Tensor-Based Models for Blind DS-CDMA Receivers by
Dimitri Nion and Lieven De Lathauwer ETIS Lab., CNRS UMR 8051 6 avenue du Ponceau, 95014 CERGY FRANCE
ASILOMAR 2007 November 4-7 2007, Pacific Grove, USA
Context Research Area: Blind Source Separation (BSS) Application:
Wireless Communications (DS-CDMA system here)
System:
Multiuser DS-CDMA, uplink, antenna array receiver
Propagation:
P1 Instantaneous channel (single path) P2 Multipath Channel with Inter-Symbol-Interference (ISI) and farfield reflections only (from the receiver point of view) P3 Multipath Channel (ISI) and reflections not only in the far-field (specular channel model)
Assumptions:
No knowledge of the channel, neither of CDMA codes, noise level and antenna array response (BLIND approach)
Objective:
Estimate each user’s symbol sequence
Method:
- Deterministic: relies on multilinear algebra - How? store observations in a third order tensor and decompose it in a sum of users’ contributions
Idea:
- Tensor Model « richer » than matrix model
2
Introduction
DSDS-CDMA system: cooperative vs. blind User 1
User R
Channel 1
Channel R
antennas
yk(t) Cooperative case: and
are known of of
Equalization and Separation Blind case:
3 are unknown
Introduction
Blind Approach: Why? Several motivations among others:
Elimination or reduction of the learning frames: more than transmission rate devoted to training in UMTS
40 % of the
Training not efficient in case of severe multipath fading or fast time varying channels
Applications: eavesdropping, source localization, …
If learning sequence unavailable or partially received 4
Introduction
Blind Approach: How? (1) K receive antennas
Chip-Rate Sampling Observation during J.Ts where Ts = symbol period
I=spreading factor
Spatial Diversity
Build the 3rd order tensor of observations
Temporal diversity Code diversity
Y
Numerical processing: Blind Equalization and Separation performed by decomposition of Y
5
Introduction
Blind Approach: How?
(2)
J
J
J
K
K
=
I
Y
K
+ …+
I
I
Y1
YR
Decomposition of Y : sum of R users’ contributions Algebraic structure of Yr ?
Estimation of Yr ?
Identifiability of Yr ?
Different according to the propagation scenario
Goal: Blind Separation and equalization
Uniqueness of tensor decompositions
Build different tensor decompositions
Build algorithms to compute tensor decompositions
Constraints on the number of users
Part I
Part II
Not in this talk
Introduction I.
Tensor Decompositions 1. Single path only (instantaneous channel): PARAFAC decomposition 2. Multipath Channel with ISI and far-field reflections only : Block-Component-Decomposition in rank-(L,L,1) terms : BCD(L,L,1) 3. Multipath Channel with ISI and reflections not only in the far-field: Block-Component-Decomposition in rank-(L,P,.) terms : BCD(L,P,.)
II.
Algorithms to compute tensor decompositions
II.
Simulation Results
Conclusion and Perspectives
7
Part I: Tensor Decompositions
PARAFAC decomposition If single path only (instantaneous mixture), Y follows a PARAFAC decomposition [Sidiropoulos, Giannakis & Bro, 2000].
Analytic Model:
R
y ijk =
∑hc r
ir
s jr a kr
r=1
Algebraic Model:
a1
J
K
= h1
I
s1
aR +
…
+
c1
Y
Y1 (User 1)
hR
sR cR
YR (User R)
cr holds the I ‘chips’ rth user’s spreading code ar holds the response of the K antennas sr holds the J consecutive symbols transmitted by user r hr fading factor of the instantaneous channel
8
Part I: Tensor Decompositions
BCDBCD-(L,L,1)
If multi-paths in the far field + ISI , Y follows a « Block Component Decomposition in rank-(L,L,1) terms », BCD-(L,L,1)
[De Lathauwer & De Baynast, 2003], [Nion & De Lathauwer, SPAWC 2007].
