Two-Way MIMO Decode-and-Forward Relaying Systems with Tensor Space-Time Coding

1. Two-Way MIMO Decode-and-Forward Relaying
Systems with Tensor Space-Time Coding
W. C. Freitas(1)
, G. Favier(2)
, A. L. F. de Almeida(1)
, and Martin Haardt(3)
September 4, 2019
(1)
Wireless Telecom Research Group
Federal University of Cear´a
Fortaleza, Brazil
http://www.gtel.ufc.br
(2)
Laboratoire I3S
Universit´e Cˆote d’Azur
Nice, France
http://www.i3s.unice.fr
(3)
Comm. Research Lab.
Ilmenau University of Technology
Ilmenau, Germany
http://tu-ilmenau.de/crl
1/17

2. Agenda
• Introduction
• System model
• Decode-and-Forward (DF) closed-form semiblind receiver
• Simulation results
• Conclusions
2/17

3. Introduction
• Tensor-based receivers have been successfully used for joint symbol and channel
estimation in cooperative MIMO communications
• Previous works derived two-way cooperative MIMO receivers, most assuming
amplify-and-forward (AF) relays
• Proposal: a new closed-form semiblind receiver for a two-way DF relaying
system that:
1. avoids error propagation problems;
2. exploits a cross-coding structure that suppresses interference between sources
and simplifies the receiver.
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4. System Model
Two-way MIMO Relaying System
Source i
. . .
Ms
Relay
. . .
Mr
Source j
. . .
Ms
H(sir)
H(rsi)
H(sj r)
H(rsj )
• Two-way MIMO: each source aims to estimate the signals sent by the other
• Uplink phase: sources transmit to the relay
• Downlink phase: relay transmits re-encoded data to the sources
• Ms and Mr denote the number of antennas at sources and relay
• H(sir)
∈ Mr×Ms
and H(rsi)
∈ Ms×Mr
, denote the channels during the
uplink and downlink phases, respectively
4/17

5. Tensor of Signals Received at the Relay
Uplink Transmission
Source i
. . .
Ms
Relay
. . .
Mr
Source j
. . .
Ms
H(sir)
H(sj r)
• Symbol matrix transmitted by source i: S(i)
∈ N×R
containing N data
symbols in R data-streams
• Tensor space-time code (TSTC) at source i: C(i)
∈ R×Ms×P
, P the time
spreading length of the codes at the sources
• The signals received at the relay form a tensor X ∈ N×Mr×P
X = C(i)
×1 S(i)
×2 H(sir)
+ C(j)
×1 S(j)
×2 H(sj r)
(1)
⇓
Block Tucker-2 model5/17

6. Relay Processing
Downlink Transmission
Source i
. . .
Msi
Relay
. . .
Mr
Source j
. . .
Msj
H(rsi)
H(rsj )
• Assuming a DF protocol, the relay estimates and decodes the symbol matrices
ˆS(i)
and ˆS(j)
• A cross-coding scheme is proposed at the relay to suppresses interference
between sources, using C(j)
to encode ˆS(i)
, while C(i)
is used to encode ˆS(j)
6/17

7. Relay Processing
Symbols Estimation and Projection on the Alphabet
• From the flat 3-mode unfolding of X (disregarding the noise)
XP ×NMr = C
(i)
3 S(i)
⊗ H(sir)
T
+ C
(j)
3 S(j)
⊗ H(sj r)
T
. (2)
• C
(i)
3 , C
(j)
3 ∈ P ×RMs
are chosen as two blocks extracted from a P × 2RMs
DFT matrix, i.e., C
(i)H
3 C
(i)
3 = IRMs and C
(i)H
3 C
(j)
3 = 0RMs
• Least-square (LS) estimation of the Kronecker products from Eq. (2) given by
Z
(i)
RMs×NMr
∼= C
(i)H
3 XP ×NMr = S(i)
⊗ H(sir)
T
Z
(j)
RMs×NMr
∼= C
(j)H
3 XP ×NMr = S(j)
⊗ H(sj r)
T
.
(3)
7/17

8. Identifiability
Ambiguity Suppression and Complexity
• Kronecker product factorization algorithm can be used to estimate S(i)
and
H(sir)
for the source i from Z
(i)
RMs×NMr
• Uniqueness of Kronecker factors and S(i)
and H(sir)
estimated at source i up
to scalar ambiguity
• Element s
(i)
1,1 is known and equal to 1. Final estimates of the channels and
symbol matrices is (λS(i) = 1/ˆs
(i)
1,1)
ˆS(i)
← ˆS(i)
λS(i) , ˆH
(sir)
Mr×Ms
← ˆH
(sir)
Mr×Ms
λ−1
S(i) ,
• Complexity associated with the SVD calculation of matrix J × K ⇒
Ç(min(J, K)JK)
• Relay complexity: Ç(min(NMr, RMs)NMrRMs)
8/17

