A TENSOR-BASED BLIND DS-CDMA RECEIVER USING SIMULTANEOUS MATRIX DIAGONALIZATION Dimitri (1)
(1) Nion ,
Lieven De
(1) Lathauwer ,
ETIS, UMR 8051 (CNRS,ENSEA,UCP), Cergy-Pontoise, France E-mail: {nion, delathau}@ensea.fr
We present a deterministic tensor-based technique for the blind separation-equalization of DSCDMA signals received by an antenna array, in the context of far-field reflections only. Our method relies on the decomposition in terms of rank-(L,L,1) of a third-order tensor. We show that this decomposition can calculated by means of simultaneous diagonalization of a set of matrices, which is more accurate than the standard ALS algorithm. Communication System • Parameters
• How
– The matrix E is (JI × R). We denote by Er the (I × J) matrix representation of the rth column of E. We have:
and propagation model:
R
– R: Nb of users, transmitting at the same time within the same bandwidth. – I: Spreading Factor of CDMA codes. – J: Duration of the observation window (in Symbol Periods). – K: Nb of receiving Antennas. – I × J × K samples collected at the receiver – Multipath propagation: far-field reflections only (no angular spread) and ISI over L consecutive symbols (large delay spread). • Chip-Rate
to find W ∈ CR×R ?
JI
R
R
W
R
ˆ 1 ) ... vec(X ˆ R) = vec(X
JI
vec(E1 ) ... vec(ER )
For r = 1 . . . R, we thus have
Sampled Received Signal: Analytic Form
ˆr X
I
=
W1r
E1
+...+
WRr
ER I
I J
J
J
Rank L matrix
– The coefficients of W are those of linear combinations of the matrices Er that yield the ˆ r. rank-L matrices X ˆ ∈ CI×J , – Tool: mapping for rank-L detection (cf paper). Let X
• Chip-Rate
ˆ is at most rank − L ˆ X, ˆ . . . , X) ˆ = 0 if and only if X φ (|X, {z } (L + 1) times – After some algebraic manipulations (see paper for details), we can show that the matrix W can be estimated by simultaneous diagonalization of the following set of matrices:
Sampled Received Signal: Algebraic Form
D1
R
a1
aR
K
R
=
0 W
0
WT
K K
K J J I
M1
=
Y
L I
J
L
+...+
ST1
L I
H1
L
J
+
STR
I
HR
User 1
User R
N
R
• The matrices Sr are Toeplitz.
Computation of the Decomposition by Simultaneous Diagonalization
W
0
WT
diagonal matrices
Simulation Results • Parameters: codes of length I = 8, J = 50 QPSK symbols collected, K = 4 antennas, L = 2 interfering symbols, R = 4 users.
• Optimization problem: From the knowledge of the tensor of observations Y only, estimate the unknowns Hr , ar and Sr by minimization of the cost function ˆ r •2 S ˆ r •3 a ˆ r k2F H
=
– The system can be solved by any algorithm for joint-diagonalization by congruence of a set of matrices. ˆ = V∗ · W−T . The columns of H ˆ r can – Once W is found, the estimation of A is given by A be estimated as the L left singular vectors associated with the L largest singular values of ˆ r then corresponds to the product of the first L singular values and the L Xr . The matrix S associated right singular vectors.
• Each term of the decomposition contains the information related to one particular user (channel, antenna response and symbols).
ˆ 2 = kY − f (H, S, A) = kY − Yk F
MR
0
known matrices
AWGN
Decomposition in rank-(L,L,1) terms
R X
DR
R
0
(1)
0
10
10
r=1
−1
−1
10
on the dimensions:
est
Mean value of ||Y−Y ||
• Assumptions
10
−2
• Reformulation
I≥L J ≥L . min(IJ, K) ≥ R
(2)
SER
10
−3
10
MMSE ALS with 2 init ALS with 1 init ALS with 10 init SD
−4
10
of the problem:
−5
10
Yˆ =
R X
ˆ r •3 a ˆr, X
r=1
−4
ALS with 1 init ALS with 2 init ALS with 10 init SD MMSE
10
−5
−6
0
2
4
6
8
10
12
14
16
18
10
0
1
SNR(dB)
ˆ r result from X ˆr = H ˆr = H ˆ r •2 S ˆr·S ˆT . in which the (I × J) matrices X r ˆ ∈ CJI×K ˆ Y – Consider one matrix representation of Y:
ˆ – SVD of Y:
−3
10
10
−6
10
−2
10
˜ ·A ˆT ˆ ˆT = X ˆ ˆ Y = vec(X1) · · · vec(XR) · A
(3)
Y = U · Σ · VH = E · VH
(4)
−log10 (σ)
3
4
5
SD ), more relaxed than • The SD technique implies a new bound for the number of users (Rmax SC ). the sufficient condition previously derived (Rmax (SC) (SD) I J K L Rmax Rmax
– Combine Eqs. (3) and (4): there exists an a priori unknown non-singular matrix W ∈ CR×R that satisfies ˜ = E·W X . (5) T −1 H ˆ A = W ·V
2
4 4 8 2 4 5 8 2 4 6 8 2
2 2 3
4 5 7