Analytic Model:
y ijk =
R
L
∑
r =1
a kr
K
= Y J
l =1
L interfering symbols
Algebraic Model:
I
∑
h r ( i + ( l − 1 ) I ) s (j r− )l + 1
r =1
K
J
R
∑
ar
L I
Hr
L
s0 s1 s2 ……………. sJ-1 s-1 s0 s1 s2 …………… sJ-2
SrT
Toeplitz structure because of ISI 9
Part I: Tensor Decompositions
BCDBCD-(L,P,.)
If multi-paths not only in the far-field + ISI , Y follows a BCD-(L,P,.)
[Nion & De Lathauwer, ICASSP 2005].
Analytic Model: R
y ijk =
P
L
(r) a (θ ) h (i + (l − 1 )I)s ∑ ∑ k rp ∑ rp j − l +1 r=1 p=1
l =1
1 path = 1 delay, 1angle of arrival and 1 fading coefficient Algebraic Model:
P paths
K P
K
R Y
I
=
∑ r= 1
J
P
Ar J L
I Hr L
s0 s1 s2 ……………. sJ-1 s-1 s0 s1 s2 …………… sJ-2
SrT
Toeplitz structure (IES) 10
Part I: Tensor Decompositions
Unknowns for each decomposition J
K
Y
I
∑
Y
I
J
A ∈ C K ×R
ar L
I
J
A ∈ C K ×R
L
=∑
r= 1
P
P
I L
Hr
BCD-(L,P,.)
Ar J
L
H ∈ C I ×RL S ∈ C J ×RL Block-Toeplitz
SrT
Hr
R
Y
BCD-(L,L,1)
J
K K
H ∈ C I ×R S ∈ C J ×R
hr
K
=∑
r= 1
sr
r= 1
R
K I
=
PARAFAC
ar
R
SrT
H ∈ C I ×RPL S ∈C
J × RL
Block-Toeplitz
A ∈ C K ×RP
11
Introduction I.
Les décompositions tensorielles
II.
Algorithms to compute Tensor Decompositions 1. Algorithm 1: ALS (“Alternating Least Squares”) 2. Algorithm 2: ALS + LS (“Line Search”) 3. Algorithm 3: LM (“Levenberg-Marquardt”)
III.
Simulation Results Conclusion et Perspectives
12
Part II: Algorithms
Objective of the proposed algorithms Decomposition of Y
Estimation of components A, S and H
Minimize frobenius norm of residuals. Cost function:
ˆ) ˆ , Sˆ , A Φ = Y − Tens ( H
2 F
Tens = PARAFAC or DCB-(L,L,1) or DCB-(L,P,.)
Useful Tool: « Matricize » the tensor of observations Code Diversity
Y
Yk
YI×KJ = cat[Yk ]
Yi
YJ×IK = cat[Yi ]
I Spatial Diversity
K
Temporal Diversity J
Yj
YK ×JI = cat[Yj ] 3 matrix representations of the same tensor
13
Part II: Algorithms
Algorithm 1: ALS « Alternating Least Squares » Principle: Alternate between least squares update of the 3 matrices A=[A A1,…,A AR], S=[S S1,…,S SR] et H=[H H1,…,H HR].
ˆ ( 0) , H ˆ ( 0) , k = 1 Initialization : A while Φ ( k −1) − Φ ( k ) > ε (e.g. ε = 10-6 )
[ [ ˆ ⋅ [Z (H
ˆ ( k −1) , H ˆ ( k −1) ) Sˆ ( k ) = YJ×IK ⋅ Z1( A ˆ ( k −1) ) ˆ ( k ) = YI×KJ ⋅ Z 2 (Sˆ ( k ) , A H ˆ (k ) = Y A K ×JI k ← k +1
3
(k )
, Sˆ ( k ) )
]
]
]
(1) ( 2) (3)
14
Part II: Algorithms
Convergence of ALS « Easy » Problem
«Difficult» Problem
Long swamp DCB-(L,P,.)
DCB-(L,P,.)
