Nonlinear Precoding for ISI Channels Frederick Lee Stanford University October 21, 2004 AIM Workshop on Time-Reversal Communications in Richly Scattering Environments

H(z) wkwk ykyk nknk ŵkŵk H(z) Rx: ZF-LE Motivation of ZF-THP (1) whitened matched filtered channel (causal & minimum-phase) Problem: Noise Enhancement

H(z) wkwk ykyk nknk ŵkŵk H(z) Rx: ZF-DFE Motivation of ZF-THP (2) Problem: Error Propagation

Tx: ZF-LE H(z) H(z) - 1 wkwk xkxk ykyk nknk ŵkŵk - Motivation of ZF-THP (3) Problem: Increase in average & peak power of x k

Motivation of ZF-THP (4) Power of x k only increases slightly after modulo operator H(z) H(z) - 1 Mod wkwk xkxk ykyk nknk ŵkŵk - zkzk

ZF-THP: 1-D Constellation (1) 13M-1-M+1 -3 M-ary PAM w k - z k wkwk xkxk H(z) - 1 Mod 2M wkwk xkxk - H(z) - 1 wkwk xkxk - vkvk akak zkzk zkzk (some integral multiple of 2M)

ZF-THP: 1-D Constellation (2) 13M-1-M+1 -3 Extended signal set V w k - z k wkwk xkxk ykyk H(z) ykyk nknk Mod ŵkŵk H(z) - 1 wkwk xkxk - vkvk akak

ZF-THP: 2-D Constellation (1) H(z) - 1 Mod w k I + jw k Q - x k I + jx k Q u k I + ju k Q ukIukI Mod 2M ukQukQ xkIxkI xkQxkQ j Mod x k I + jx k Q u k I + ju k Q Square QAM = (M-ary PAM) 2

ZF-THP: 2-D Constellation (2) 16-QAM

ZF-THP: 2-D Constellation (3) Construct a 2-D lattice = {k 1 v 1 + k 2 v 2 }, where v 1, v 2 are linearly independent vectors & k 1, k 2 are integers Find a region R such that R+ fills up entire 2-D space with no gap

ZF-THP: 2-D Constellation (4)

ZF-FLP (1) (Assume f k 2 ) H(z) - 1 w k = x k ykyk nknk ŵkŵk fkfk H(z) k vkvk (w k 2 )

ZF-FLP (2) v1v1 v2v2 Voronoi region w k = f k = v k = y k =

ZF-FLP (3) H(z) - 1 wkwk xkxk ykyk nknk ŵkŵk - fkfk mkmk - k x'kx'k vkvk

ZF-FLP (4) v1v1 v2v2 Voronoi region w k = f k = v k = y k = x k = x' k =

ZF-FLP (5) H(z) H(z) - 1 wkwk xkxk ykyk nknk ŵkŵk - Mod fkfk mkmk H(z) k x'kx'k

Comparisons of THP and FLP THP Dependent on boundary region R and constellation size No restrictions for channels with spectral zeros FLP Dependent on signal lattice (Independent of R and constellation size) Receiver filter unstable for channels with spectral zeros, which leads to error propagation

Performance Example Source: R. Fischer & J. Huber, Comparison of precoding schemes for digital subscriber lines, IEEE Trans. Commun., Mar. 1997

THP for Multi-User Broadcasting Channels s1's1' mod I - B HF sK'sK' mod n x y1y1 yKyK... mod sksk Element-Wise Operation Feedback Filter (Triangular) Channel (Flat or ISI) Feedforward Filter Joint (vector/matrix) processing at BSIndividual (scalar) processing for each user