A! Aalto UniversityComnet

IMANET+ Seminar

Joint Grassmann-Stiefel Codebooksfor Base Station Cooperation

Renaud-Alexandre PitavalProf. Olav Tirkkonen’s group

Aalto University, Department of Communications and Networking, Finland

A! Introduction and motivation• Codebook(CB)-based precoding (beamforming) for CoMP.

• Focus: codebook construction.

• Single-cell MIMO: Grassmannian CBs.− > there is an infinity of CB with same performance.

• CoMP aggregate channel: fluctuating number of Tx antennas andpath loss effects.− > Suggestion: product CB reusing per-cell CB.

• Here, novel joint Grassmann-Stiefel CB design for product CB.

• Also, some low-complexity codeword searches discussed.

Timetable

I System Model (3-8)II Codebook Design (9-12)

III Explicit Codebook Constructions (10-19)IV Codeword Selection (20-25)

IMANET Seminar - R-A Pitaval 2 (26)

A! Closed Loop MIMO Concept

• General MIMO signal model y = HWx + n

• 1. Receiver and transmitter share codebook C = {C1, . . . ,Cnb}2. Receiver selects best codeword of C and feeds back index3. Transmitter constructs precoder W based on CSI received• Orthonormal matrices, i.e. CH

i Ci = I ∀i⇒ Stiefel codebook• In many scenarios, rate I(Ci) ≡ I(CiU)⇒ Grassmann codebook

IMANET Seminar - R-A Pitaval 3 (26)

A! Stiefel and Grassmann Manifolds

• Stiefel manifold VCnt,ns:• Space of rectangular nt × ns unitary

matrices• Chordal distance:ds(X,Y) = ‖X−Y‖F

• Grassmann Manifold GCnt,ns:• Space of all p-dimensional subspaces of Cnt

• GCnt,ns ∼= VCnt,ns/Uns: set of equivalenceclasses of nt × ns Stiefel-matrices

An element in GCnt,ns is [Y] = {YU | U ∈ Uns} .Chordal distance: dc([X], [Y]) = 1√

2‖XXH −YYH‖F

− > Chordal distances: Euclidean distance from sphericalembeddings

IMANET Seminar - R-A Pitaval 4 (26)

A! Grassmannian Codeword

• Grassmann codeword invariant under any Uns rotation.

• By necessity, one has to choose a Stiefel representative.

• Reciprocally, any Stiefel matrix generates a Grassmann plane.

IMANET Seminar - R-A Pitaval 5 (26)

A! Network MIMO

• CoMP: coordinated multipoint /Base stations cooperation: case c)

IMANET Seminar - R-A Pitaval 6 (26)

A! System Model

• Received signal from nbs base stations each with nt antennas:

y = HlsWlsx + n

• Hls = HssG: large scale aggregate channel• Hss = [H1, . . . ,Hnbs]: small-scale i.i.d Rayleigh channel• G = diag(α1Int, . . . , αnbsInt): large scale path gains• Vls and Vss: ns-largest RSV of Hls and Hss, respectively.

• Optimum precoding: Wls,opt = Vls (up to right-unitary rotation).

• Assume a single CB C = {Ci}ncbi=1 implemented at every Rx.

• C independent of nbs and G.

• To deal with the heterogeneous path loss effects, assume G

known at Txs so that Rx quantizes Vss rather than Vls.

• Wls,opt ∝ GVss⇒Wss,opt = Vss (up to right-unitary rotation).

IMANET Seminar - R-A Pitaval 7 (26)

A! Product Codebooks

• Per-cell codebook: C of (nt × ns)-Stiefel matrices

• Product codebook: Cpr = 1√nbsC ⊗ · · · ⊗ C

A A

A

A

B

B

C B

B

A

C

C

A B C

AAAA A A

B

B

B

B BA

A

A A

C CA

C

C

C

Per-cell CB

Product CBnbs=2

Product CBnbs=3

• Proposed for CoMP, but have also some benefits for large MIMO• One single codebook to be implemented per-transmission rank.

• Easier to design because we focus on discretizing smaller spaces.

• Alphabet and power constraints propagate to product CB.

IMANET Seminar - R-A Pitaval 8 (26)

A! Per-cell Codebooks Design• Even, if entire codeword invariant under Uns rotation

• Part of codeword from BS2 NOT invariant under rotation

• Per-cell component: Wopt = [WHopt,1, . . . ,W

Hopt,nbs

]H .

• For i.i.d channel, [Wopt] Haar distributed on Grassmann manifold.

• Polar decomposition: Wopt,i = ViPi then Vi ∈ VCnt,ns is Haar.

• Thus, the per-cell codebook should be• for the first BS: a uniform Grassmann codebook.

• for the other BSs: a uniform Stiefel codebook.