I=8, J=50, K=6,
I=8, J=50, K=6,
L=2, P=2, R=3
L=2, P=2, R=3
Because of long swamps that might occur, we propose 2 algorithms that improve convergence speed. 15
Part II: Algorithms
Algorithm 2: Insert a Line Search step in ALS For each iteration, perform linear interpolation of the 3 components A, H and S from their values at the 2 previous iterations. Iteration k Directions of research
1.Line Search:
Sˆ ( new ) = Sˆ ( k −2 ) + ρ (Sˆ ( k −1) − Sˆ ( k −2 ) ) ˆ ( new ) = A ˆ ( k −2 ) + ρ ( A ˆ ( k −1) − A ˆ ( k −2 ) ) A
Choice of step ρ important
ˆ ( new ) = H ˆ ( k −2 ) + ρ (H ˆ ( k −1) − H ˆ ( k −2 ) ) H 2. ALS update
[
Can be optimally calculated with
]
~ ˆ ( new ) ˆ ( new ) ˆs( k ) = Z1 ( A ,H ) ⋅ YJIK ˆ ( new ) ) ˆ ( k ) = Y ⋅ Z (Sˆ ( k ) , A H I×KJ
ˆ (k ) = Y A K ×JI k ← k +1
[ ˆ ⋅ [Z (H 2
3
(k )
, Sˆ ( k ) )
]
]
« Enhanced Line Search with Complex Step» (ELSCS)
16
Part II: Algorithms
Algorithm 3: LM « LevenbergLevenberg-Marquardt » Concatenate vectorized unknowns vec(A A), vec(H H) and s in a long vector p Update p: Gauss-Newton: Levenberg-Marquardt:
p (k + 1 ) = p (k) + ∆p (k)
(1)
( J H J ) ∆p (k) = − g
(2)
( J H J + λ I)∆p(k) = −g
(3)
The matrix ( J H J + λ I) is positive definite: solve (3) by Cholesky decomposition and Gaussian elimination. According to the condition number of JHJ + λ I, update λ in each iteration. If ( J J + λ I) ill-conditioned then increase λ : H
get closer to gradient descent update ∆p
(k)
1 ≈ − g λ
If ( J H J + λ I) well-conditioned then decrease λ: get closer to Gauss-Newton update
(J H J)∆p(k) ≈ −g 17
Part II: Algorithms
Convergence of algorithms ALS, ALS+LS et LM «easy» problem
«difficult» problem
Gradient Descent
Gauss Newton (quadratic convergence)
LM and ALS+ELSCS converge much faster than standard ALS, especially for difficult problems: the length of swamps is considerably reduced.
18
Introduction I.
Tensor Decompositions
II.
Algorithms to compute Tensor Decompositions
III.
Simulation Results Conclusion et Perspectives
19
Part III: Simulation Results
Impact of number of antennas BCD-(L,P,.) with: spreading factor I=12, J=100 symbols, L=2 interfering symbols, P=2 paths per user and 10 random initializations, + AWGN
K=4 antennas and R=5 users
K=6 antennas and R=3 users
20
Part III: Simulation Results
Impact of Near-Far effect R
Y = ∑αr r =1
Yr Yr
+B
κ (Y) =
max(α r ) min(α r )
F
BCD-(L,L,1) with spreading factor I=4, J=100 symbols, K=4 antennas, L=2 interfering symbols, R=5 users and 10 random initializations, + AWGN Note: more users than antennas (R>K) and overloaded system (R>I)
κ (Y) = 1
κ (Y) = 5
21
Over-estimation of the number of paths P Y built with P=3 paths for each user. Decomposition calculated with over-estimation of P (P=4 and P=5) and underestimation of P (P=2). MSE of symbol matrix vs. SNR
22
Conclusion Tensor Models: PARAFAC receiver: ok if single path (instantaneous mixture) BCD receivers: multipaths + ISI (blind separation and equalization) Approach: Deterministic, exploits multi-linearity of received signal, i.e. algebraic structure of tensor of observations. 1 diversity = 1 dimension of this tensor. Algorithms: standard ALS sensitive to swamps that appear with ill-conditioned data or severe Near-Far effect ALS+ELSCS and LM offers much better performance. Performances: Blind BCD receivers potentially very close to MMSE, provided that enough diversity is exploitable. Uniqueness (not in this talk): Maximum number of users admissible in the system depends on the dimensions of 23 the problem.
