Performance Bounds in OFDM Channel Prediction Ian C. Wong and Brian L. Evans Wireless Networking and Communications Group The University of Texas at Austin December 1, 2018

Adaptive Orthogonal Frequency Division Multiplexing (OFDM) Adjust transmission based on channel information Maximize data rates and/or improve link quality Problems Feedback delay - significant performance loss [Souryal & Pickholtz, 2001] Volume of feedback - power and bandwidth overhead Internet Back haul Base Station Doubly-selective Wireless Channel Mobile Feedback channel information December 1, 2018

Prediction of Wireless Channels Use current and previous channel estimates to predict future channel response Overcome feedback delay Adaptation based on predicted channel response Lessen amount of feedback Predicted channel response may replace direct channel feedback … Put description of delta December 1, 2018

Previous Work Prediction on each subcarrier [Forenza & Heath, 2002] Each subcarrier treated as a narrowband autoregressive WSS process [Duel-Hallen et al., 2000] Prediction using pilot subcarriers [Sternad & Aronsson, 2003] Used unbiased power prediction [Ekman, 2002] Prediction on time-domain taps [Schafhuber & Matz, 2005] Used adaptive prediction filters Applied to predictive equalization December 1, 2018

Previous Work Comparison of prediction approaches using unified framework [Wong et al, 2004] Time-domain approach gives best MSE performance vs. complexity tradeoff Prediction using high-resolution frequency estimation [Wong & Evans, 2005] Shown to significantly outperform previous methods with same order of complexity Key idea – 2-step 1-dimensional frequency estimation Remove articles December 1, 2018

Summary of Main Contributions Simple, closed-form expression for MSE lower bound in OFDM channel prediction for any unbiased channel estimation/prediction algorithm Yields important insight into designing OFDM channel predictors without extensive numerical simulation Simple, closed-form expression for MSE lower bound in OFDM channel prediction using 2-step 1-dimensional frequency estimation December 1, 2018

System Model OFDM baseband received signal Perfect synchronization and inter-symbol interference elimination by the cyclic prefix Flat passband for transmit and receiver filters over used subcarriers Deterministic wideband wireless channel model Far-field scatterer (plane wave assumption) Linear motion with constant velocity Small time window (a few wavelengths) Put legend when with equation December 1, 2018

Pilot-based Transmission Comb pilot pattern Least-squares channel estimates t f … Dt Df Get rid of pattern, legend of LS estimates, remind key parameters, Nt nf,… December 1, 2018

Prediction as parameter estimation Channel is a continuous non-linear function of the 4M-length channel parameter vector Deterministic channel prediction premise Estimate parameters of channel model from the least-squares channel estimates 2-dimensional sum of complex sinusoids in white noise Extrapolate the model forward December 1, 2018

Cramer-Rao Lower Bound (CRLB) CRLB for narrowband case [Barbarossa & Scaglione, 2001] [Teal, 2002] First-order Taylor approximation Expensive numerical evaluations necessary Monte-Carlo generation of parameter vector realizations CRLB for function of parameters [Scharf, 1991] Fix this slide… add legend for notation December 1, 2018

Closed-form asymptotic MSE bound Using large-sample limit of CRLB matrix for general 2-D complex sinusoidal parameter estimation [Mitra & Stoica, 2002] Much simpler expression Achievable by maximum-likelihood and nonlinear least-squares methods Monte-Carlo numerical evaluations not necessary Add box to equation , legend for notation Nf Nt Dt Df, n, k kbar, add bullet lead in to equation December 1, 2018

Insights from the MSE expression Amplitude & phase error variance Doppler frequency & phase cross covariance Doppler frequency error variance Time-delay & phase cross covariance Time-delay error variance Linear increase with 2 and M Dense multipath channel environments are the hardest to predict [Teal, 2002] Quadratic increase in n and |k| with f and  estimation error variances Emphasizes the importance of estimating these accurately Nt, Nf, Dt and Df should be chosen as large as possible to decrease the MSE bound Fix equations, put arrows or animation to explain quadratic increase, say what is new and known, explain some tradeoffs… December 1, 2018

High-performance OFDM channel prediction algorithm [Wong & Evans, 2005] In wireless channels, a number of sinusoidal rays typically share a common time delay Proposed 2-step 1-D estimation Lower complexity with minimal performance loss Rich literature of 1-D sinusoidal parameter estimation Allows decoupling of computations between receiver and transmitter December 1, 2018

Asymptotic MSE Lower Bound for 2-step estimation Amplitude & phase error variance Doppler frequency & phase cross covariance Doppler frequency error variance Time-delay error variance Used asymptotic CRLB matrix for 1-D sinusoidal parameter estimation [Stoica et al., 1997] Complex amplitude estimation error variance of first step used as the “noise variance” in second step For large prediction lengths, i.e. large n Put it in dB after 2.5 December 1, 2018

IEEE 802.16 Example December 1, 2018

MSE vs. SNR, n=500 Legend in order as they appear in plot December 1, 2018

MSE vs. n, SNR=10 dB Same comment as previous,say prediction length in symbols December 1, 2018

Conclusion Derived simple, closed-form expressions for MSE lower bound for OFDM channel prediction Expensive numerical evaluation unnecessary Yields valuable insight into design of channel predictors Block lengths and downsampling factors should be made as big as possible Estimation of Doppler frequencies/time delays very important Dense multipath channels may not be predictable MSE Lower bound for 2-step OFDM channel prediction Small penalty compared to above bound Basis for a high-performance channel prediction algorithm Proposed 2-step 1-D prediction algorithm is close to the lower bound December 1, 2018