IMANET Seminar - R-A Pitaval 9 (26)

A! Codebook Designs Criteria• From Grassmann beamforming literature:• Several non-equivalent distances can be defined.• Several codebook criteria exists providing notion of uniformity.• All different theoretical problems, but in practice routhly same performance.

• Quantization theory: minimize average distortion

DM(C) = E[

d2(V,Cq(V ))]

. (1)

• Classical discrete maths problems:• Paking problem: maximize minimum distance δ = arg min

1≤i,j≤ncbd(Ci,Cj).

• Thomson problem: maximizing the p-mean distance.

• Grassmann chordal distance of product CB related to Grassmannand Stiefel chordal distance of per-cell CB

δ2g(Cpr) ≥ min

{

δ2g(C),δ2g(C) + (nbs − 1)δ2s(C)

n2bs

}

.

IMANET Seminar - R-A Pitaval 10 (26)

A! Impact on Spectral Efficiency (1/2)

1 1.5 2 2.5 3 3.5 43

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Number of BS

Spe

ctra

l effi

cien

cy [b

ps/H

z]

Global Grassmann CBProduct CB with proposed designProduct CBs averaged over StiefelProduct CB with good Grassmannquantization but bad Stiefel quantization

(nbs

x4)x2

(nbs

x2)x1

• 2/4/6/8x1 and 4/8/12/16x2 MIMO at 10 dB SNR• One feedback bit per transmit antenna

IMANET Seminar - R-A Pitaval 11 (26)

A! Impact on Spectral Efficiency (2/2)

1 1.5 2 2.5 3 3.5 4 4.5 53

3.5

4

4.5

5

5.5

6

6.5

7

Per−cell codebook size in bits

Spe

ctra

l effi

cien

cy [b

ps/H

z]

Global Grassmann CB Product CB with proposed designProduct CBs averaged over StiefelProduct CB with good Grassmannquantization but bad Stiefel quantization

(2x4)x2

(2x2)x1

• 4x1 and 8x2 MIMO with 1- to 5-bit 2x1 and 4x2 per-cell codebook

IMANET Seminar - R-A Pitaval 12 (26)

A! Joint Grassmann-Stiefel Codebooks

• Take Grassmann CB⇒ Pick best Stiefel representative of eachGrassmannian codeword.

• For max-min-dist and max-mean-dist, Monte-Carlo and bruteforce search on constrainted alphabet may be used.

IMANET Seminar - R-A Pitaval 13 (26)

A! Joint G-S CB: Real-valued Example for 3 Tx

VC3,1 ∼= S2 (real sphere)

GC3,1: set of antipodal spherical codes

Square 1√3

111

1−11

−111

−1−11

Tetrahedron 1√3

111

−11−1

1−1−1

−1−11

IMANET Seminar - R-A Pitaval 14 (26)

A! Low Complexity Example for 2Tx

Square CB 1√2

{[

11

] [

1−1

] [

1i

] [

1−i

]}

Stiefel-improved CB 1√2

{[

11

] [

1−1

] [

−1−i

] [

−1i

]}

Squared Grass. dist.Squared Stief. dist.

Square CB Stiefel-improved CB

1

1� 2

1 �2

1 �2

1

1� 2

2

1

1

1

2

1

2

3

3

3

2

3

• Improved CB maximizes min-dist. and mean-dist.• No additional implementational cost.• Performance of product CB close to global Grassmann CB

IMANET Seminar - R-A Pitaval 15 (26)

A! Loyld Algorithm and Centroid• Minimize average distortion by iteration of 2 key steps:• Nearest Neighbor rule (NN): Partitioning in Voronoi cells

Rk = {V ∈M| k = q(V)}. (2)

• Centroid Computation (CC): Finding the centroids ofRks

Zk = arg minZ∈M

E[

d2(V,Z) | V ∈ Rk

]

. (3)

• Centroid for surface embedded in Euclidean space:– Center of mass in ambiant space: Mk = E [V | V ∈ Rk]– Centroid is normal projection of Mk ontoM

IMANET Seminar - R-A Pitaval 16 (26)

A! Lloyd-type Algorithm on Stiefel Conditionedon a Grassmannian CB

1. Initialization: Take Stiefel CB C = {C1, . . . ,Cncb} ⊂ VCnt,ns,representative of desired Grassmann CB

2. NN: Partition VCnt,ns in Voronoi cells {R1, . . . ,Rncb}3. For all k perform the following

(a) Centroid: Compute Stiefel centroid Zk:Center of mass: Mk = E [V | V ∈ Rk].Polar decomposition: Mk = ZkPk.

