960 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015 Characterizing the Impact of Feedback Delays on Wideband Rate Adaptation Jobin Francis and Neelesh B. Mehta, Senior Member, IEEE Abstract—In contemporary orthogonal frequency division mul- tiplexing (OFDM) systems, such as Long Term Evolution (LTE), LTE-Advanced, and WiMAX, a codeword is transmitted over a group of subcarriers. Since different subcarriers see different channel gains in frequency-selective channels, the modulation and coding scheme (MCS) of the codeword must be selected based on the vector of signal-to-noise-ratios (SNRs) of these subcarriers. Exponential effective SNR mapping (EESM) maps the vector of SNRs into an equivalent flat-fading SNR, and is widely used to simplify this problem. We develop a new analytical framework to characterize the throughput of EESM-based rate adaptation in such wideband channels in the presence of feedback delays. We derive a novel accurate approximation for the throughput as a function of feedback delay. We also propose a novel bivariate gamma distribution to model the time evolution of EESM between the times of estimation and data transmission, which facilitates the analysis. These are then generalized to a multi-cell, multi-user scenario with various frequency-domain schedulers. Unlike prior work, most of which is simulation-based, our framework encom- passes both correlated and independent subcarriers and various multiple antenna diversity modes; it is accurate over a wide range of delays. Index Terms—OFDM, adaptation, Exponential Effective SNR Mapping (EESM), feedback delay, scheduling, co-channel inter- ference, bivariate gamma distribution. I. I NTRODUCTION C URRENT and next generation wireless systems, such as Long Term Evolution (LTE), LTE-Advanced, and WiMAX, have been designed to meet the incessant demand for higher data rates. They employ orthogonal frequency division multiplexing (OFDM) because it avoids inter-symbol and intra- cell interference. OFDM divides the available bandwidth into narrowband orthogonal subcarriers. To efficiently utilize the scarce bandwidth, adaptive modulation and coding (AMC), in which rate is adapted, and scheduling, in which the user that is transmitted to is adapted, are extensively utilized. In these OFDM systems, a codeword is transmitted over a group of subcarriers [2]. Due to the frequency-selective nature of the channel, different subcarriers see different gains. Thus, the block error rate (BLER) is a function of the vector of Manuscript received April 15, 2014; revised August 16, 2014; accepted October 4, 2014. Date of publication October 15, 2014; date of current version February 6, 2015. This research is partially supported by a philanthropic grant from the Broadcom Foundation, USA. The associate editor coordinating the review of this paper and approving it for publication was L. Lai. The authors are with the Department of Electrical Communication Engi- neering, Indian Institute of Science (IISc), Bangalore 560 012, India (e-mail: jobin@ece.iisc.ernet.in; nbmehta@ece.iisc.ernet.in). Digital Object Identifier 10.1109/TWC.2014.2363083 signal-to-noise-ratios (SNRs) of the subcarriers assigned to the codeword. Consequently, the AMC scheme must select the modulation and coding scheme (MCS) based on this vector of SNRs. Further, no closed-form expression for the BLER in vector channels is available. This is unlike the well-studied problem of AMC over narrowband channels, in which the MCS choice is based on just one SNR [3]. An important practical issue in these systems is the delay between the time of estimation of the channel gains and the time of data transmission. This delay ranges from milliseconds to tens of milliseconds [2]. It degrades the throughput because either the AMC scheme may overestimate or underestimate the rate or a sub-optimal user may get scheduled. Characterizing the effect of feedback delay on AMC over wideband channels is the focus of this paper. AMC over frequency-selective fading channels, in principle, requires a cumbersome multi-dimensional lookup table to map the vector of SNRs to the MCS. Hence, link quality metrics (LQMs) such as exponential effective SNR mapping (EESM) have been developed to simplify this problem [4]–[6]. EESM maps the vector of subcarrier SNRs seen by the codeword into an effective flat-fading SNR, which is interpreted to be the equivalent SNR in an additive white Gaussian noise (AWGN) channel for that MCS. Thus, EESM reduces the problem of AMC over a frequency-selective channel to that over a frequency-flat channel. If γ i (t) denotes the SNR of the ith subcarrier at time t, for 1 ≤ i ≤ N sc , then the effective SNR γ (m) eff (t) for MCS m is defined as γ (m) eff (t)= −β m log  1 N sc N sc  i=1 exp  − γ i (t) β m   , (1) where β m > 0 is an MCS-dependent scaling parameter. Its ac- curacy has been conclusively established in several prior works [5], [6]. It has been widely used in system-level simulations of LTE and WiMAX [7], [8] and to generate feedback to the base station (BS) [9]. A. Related Literature We first survey the simpler problem of characterizing the effect of feedback delay in narrowband systems, which has been widely studied in the literature. The effect of channel estimation errors, feedback errors, and feedback delay on throughput is analyzed in [10] for a single cell, multi-user system with a greedy scheduler. However, subcarrier-level scheduling and adaptation is assumed. Thus, the techniques for adaptation 1536-1276 © 2014 IEEE. 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