Efficient Vector Perturbation in MultiAntenna MultiUser Systems Based on Approximate Integer Relatio

1
Efficient Vector Perturbation in Multi-Antenna
Multi-User Systems Based on Approximate Integer
Relations

• D. Seethaler and G. Matz

2
Outline

• System model
• Vector perturbation using lattice reduction
  (LR)
• Efficient LR with approximate integer relations
• Bruns algorithm
• Simulation results

3
Multi-Antenna Broadcast System (Downlink)

• System model

user 1
channel
user k
. . .
. . .
precoding
. . .
user K
K data symbols
M Tx antennas
users, each with one antenna
Users cannot cooperate precoding with
full CSI at Tx
with
Input/output relation
4
Vector Perturbation (Peel et al.)
Tx vector

• here, and is an integer
  perturbation vector
• is a fixed real-valued scaling factor
• precoder performs channel inversion and vector
  perturbation

Received symbols

• follows from
• Rx-SNR is determined by

• get rid of via modulo operation
• quantization to symbol alphabet

Remaining Rx processing
5
Choice of Perturbation Vector

• Optimum vector perturbation maximizes Rx-SNR

Can be implemented using sphere encoding

• Efficient (suboptimum) technique
  Tomlinson-Harashima precoding (THP)

• For channels with large condition number
• Sphere encoding has high complexity
• Suboptimum techniques have poor performance
• Small condition number All methods work fast
  and well

6
Vector Perturbation Using Lattice Reduction (LR)

• View as basis of a lattice

• Try to find better ( reduced) basis for
  same lattice

• All lattice bases are related via unimodular
  matrix , i.e.

• LR-assisted vector perturbation (
  , )
• Cost function
• Solve
  (or use any approximation)
• Use as perturbation
  vector

7
Lattice Reduction

• Orthogonality defect (quality of reduced basis
  for )

• LR Find achieving
  small and thus short

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

• LLL-LR assisted THP achieves full diversity
• But LLL can be computationally intensive

8
Basic Idea of Integer Relation (IR) Based LR

• Goal More efficient LR method

• To achieve small
  

channel singular values
left channel singular vectors
vectors must be sufficiently orthogonal to
singular vectors with small singular values

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

Proposed method Find integer vectors that
are
sufficiently orthogonal to

• Approximate IR Achieve small
  with as short as possible

• Can be realized very efficiently using Bruns
  algorithm

9
Bruns Algorithm

• Initialization

• Find

• Calculate

repeat until termination condition is satisfied

• Replace

(update of )

• Very simple Scalar divisions, quantizations,
  and vector updates

• is also updated recursively and
  can be made arbitrarily small

10
Performance of Bruns Algorithm
Example using and averaging over 1000
randomly picked
average
average
number of iterations
11
Lattice Reduction via Bruns 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 Bruns algorithm to any column of
12
Simulation Results

• iid Gaussian channel
• 4-QAM
• Iterations on average

• Brun 4.8
• LLL 42

Symbol Error Rate
THP w. LLL
THP

• A Brun iteration is less
• complex than an
• LLL iteration

THP w. Brun
Sphere encoding (optimal)
SNR
LR using Bruns algorithm can exploit large part
of available diversity
13
Summary Conclusions

• We proposed an efficient vector perturbation
  algorithm for multi-
• antenna multi-user systems
• Vector perturbation is implemented using
  reduced lattice basis
• Reduced lattice basis is obtained via Bruns
  algorithm for finding
• approximate integer relations
• Properties of proposed algorithm
• Exploits large part of available diversity
• Very low complexity