(b) Find the Grassmannian plane from C closest to Zk:

i = arg min1≤l≤ncb

dc(Cl,Zk). (4)

(c) Procrutes problem: Find rotation between the centroid and the Stiefelmatrix generating the closest Grassmannian plane Ci:

R = arg minU∈Uns

ds(Zk,CiU). (5)

(d) Update: Replace codeword i

Ci ← CiR (6)

4. Loop back to Step 2) until convergence.

IMANET Seminar - R-A Pitaval 17 (26)

A! Algorithm Illustration

1) 2)-3)

k = ’orange’

Centroidi = ’red’

Closest Grass. line Update ’red’

R

Converge to

a) b) c)-d)

4)

4)k=’red’

k=’blue’

5)

IMANET Seminar - R-A Pitaval 18 (26)

A! Grassmann-Stiefel Distortion

2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

Gra

sssm

ann

dist

ortio

n

2 4 6 8 10 12 14 16

0.5

1

1.5

2

2.5

3

3.5

Codebook cardinality ncb

Stie

fel d

isto

rtio

n

Lloyd on GrassmannianLloyd on Stiefel manifoldCascade: − Lloyd on Grassmannian − Algorithm 1

2x1

4x2

2x1

4x2

IMANET Seminar - R-A Pitaval 19 (26)

A! Codeword Selection: Multi-cell

Joint codeword selection

Wss = Qjs(Vss) = arg minCpr∈Cpr

dg(Cpr,Vss)

Joint codeword selection with transformed codebook

Wss = Qjs/tr(Vls) = arg minCpr∈Cpr

dg(GCpr,Vls).

− > Both based on Grassmann distance

− > The two joint selection methods provide similar performance forcell edge user, where G ∝ I.

− > High complexity due to the size of the exhaustive searchrequired

IMANET Seminar - R-A Pitaval 20 (26)

A! Codeword Selection: Per-cell

To decrease complexity, single cell channel matrix could be quantizedindependently.

Complexity reduced from O(

nnbscb

)

to O (nbsncb)

Independent codeword selection :

Wss,k = Qind(Vss,k) = argminC∈C

dg(C,Vss,k)

− > Based on Grassmann distance − > Large loss of performance

IMANET Seminar - R-A Pitaval 21 (26)

A! Stiefel-Grassman Per-cell CW Selection

• First quantize strongest channel using Grassmann distance

Wss,1 = Qind(Vss,1) = argminC∈C

dg(C,Vss,1) .

• Then channels from other BSs using Stiefel distance:

Wss,k = Qstief(Vss,k) = argminC∈C

ds(C,Vss,kRH)

given the polar decomposition WHss,1Vss,1 = RP where R ∈ Uns

and P is a positive-semidefinite Hermitian matrix.

IMANET Seminar - R-A Pitaval 22 (26)

A! Serial Codeword Selection• Order channels α1 ≥ . . . ≥ αnbs

• First channel quantized as previously using Grassmann distance

• Then the per-cell components are selected sequentially:

Given first (k − 1) per-cell codewords, the kth codeword is

Wss,k = argminC∈C

dg(C1→k,Vls,1→k)

with

C1→k =[

α1WHss,1, . . . , αk−1WH

ss,k−1, αkCH]H

Vls,1→k =[

Iknt0knt,nt(nbs−k)]

Vls

IMANET Seminar - R-A Pitaval 23 (26)

A! Comparison Between Selection Methods

0 5 10 15 20

1

2

3

4

5

6

7

8

9

10

11

12

SNR [dB]

Spe

ctra

l effi

cien

cy [b

ps/H

z]

Perfect precodingJointIndep Grass−StiefSerialIndep GrassNo precoding

(2x2)x1

(2x4)x2

4× 1 and 8× 2 systems using 2× 1 and 4× 2

Codebooks with one feedback bit per transmit antenna.

IMANET Seminar - R-A Pitaval 24 (26)

A! With Large Scale Path Gain Imbalance

0 0.2 0.4 0.6 0.8 1

2

2.5

3

3.5

4

4.5

Large scale path loss imbalance α2/α

1

Spe

ctra

l effi

cien

cy [b

ps/H

z]

Perfect precodingJoint transformedJointIndep Grass−StiefSerialIndep Grass

(2x4)x2

(2x2)x1

4× 1 and 8× 2 systems using 2× 1 and 4× 2 codebooks

Strongest channel fixed at SNR = 6 dB. 1 fdbck bit per Tx antenna.

IMANET Seminar - R-A Pitaval 25 (26)

A! Summary• Product codebook quantization for CoMP-JT.

• Product codebook structure may also be of interest for largeMIMO

• Single CB to be implemented− > same single-cell performance,− > near-optimal multi-cell performance (with proposed design)

• New coding problem: joint Grassmannian-Stiefel codebook.

⇒ Joint discretization of quotient space and linear representation

• Product structure can be exploited to decrease codewordselection complexity trading only small performance loss

Work submitted to IEEE TWC and partly presented in VTC Spring2012.

IMANET Seminar - R-A Pitaval 26 (26)