INSTITUT FÜRNACHRICHTENTECHNIK UND HOCHFREQUENZTECHNIK

MIMO Communications and Algorithmic Number Theory

G. Matz

joint work with D. Seethaler

Institute of Communications and Radio-Frequency EngineeringVienna University of Technology (VUT)

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 2 –

RX antennas

Setting the Stage

• MIMO communications:

• Algorithmic number theory (ANT) is the study of algorithms that perform number theoretic computations

(Source: Wikipedia)

Examples: primality test, integer factorization, lattice reduction

TX RX. . .

. . . channel

TX antennas

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 3 –

Outline

• MIMO detection and ANT

• MIMO precoding and ANT

• Precoding via Vector Perturbation

• Approximate Vector Perturbation using Lattice Reduction

• Lattice Reduction using ANT: Brun‘s Algorithm

• Simulation Results

• Conclusions

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 4 –

Multi-Antenna Broadcast (Downlink)

System model:pr

ecod

ing

user #1

. . .

. . . channel

M TX antennas

user #k

user #K

users, each with one antenna

Users cannot cooperate shift MIMO processing to TX precoding

withMIMO I/O relation:

K symbols

CSI at TX required

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 5 –

• here, and is an integer perturbation vector

• precoder performs channel inversion and vector perturbation

Vector Perturbation (Peel et al.)

TX vector:

Receive symbols:

• follows from

• RX-SNR equals 1/ choose z such that s(z) is “short“

Remaining RX processing:

• get rid of z via modulo operation

• quantization w.r.t. symbol alphabet

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 6 –

• Optimum vector perturbation maximizes RX-SNR:

Choice of Perturbation Vector

• Suboptimum precoding: e.g. Tomlinson-Harashima precoding (THP)

• For channels with large condition number

- sphere encoding has high complexity

- THP etc. have poor performance

• Small condition number: all methods work fast and well

• Implementation: sphere encoder

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 7 –

Structure of Channel Singular Values

104

103

2

101

0 20 40 60 80 100

104

03

102

101

condition number0 20 40 60 80 100

smallest singular value and associated singular vector v cause problems

M=K=4

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 8 –

Vector Perturbation for Poor Channel Condition

Example: BPSK, real-valued channel & noise, M=K=2, = 2

TX vector perturbed versions of TX vector

search TX vector that - is integer - has small length - is orthogonal to v

approximate integer relation (ANT)

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 9 –

Relation of MIMO and ANT

Precoding at Tx Detection at Rx

Approximate Integer

Relations

Simultaneous Diophantine

Approximations

duality

duality

MIMO

ANT

poorly conditioned channels

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 10 –

• Try to find “better“=reduced lattice basis

Vector Perturbation Using Lattice Reduction (LR)

• LR-assisted vector perturbation ( , )

- cost function:

- solve or use THP approximation

- use as perturbation vector

• View as basis of a lattice

• All lattice basis are related via a unimodular matrix:

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 11 –

Lattice Reduction

• Orthogonality defect (quality of lattice basis):

left channelsingular vectors

channel singular values

- LLL-LR assisted THP achieves full diversity

- but LLL can be computationally intensive

• Most popular LR method: Lenstra-Lenstra-Lovász (LLL) algorithm

• LR: find achieving small and thus small

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 12 –

Integer Relation Based LR

• For poorly conditioned channels, only one singular value is small

• To achieve small , vectors must be

sufficiently orthogonal to singular vectors with small singular values

find integer vectors that are sufficiently orthogonal to

• This is the approximate integer relation (IR) problem in ANT

• IR-LR focuses on one singular vector (in contrast to LLL-LR)

- some performance loss

- significantly smaller complexity

• Goal: more efficient LR method

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 13 –

- small

- small

• can be made arbitrarily small using long vectors

Approximate Integer Relations

• Approximate IR: achieve small with as short as possible

• Tradeoff:

governed by channel singular values

• Can be realized very efficiently using Brun‘s algorithm

• large will increase

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 14 –

Brun’s Algorithm

• Initialization:

• Find

• Calculate

• Replace

• is also updated recursively and can be made arbitrarily small

(update of )

• Very simple: scalar divisions, quantizations, and vector updates

repe

at u

ntil

term

inat

ion

con

diti

on is

sa

tisfie

d

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 15 –

Performance of Brun’s Algorithm

average

no. of iterations

average

• Example using and averaging over 1000 randomly picked

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 16 –

terminate if update of does not decrease

Lattice Reduction via Brun’s Algorithm

• at each iteration, is a basis for

• recall: LR aims at minimizing

• we are just interested in channels with one small singular value

• in this case,

apply Brun’s algorithm to any column of

Termination condition

Calculation of

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 17 –

Simulation Results (1)

THP w. LLL

SNR

THP w. Brun

•

• iid Gaussian channel

• 4-QAM

• Iterations on average:

Sym

bol

Err

or

Ra

te

LR using Brun‘s algorithm can exploit large part of available diversity

Sphere encoding(optimal)

THP - Brun: 2.5 - LLL: 12.9

• A Brun iteration is less complex than an LLL iteration

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 18 –

Simulation Results (2)

THP w. LLL

SNR

THP w. Brun

Sym

bol

Err

or

Ra

te

Sphere encoding(optimal)

THP

•

• iid Gaussian channel

• 4-QAM

• Iterations on average:- Brun: 4.8 - LLL: 42

• A Brun iteration is less complex than an LLL iteration

LR using Brun‘s algorithm can exploit large part of available diversity

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 19 –

Conclusions

• Algorithmic number theory provides useful tools for MIMO

detection and MIMO precoding

• Here: proposed vector perturbation using lattice reduction

based on integer relations

• Efficient implementation: Brun‘s algorithm

• Good performance at very small complexity

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 20 –

• ML detector:

MIMO Detection

- exact implementation: sphere decoder

- suboptimum detectors: ZF, MMSE, V-BLAST, …

• If is poorly conditioned:

• Everything is fine if is close to orthogonal

- poor performance (ZF, MMSE, V-BLAST, ...)

- or high complexity (ML)

• RX vector:

NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006 – 21 –

ML

d1

d2

ML

d1

d2

Detection for Poor Channel Condition

Example: BPSK, real-valued channel & noise, M=K=2

v

ZF-domain Rx vector:

search TX vector that is - integer - close to line y+v

simultaneous Diophantine

approximation(ANT)