April 22, 2023

Performance Bounds inOFDM Channel Prediction

Ian C. Wong and Brian L. EvansWireless Networking and Communications Group

The University of Texas at Austin

April 22, 2023

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

InternetBack haul

Base Station

Doubly-selective Wireless Channel

Mobile

Feedback channel information

April 22, 2023

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

…

April 22, 2023

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

April 22, 2023

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

April 22, 2023

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-step1-dimensional frequency estimation

April 22, 2023

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)

System Model

April 22, 2023

Comb pilot pattern

Least-squares channel estimates

Pilot-based Transmission

t

f …

Dt

Df

April 22, 2023

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

April 22, 2023

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]

April 22, 2023

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

April 22, 2023

Insights from the MSE expression

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

Amplitude & phase error variance

Doppler frequency & phase cross covariance

Doppler frequency error variance

Time-delay & phase cross covariance

Time-delay error variance

April 22, 2023

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

April 22, 2023

Asymptotic MSE Lower Bound for 2-step estimation

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

Amplitude & phase error variance

Doppler frequency & phase cross covariance

Doppler frequency error variance

Time-delay error variance

April 22, 2023

IEEE 802.16 Example

0 0.5 1 1.5 2 2.5 3x 10

-6

0

0.1

0.2

0.3

0.4

0.5

Time delay

Pat

h po

wer

April 22, 2023

MSE vs. SNR, n=500

10 15 20 25 30 35-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

SNR in dB

NM

SE

in d

B

High-performance predictionMSE Lower Bound, 2-StepMSE Lower Bound

April 22, 2023

MSE vs. n, SNR=10 dB

50 100 150 200 250 300 350 400 450-20

-18

-16

-14

-12

-10

Prediction length in symbols (n)

NM

SE

in d

B

High-performance predictionMSE Lower Bound, 2-StepMSE Lower Bound

April 22, 2023

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