Differential Distributed Space-Time Coding with Imperfect Synchronization in Frequency-Selective Channels

1. Introduction
D-OFDM DSTC
Simulation
Summary
Differential Distributed Space-Time Coding with
Imperfect Synchronization in Frequency-Selective
Channels
M. R. Avendi
Center for Pervasive Communications and Computing
Department of Electrical Engineering & Computer Science
University of California, Irvine
April, 2014
1

2. Introduction
D-OFDM DSTC
Simulation
Summary
Outline
1 Introduction
2 D-OFDM DSTC
3 Simulation
4 Summary
2

3. Introduction
D-OFDM DSTC
Simulation
Summary
Cooperative Communications
Phase I: Source transmits, Relays listen
Phase II: Relays re-broadcast their received signal to
Destination
Virtual antenna array, improving diversity
q1
q2
qR
g1
g2
gR
Source
Destination
Relay 1
Relay 2
Relay R
3

4. Introduction
D-OFDM DSTC
Simulation
Summary
Multipath Fading
Mobile Phone
Base Station
4

5. Introduction
D-OFDM DSTC
Simulation
Summary
Fading Effects
Ts
Ts
Ts + Tm
Ts + Tm
timetime
timetime
Flat-Fading
Frequency-Selective
5

6. Introduction
D-OFDM DSTC
Simulation
Summary
Channel Models
Flat-fading channel, one tap filter h[n] = h0:
y[n] = h0x[n] + n[n]
Frequency selective channel, multiple taps filter:
h[n] =
L−1
l=0
hl δ[n − l]
y[n] = x[n] ∗ h[n] =
L
l=0
hl x[n − l]
Inter Symbol Interference (ISI)
Circular convolution
y[n] = x[n] ⊗ h[n] =
L
l=0
hl x[n − l]N
DFT/IDFT: Y [m] = X[m]H[m]6

7. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
Differential OFDM (D-OFDM) DSTC
Source, R Relays, Destination
Frequency-Selective Channels: {qi,l},{gi,l } for i = 1, · · · , R,
l = 0, · · · , L − 1
Source
Relay 1
Relay 2
Relay R
Destination
{q1,l }
{q2,l }
{qR,l }
{g1,l }
{g2,l }
{gR,l }
Figure: Cooperative network under consideration, Source
communicates with Destination through R relays.
7

8. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-DSTC OFDM: Source
{v1[n]}
{vR [n]}
{V[n]}
USTC
Differential
Encoding
IDFT
IDFT
Add Ncp1
Add Ncp1
{s1[n]}
{sR [n]}
{S1[m]}
{SR [m]}
Figure: Encoding process at Source8

9. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
Encoding at Source: R = 2 Relays
Consider 2N symbols: {v1[n]}, {v2[n]} for n = 0, · · · , N − 1.
Encode to Unitary Space-Time Codes (USTC)
V[n] =
1
|v1[n]|2 + |v2[n]|2
v1[n] −v∗
2 [n]
v2[n] v∗
1 [n]
, (1)
for n = 0, · · · , N − 1.
Differential Encoding
s[n](k)
= V[n](k)
s[n](k−1)
= [s1[n], · · · , sR [n]],
s[n](0)
= [ 1 0 · · · 0 ]t
, n = 0, · · · , N − 1,
(2)
Apply IDFT: Sr [m] = IDFT{sr [n]} , for r = 1, · · · , R and
m = 0, · · · , N − 1
Add Cyclic Prefix Ncp1
≥ (L − 1)
9

10. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-DSTC OFDM: Relays
Xi,1[m]
Xi,R[m]
Zi,1[m]
Zi,R[m]
Add Ncp2
Add Ncp2
Remove Ncp1
Remove Ncp1
STC Configure
Figure: Configuration process at Relay i, i = 1, · · · , R.
10

11. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-DSTC OFDM: Relays
Remove Cyclic Prefix
Zi,r [m] = P0R (qi [m] ⊗ Sr [m]) + Ψi,r[m], (3)
Ψi,r[m] ∼ CN(0, N0).
STC configuration



Xi,1[m]
...
Xi,R[m]


 = A



Bi



Zi,1[m]
...
Zi,R[m]


 + Ci




◦
Z∗
i,1[m]
...
◦
Z∗
i,R[m]







 (4)
where A amplification factor and Bi , Ci combining matrices
11

12. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
Continue · · ·
circular time-reversal
◦
Zi,r[m] =
Zi,r[0], m = 0
Zi,r[N − m], otherwise,
(5)
for i, r = 1, · · · , R and m = 0, · · · , N − 1.
Add cyclic prefix: Ncp2
≥ (L + dmax), dmax maximum sync
delay
12

13. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
Configuration: R = 2 Relays
Combining Matrices
B1 =
1 0
0 1
, C1 = 0, B2 = 0, C2 =
0 −1
1 0
. (6)
STC configuration
X1,1[m] = AZ1,1[m],
X1,2[m] = AZ1,2[m],
X2,1[m] = −A
◦
Z∗
2,2[m],
X2,2[m] = A
◦
Z∗
2,1[m].
(7)
for m = 0, · · · , N − 1.
13

14. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Destination
X1,r [−1] X1,r [0] X1,r [1] X1,r [2]
Xi,r[−di − 1] Xi,r[−di ] Xi,r[−di + 1] Xi,r[−di + 2]
XR,r [−dR − 1] XR,r [−dR] XR,r [−dR + 1] XR,r [−dR + 2]
Ts
τi
τR
Figure: Received signals from the relays in the first path at Destination,
when Relay i is di Ts + τi seconds late, di ∈ Z and 0 ≤ τi ≤ Ts.
14

15. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Destination
Effective pulse-shape: Raised-Cosine, β roll-off factor
p(t) = sinc(t/Ts)
cos(πβt/Ts )
(1 − 4β2t2/T2
s ),
Ts
Basedband Signal Sampled Signal
Matched Filter
Figure: Filtering and Sampling.
15

16. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Destination
−2 −1 0 1 2
Relay 1
Relay 2
τ
X1,r [0]
X1,r [1]
X2,r [−d2 − 1]
X2,r [−d2]
X2,r [−d2 + 1]
t/Ts
Figure: Received signals at Destination after the matched-filter using a
raised-cosine filter with roll-off factor β = 0.9, when
τ1 = 0, τ = τ2 = 0.3Ts.16

17. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Destination
{v1[n]}
{vR [n]}
Differential
Decoding
DFT
DFT
Remove Ncp2
Remove Ncp2
{Y1[m]}
{YR[m]}
{y1[n]}
{yR [n]}
Figure: Decoding process at Destination17

18. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Destination
Remove Cyclic Prefix
Yr [m] =
R
i=1
p(τi ) (gi [m] ⊗ Xi,r[m − di ]) + Φr [m]
+
R
i=1
p(Ts − τi ) (gi [m] ⊗ Xi,r[m − 1 − di ]) ,
(8)
for m = 0, · · · , N − 1 and r = 1, · · · , R, where
Φr [m] ∼ CN(0, N0).
18

19. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Destination
Take DFT
yr [n] =
R
i=1
Gi [n]xi,r[n] + φr [n] (9)
for r = 1, · · · , R, n = 0, · · · , N − 1, where
Gi [n] =
L−1
l=0
gi,l e−j 2πnl
N ,
Gi [n] = p(τi ) + p(Ts − τi )e−j 2πn
N Gi [n]e−j
2πndi
N ,
xi,r[n] = DFT{Xi,r[m]}, φr [n] = DFT{Φr [m]}.
(10)
Note that φr [n] ∼ CN(0, N0).
19

20. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Destination
In the matrix form:
y[n] = A P0RS[n]H[n] + w[n], (11)
where
S[n] = B1s1[n], · · · , BRsR [n] ,
H[n] = [ H1[n], · · · , HR [n] ]t
,
Hi [n] = Qi [n]Gi [n],
w[n] =
R
i=1
Gi [n]Bi ψi [n] + φ[n],
(12)
for i, r = 1, · · · , R and 0 ≤ n ≤ N − 1. It is noted that
ψi,r [n] ∼ CN(0, N0).
20

21. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
Example: R = 2 relays
R = 2 relays using Alamouti STC
y[n] = A 2P0
s1[n] −s∗
2 [n]
s2[n] s∗
1 [n]
H1[n]
H2[n]
+
w1[n]
w2[n]
, (13)
with
H1[n] = Q1[n]G1[n], H2[n] = Q∗
2 [n]G2[n],
w1[n] = A G1[n]ψ1,1[n] − G2[n]ψ∗
2,2[n] + φ1[n],
w2[n] = A G1[n]ψ1,2[n] + G2[n]ψ∗
2,1[n] + φ2[n].
(14)
for 0 ≤ n ≤ N − 1.
21

22. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: Received SNR
Noise, for given {gi,l }, w[n] ∼ CN(0, σ2[n]IR )
σ2
[n] = N0 1 + A2
R
i=1
|Gi [n]|2
. (15)
Received SNR for given {gi,l }
γ n, {τi }R
i=1 =
A2P0
R
i=1
|Gi [n]2|
N0 1 + A2
R
i=1
|Gi [n]|2
, (16)
for n = 0, · · · , N − 1.
22

23. Introduction
D-OFDM DSTC
Simulation
Summary
Source
Relays
Destination
D-OFDM DSTC: RX SNR
0 7 14 21 28 35 42 49 56 63
17
17.5
18
18.5
19
19.5
20
τ=0 & 1
τ=0.2 & 0.8
τ=0.4 & 0.6
τ=0.5
n
γ(n,τ2),dB
Figure: Average received SNR vs. n and τ = τ2 for a network with two
relays over flat-fading channels, when N = 64, P/N0 = 25dB,
P0 = P/2, Pr = P/4, |g1,0|2
= |g2,0|2
= 1.23

24. Introduction
D-OFDM DSTC
Simulation
Summary
Simulation Setup
Networks with R = 2 and R = 4 relays
Flat-Fading and Frequency-Selective fading with length L = 6
For R = 2 relays: Alamouti STC with BPSK and QPSK
For R = 4 relays: QOSTBC with BPSK and π/2-rotated
BPSK
V[n] =
1
4
i=1
|vi [n]|2




v1[n] −v∗
2 [n] −v∗
3 [n] v4[n]
v2[n] v∗
1 [n] −v∗
4 [n] −v3[n]
v3[n] −v∗
4 [n] v∗
1 [n] −v2[n]
v4[n] v∗
3 [n] v∗
2 [n] v1[n]




(17)
24

25. Introduction
D-OFDM DSTC
Simulation
Summary
Simulation Results
0 5 10 15 20 25 30
10
−4
10
−3
10
−2
10
−1
10
0
τ = (0.4&0.6)Ts
τ = (0.2&0.8)Ts
τ = 0&Ts
D-DSTC, τ = 0
D-DSTC, τ = 0.2Ts
D-DSTC, τ = 0.4Ts
D-DSTC, τ = 0.6Ts
Coherent DSTC, τ = 0
BER
P/N0dB
Figure: Simulation BER, R = 2 relays, flat-fading channels, D-OFDM
DSTC(N = 64, Ncp = 1), D-DSTC, and coherent DSTC, using Alamouti
code and BPSK, τ1 = 0, τ2 = τ.25

26. Introduction
D-OFDM DSTC
Simulation
Summary
Simulation Results
0 5 10 15 20 25
10
−4
10
−3
10
−2
10
−1
10
0
τ = (0.4&0.6)Ts
τ = (0.3&0.7)Ts
τ = 0&Ts
D-DSTC, τ = 0
BER
P/N0dB
Figure: Simulation BER, R = 4 relays, flat-fading channels, D-OFDM
DSTC (N = 64, Ncp = 1) and D-DSTC, using QOSTBC, τ1 = 0, τi = τ
for i = 2, 3, 4.26

27. Introduction
D-OFDM DSTC
Simulation
Summary
Simulation Results
0 5 10 15 20 25 30 35
10
−4
10
−3
10
−2
10
−1
10
0
τ = 0.5Ts
τ = (0.4&0.6)Ts
τ = (0.3&0.7)Ts
τ = 0&Ts
BER
P/N0dB
Figure: Simulation BER, R = 2 relays over frequency-selective channels,
D-OFDM DSTC (N = 64, Ncp = 7), using Alamouti code and QPSK,
τ1 = 0, τ2 = τ.27

28. Introduction
D-OFDM DSTC
Simulation
Summary
Simulation Results
0 5 10 15 20 25
10
−4
10
−3
10
−2
10
−1
10
0
τ = (0.4&0.6)Ts
τ = (0.3&0.7)Ts
τ = (0.2&0.8)Ts
τ = 0&Ts
BER
P/N0dB
Figure: Simulation BER, R = 4 relays over frequency-selective channels,
D-OFDM DSTC (N = 64, Ncp = 7), using QOSTBC, τ1 = 0, τi = τ for
i = 2, 3, 4.28

29. Introduction
D-OFDM DSTC
Simulation
Summary
Summary
Relay networks in frequency-selective channels
Synchronization Errors
Differential encoding and decoding with and OFDM approach
No channel or delay requirement
3RN coherence interval required
Thank You!
29