joint grassmann-stiefel codebooks for base station...
TRANSCRIPT
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)