SNR), and the outage probability as a function of the output SNR of the system. Furthermore, a simplified closed-form expression for the effective noise PDF, in terms of the Gaussian distribution, and the noise variance are firstly derived in this paper for simplifying the performance analysis. Monte-Carlo simulation results confirm that the outcome from the derived equation and the approximation closely matched those obtained by simulation. In addition, we employ the proposed noise variance to estimate the log-likelihood ratio (LLR) instead of the approximate noise variance for the low-complexity soft-output ZF detection. The computational complexity of the proposed detection is thus significantly reduced, whereas its bit error rate (BER) is lower than that of the classical detection. Focusing on a
$10\times 200$
Coded-OFDM-M-MIMO system, 97.81% of multiplications, required for producing the LLR from the estimated symbol, were minimized by utilizing the proposed detection. Therefore, the derived equations can be efficiently used for analyzing the performance of OFDM-M-MIMO systems, and reducing the computational complexity of the soft-output ZF detection.