9. Cross-Coding at the Relay
Relay Processing
• Relay re-encodes the estimated signals to be transmitted using tensor
space-time codes C(i)
and C(j)
∈ R×Ms×P
• Relay re-encodes the estimated signals using cross-coding: ˆS(i)
using C(j)
, and
ˆS(j)
using C(i)
• 3-mode unfoldings C
(i)
3 , C
(j)
3 ∈ P ×RMs
are chosen as two blocks extracted
from a P × 2RMs DFT matrix as
C
(j)H
3 C
(i)
3 = 0RMs (4)
C
(j)H
3 C
(j)
3 = IRMs (5)
• Such orthogonal design allows to derive the closed-form semiblind receiver
9/17

10. Tensor of Signals Received at the Receiver
Downlink Transmission
Source i
. . .
Msi
Relay
. . .
Mr
Source j
. . .
Msj
H(rsi)
H(rsj )
• Tensor of signals received at source i: Y(i)
∈ N×Ms×P
(Mr = Ms)
Y(i)
= C(i)
×1
ˆS(j)
×2 H(rsi)
+ C(j)
×1
ˆS(i)
×2 H(rsi)
(6)
⇓
Block Tucker-2 model
10/17

11. Proposed Closed-Form Semiblind Receiver
Source i Processing
• The flat 3-mode unfolding of the received signals tensor Y(i)
Y
(i)
P ×NMs
= C
(i)
3
ˆS(j)
⊗ H(rsi)
T
+ C
(j)
3
ˆS(i)
⊗ H(rsi)
T
. (7)
• LS estimate of the Kronecker product is given by
R
(i)
RMr×NMs
∼= C
(i)H
3 Y
(i)
P ×NMs
= ˆS(j)
⊗ H(rsi)
T
. (8)
• Kronecker product factorization algorithm can be used to estimate S(j)
for the
source i from R(i)
• Source complexity: Ç(min(NMs, RMr)NMsRMr).
• The same approach can be followed by source j to estimate S(i)
from R(j)
11/17

12. Closed-form DF Semiblind Receiver
Inputs: X , Y(i)
, C(i)
and C(j)
.
• Relay Processing
1. Compute the LS estimate of Z
(i)
RMs×NMr
= S(i)
⊗ H(sir)
T
using Eq. (3).
2. Use the Kronecker factorization algorithm to estimate S(i)
and H(sir)
.
3. Remove the scaling ambiguities of ˆS(i)
and ˆH(sir)
.
4. Project the estimated symbols onto the alphabet.
5. Re-encode ˆS(i)
using C(j)
, and ˆS(j)
using C(i)
.
• Source i Processing
1. Compute the LS estimate of R
(i)
RMr×NMs
= S(j)
⊗ H(rsi)
T
using Eq. (8).
2. Use the Kronecker factorization algorithm to estimate S(j)
and H(rsi)
.
3. Remove the scaling ambiguities of ˆS(j)
and ˆH(rsi)
, and project the estimated
symbols onto the alphabet.
12/17

13. Simulation Assumptions
• Performance criteria - SER and NMSE of estimated channels averaged over
4 × 104
Monte Carlo runs.
• Each run corresponds to a realization of all channel and symbol matrices, and
noise tensors
• Transmitted symbols are randomly drawn from a unit energy m-QAM symbol
alphabet
• Number of data symbols N = 4, of data-streams R = 2 and of antennas
Ms = Mr = 2
• Modulation order m and spreading length P are adjusted to ensure the same
transmission rate, equal to 4/5 bit per channel use
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14. Simulation Results
SER comparison between DF and AF receivers.
• DF receiver outperforms AF due to less sensitivity to noise amplification
0 2 4 6 8 10 12 14 16 18
Es/N0 [dB]
10-8
10-6
10
-4
10
-2
10
0
SER
AF w/ reciprocity
AF w/out reciprocity
DF
14/17

15. Simulation Results
NMSE of estimated channels and with/without reciprocity
• NMSE performance is impacted by the reciprocity assumption
0 2 4 6 8 10 12 14 16 18
Es/N0 [dB]
10-5
10-4
10-3
10-2
10-1
NMSE
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16. Simulation Results
SER comparison between DF, EF and ZF receivers
• EF Ideal - has no error propagation at the relay
• DF ZF - LS Kronecker factorization in Eq. (8) with one factor known
0 1 2 3 4 5 6 7 8 9 10
Es/N0 [dB]
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
SER
DF
EF Ideal
DF ZF
• EF Ideal - relay re-encodes
the exact symbol matrices
• DF ZF - full knowledge of
the channel phases
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17. Conclusions
• New closed-form semiblind receiver for two-way MIMO DF relaying systems
• Advantages of DF receiver compared with AF receivers:
• Orthogonal cross-coding to eliminate sources interference
• Better SER due to the smaller propagation error, at the price of an additional
decoding at the relay
• Channel reciprocity is not needed
• Compared with ALS-based receivers, closed-form is simpler
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18. Thank you for your attention.
17/17