Lattice reduction aided MIMO Precoding

Lattice-Reduction-Aided MIMO Precoding

Robert Fischer

LEHRSTUHL FUR INFORMATIONSUBERTRAGUNGFriedrich–Alexander–Universitat Erlangen–Nurnberg

Outline 1

Introduction

Decision-Feedback Equalization and Precoding

Lattice-Reduction-Aided Precoding

Simulation Results

Summary

I

Robert Fischer: Lattice-Reduction-Aided MIMO Precoding

Introduction 2

Situation: multiple–input/multiple–output (MIMO) transmission

y = Ha + n

Problem: interference of data streams transmitted in parallel

=> separation/equalization of signals required

Recently: increasing interest in schemes for

transmitter side signal processing

i.e., “pre-equalization”, “precoding”

Here: precoding for MIMO channels

broadcast scenario — downlink, no receiver side cooperation

schemes based on lattice basis reduction

duality to receiver side schemes

I


Introduction (II) 3

Receiver Side Cooperation

Yes NoTra

nsm

itte

rSid

eCooper

ation

Yes

Point-to-Point Point-to-Multipoint(Downlink)

No

Multipoint-to-Point(Uplink)

—

I


Decision-Feedback Equalization 4

Terms: decision-feedback equalization (DFE), successive cancellation, V-BLAST

H

a1

a2

a

y

n

a1

h1

y2

h2

y3

a2g2

fT1

fT2

= [h1 · · ·hNT]

aNT

g1

Successive procedure:

subtraction of interference of already detected symbols aκ

filtering by fTk (||f k||2 = 1) for suppression of residual interference

scaling by gk for unit gain signal transfer function

decision of symbol ak

Decision order can be optimized!

I


Decision-Feedback Equalization (II) 5

Redrawing the receiver: Matrix DFE

a

H

a1

a2

a

y

n

F G

B − I

P

aNT

Calculation of required matrices: (sorted QR-type decomposition)

H P = F H G−1 B

feedforward matrix F : unitary matrix with rows fTk

scaling matrix G: diagonal matrix with entries gk

feedback matrix B: lower triangular matrix with unit main diagonal

permutation matrix P : representation of decision order

I


Precoding 6

Multipoint-to-point transmission: DFE

B − I

F

a

G PH

a1

a2

a

y

n

aNT

Point-to-multipoint transmission: Precoding

P T MOD F Hx

a1y1

B − I a

aa

n1

g1

x

nNR

yNRgNR

aNR

I


Precoding (II) 7

Tomlinson-Harashima precoding:

P T MOD F Hx

a1y1

B − I a

aa

n1

g1

x

nNR

yNRgNR

aNR

Calculation of required matrices: (sorted QR-type decomposition)

P T H = G−1

B F H

feedforward matrix F : unitary matrix (short-term power constraint)

scaling matrix G: scaling factors (G = PGP T) (SNRs ∼ 1g2k)

feedback matrix B: lower triangular matrix with unit main diagonal

permutation matrix P : representation of processing order

I


Precoding (III) 8

P T F Hx

a1y1

B − I a

aa

n1

d

nNR

yNRaNR

g1

gNR

x

with x = MOD{z} = z + d

P T H = G−1

B F H => HF B−1 = PG−1

= G−1P

I


Precoding (III) 8

P T

a1y1

a

aa

n1

d

nNR

yNRaNR

g1

gNR

G−1P

with x = MOD{z} = z + d

P T H = G−1

B F H => HF B−1 = PG−1

= G−1P

I


Precoding (III) 8

a1y1

a

a

d

yNRaNR

gNRnNR

g1n1

1−1

1

−1

Precoding = linear pre-equalization+ periodic extension of signal constellation

I


Lattice Basis Reduction 9

Complex-valued MIMO channel model:

y = Ha + n

Equivalent real-valued MIMO channel model with doubled dimensionality:[Re{y}Im{y}

]=

[Re{H} − Im{H}Im{H} Re{H}

] [Re{a}Im{a}

]+

[Re{n}Im{n}

]

In shortyr = H rar + nr

Lattice structure of signals: (example NT = 1, 4–ASK)

ar H rar yr = H rar + nr

I


Lattice Basis Reduction (II) 10

ML detection:I


Lattice Basis Reduction (II) 10

Linear equalization:I


Lattice Basis Reduction (III) 11

Interpretation as lattice:I


Lattice Basis Reduction (III) 11

Equalization to rectangular grid:I


Lattice Basis Reduction (IV) 12

Procedure:

Lattice basis reduction: (e.g. LLL algorithm)

H r = H red R

with H red: “more suited” description of the latticeR: integer coefficients and det(R) = ±1

Equalization of only H red instead of H r:

yr

nr

R−1

a1

a2

aNT

a/ar

H/H r

ar/aH−1

red

I


Lattice Basis Reduction (IV) 12

Procedure:

Lattice basis reduction: (e.g. LLL algorithm)

H r = H red R

with H red: “more suited” description of the latticeR: integer coefficients and det(R) = ±1

Equalization of only H red instead of H r:

yr

nr

R−1F P

B − I

a1

a2

aNT

a/ar

H/H r

ar/a

I


Lattice-Reduction-Aided Precoding 13

Multipoint-to-point transmission:

yr

nra1

a2

aNT

a/ar

F

B − I

P R−1

H/H r

ar/a

Point-to-multipoint transmission: [Windpassinger et al. 2004], [Stierstorfer et al. 2004]

MOD Fx

a1y1

B − I

xa

n1

nNR

yNR gNR

aNR

g1

H/H r

a/ar

ar/a

P TR

I


Simulation Results 14

Parameters:

NT = 4 transmit antennas at joint transmitter

NR = 4 separated receivers

uncoded transmission;

– all parallel data streams use 4-ary QAM

– constant average transmit power

“flat block-fading channels”

– random channel matrix H (i.i.d. Gaussian entries with unit variance)

– averaging over a large number of channel realizations

optimal processing order

I


Simulation Results (II) 15

−5 0 5 10 15 20 2510

−5

10−4

10−3

10−2

10−1

100

10 log10

(Eb,TX/N0

)[dB] −→

BE

R−→

linear pre-equal.precoding (C/R)

vector precoding

LR lin. pre-equal.LR precoding

I


Summary 16

Precoding for MIMO channels

nonlinear transmitter side preequalization

of special interest in point-to-multipoint transmission (downlink)

schemes dual to that applicable in receiver side processing

=> “uplink-downlink duality”

considerable gains possible by applying the concept of

lattice-reduction-aided equalization

independent, parallel channels with periodic extension of the signal set result

=> channel coding immediately applicable

I


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Lattice reduction aided MIMO Precoding