IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 7, JULY 2015 993 Soft-Output MIMO Detectors with Channel Estimation Error Louay M.A. Jalloul , Senior Member, IEEE, Sam P. Alex , and Mohammad M. Mansour , Senior Member, IEEE Abstract—New expressions for the soft decision bit log-likelihood ratio (LLR) of a MIMO system using quadrature amplitude modu- lation (QAM) taking into account channel estimation error (CEE). The bit LLR for the maximum likelihood (ML) and the linear min- imum mean-squared error (MMSE) receivers are derived, showing in both receivers explicit scaling of the LLR that is a function of the QAM symbol and the CEE variance. These new expressions for the LLRs are used to show that only modest improvements in the link performance are achieved relative to the LLRs that do not take into account the CEE. This indicates that separating the detector design from channel estimation does not significantly impact the system performance, which leads to simplifications in the overall receiver implementation. Index Terms—Channel estimation error, log-likelihood ratio, maximum likelihood, MIMO, MMSE, QAM. I. INTRODUCTION C ONSIDER a bit-interleaved coded modulation system that uses multi-level QAM and spatial multiplexing to transmit parallel data streams from multiple antennas that are received by multiple antennas [1]. Soft-decision decoding requires the bit LLR as input to a soft-input channel decoder. Using the Max- log-MAP approximation, it was shown in [2] that the LLR can be approximated by the so-called dual-minima metric. The bit LLR was computed in [3] for a single-input single-output (SISO) coded OFDM system and multi-level QAM with the assumption the receiver has full knowledge of the channel state informa- tion (CSI). This assumption is typically not valid in a mobile radio multi-path fading channel, especially when the number of pilot symbols is scarce and the signal-to-noise ratio (SNR) may be low. Thus, not accounting for imperfect CSI at the re- ceiver leads to sub-optimal LLR generation. The LLR metric for a SISO system with QAM modulation taking into account CEE was derived in [6], [7]. An inter-symbol interference (ISI) channel was considered in [8] using a posteriori detection prob- ability and MMSE receivers with channel estimation error in the context of iterative equalization and decoding. The bit LLR was derived for MIMO systems in [9]–[11] with the assumption of Manuscript received September 18, 2014; revised November 10, 2014; ac- cepted November 19, 2014. Date of publication November 26, 2014; date of current version December 17, 2014. The associate editor coordinating the re- view of this manuscript and approving it for publication was Prof. Ron Dabora. L. M.A. Jalloul and S. P. Alex are with Broadcom Corporation, Sunnyvale, CA 94086 USA (e-mail: jalloul@ieee.org). M. M. Mansour is with the Department of Electrical and Computer En- gineering at the American University of Beirut, Beirut, Lebanon (e-mail: mmansour@ieee.org). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2014.2374425 ideal CSI. A MIMO system was studied in [12] that considered the effect of channel estimation, however only constant modulus -ary phase shift keying modulation was considered. Contributions: The contributions in this letter are the new soft bit LLR metrics for the ML and linear MMSE receivers in a MIMO system that uses multi-level QAM, taking into account the CEE. This model was considered in [13], [14] with a single stage MMSE and [15] for a two-stage parallel interference can- celling receiver. However, the weight vector derivation for the MMSE in [13]–[15] is conditioned only on the channel esti- mate. In our ML and MMSE receiver developments, we condi- tion on both the channel estimate and the received QAM symbol, leading to the improved LLR metrics. The rest of the letter is organized as follows. Section II de- scribes the system model. The LLR generation in a MIMO system with imperfect CSI is described in Section III for the ML receiver and Section IV for the linear MMSE receiver with simulation results in Section V. The effect of colored noise is treated in Section VI. Finally, concluding remarks are provided in Section VII. Notation: Lower case and upper case bold letters denote column vectors and matrices, respectively; denotes the identity matrix of size ; and denote the absolute value and -norm, respectively; denotes the complex conjugate transpose operation and denotes the trace of the matrix ; and is the expectation of the random variable . II. SYSTEM MODEL For each code word, the input information bits are channel encoded, bit interleaved, mapped into QAM symbols and then transmitted on MIMO spatial layers. The input-output rela- tionship is described as (1) where is a complex-valued vector, denotes the number of receive antennas, is a matrix which denotes the complex channel frequency response with the assumption that . The transmitted symbol is a complex-valued vector, where and is the -ary QAM constellation of size . Each symbol is formed from a set of coded bit-interleaved sequence over GF(2), and normalized such that . The noise is modeled as a zero-mean complex Gaussian circularly symmetric (ZMCGCS) random vector with covariance . This system model applies to single-carrier systems over flat fading channels (ISI-free) as well as OFDM systems over frequency-selective channels where (1) applies to each subcarrier. 1070-9908 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.