MIMO Communications and
Algorithmic Number Theory
G. Matz
joint work with D. Seethaler
Institute of Communications and Radio-Frequency Engineering
Vienna University of Technology (VUT)
INSTITUT FÜR
NACHRICHTENTECHNIK UND
HOCHFREQUENZTECHNIK
Setting the Stage
channel
TX
antennas
...
TX
...
• MIMO communications:
RX
RX
antennas
• Algorithmic number theory (ANT) is the study of algorithms
that perform number theoretic computations
(Source: Wikipedia)
Examples: primality test, integer factorization, lattice reduction
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
–2–
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
–3–
Multi-Antenna Broadcast (Downlink)
K symbols
user
#1
user
#k
channel
...
...
precoding
System model:
user
#K
M TX
antennas
users, each with one antenna
Users cannot cooperate
shift MIMO processing to TX
MIMO I/O relation:
precoding
with
CSI at TX required
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
–4–
Vector Perturbation (Peel et al.)
TX vector:
• here,
and
is an integer perturbation vector
• precoder performs channel inversion and vector perturbation
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
–5–
Choice of Perturbation Vector
• Optimum vector perturbation maximizes RX-SNR:
• Implementation: sphere encoder
• 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
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
–6–
Structure of Channel Singular Values
M=K=4
4
10
1003
102
10 1
0
20
60
40
condition number
80
100
smallest singular value and associated singular vector v cause problems
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
–7–
Vector Perturbation for Poor Channel Condition
Example: BPSK, real-valued channel & noise, M=K=2,  =2
search TX vector that
- is integer
- has small length
- is orthogonal to v
approximate integer
relation (ANT)
TX vector
perturbed versions of TX vector
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
–8–
Relation of MIMO and ANT
MIMO
Precoding at Tx
duality
Detection at Rx
poorly conditioned channels
ANT
Approximate
Integer
Relations
duality
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
Simultaneous
Diophantine
Approximations
–9–
Vector Perturbation Using Lattice Reduction (LR)
• View
as basis of a lattice
• Try to find “better“=reduced lattice basis
• All lattice basis are related via a unimodular matrix:
• LR-assisted vector perturbation (
,
)
- cost function:
- solve
- use
or use THP approximation
as perturbation vector
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 10 –
Lattice Reduction
• Orthogonality defect (quality of lattice basis):
• LR: find
achieving small
channel singular
values
and thus small
left channel
singular vectors
• Most popular LR method: Lenstra-Lenstra-Lovász (LLL) algorithm
- LLL-LR assisted THP achieves full diversity
- but LLL can be computationally intensive
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 11 –
Integer Relation Based LR
• Goal: more efficient LR method
• To achieve small
, vectors
must be
sufficiently orthogonal to singular vectors with small singular values
• For poorly conditioned channels, only one singular value is small
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
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 12 –
Approximate Integer Relations
•
can be made arbitrarily small using long vectors
• large
will increase
• Tradeoff:
- small
governed by channel singular values
- small
• Approximate IR: achieve small
with
as short as possible
• Can be realized very efficiently using Brun‘s algorithm
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 13 –
Brun’s Algorithm
repeat until termination
condition is satisfied
• Initialization:
• Find
• Calculate
• Replace
(update of
)
• Very simple: scalar divisions, quantizations, and vector updates
•
is also updated recursively and can be made arbitrarily small
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 14 –
Performance of Brun’s Algorithm
• Example using
and averaging over 1000 randomly picked
average
average
no. of iterations
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 15 –
Lattice Reduction via Brun’s Algorithm
Termination condition
• at each iteration,
is a basis for
• recall: LR aims at minimizing
terminate if update of
does not decrease
Calculation of
• we are just interested in channels with one small singular value
• in this case,
apply Brun’s algorithm to any column of
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 16 –
Simulation Results (1)
•
Symbol Error Rate
• iid Gaussian channel
• 4-QAM
• Iterations on average:
THP w. LLL
THP
Sphere encoding
(optimal)
- Brun: 2.5
- LLL: 12.9
THP w. Brun • A Brun iteration is less
complex than an
LLL iteration
SNR
LR using Brun‘s algorithm can exploit large part of available diversity
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 17 –
Simulation Results (2)
•
Symbol Error Rate
• iid Gaussian channel
• 4-QAM
• Iterations on average:
THP w. LLL
THP
THP w. Brun
Sphere encoding
(optimal)
- Brun: 4.8
- LLL: 42
• A Brun iteration is less
complex than an
LLL iteration
SNR
LR using Brun‘s algorithm can exploit large part of available diversity
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 18 –
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
– 19 –
MIMO Detection
• RX vector:
• ML detector:
- exact implementation: sphere decoder
- suboptimum detectors: ZF, MMSE, V-BLAST, …
• If
is poorly conditioned:
- poor performance (ZF, MMSE, V-BLAST, ...)
- or high complexity (ML)
• Everything is fine if
is close to orthogonal
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 20 –
Detection for Poor Channel Condition
Example: BPSK, real-valued channel & noise, M=K=2
v
search TX vector that is
- integer
- close to line y+v
d2
simultaneous
Diophantine
approximation
(ANT)
ML
d1
ZF-domain Rx vector:
NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006
– 21 –