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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)
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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
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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
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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
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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
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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
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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
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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)
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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
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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:
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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:
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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)