Differential Dual-Hop Relaying over Time-Varying Rayleigh-Fading Channels

Motivation
System Model
Two-Symbol Detection
Multiple-Symbol Detection
Simulation
Summary
Diﬀerential Dual-Hop Relaying over Time-Varying
Rayleigh-Fading Channels
M. R. Avendi and Ha H. Nguyen
Department of Electrical & Computer Engineering
University of Saskatchewan
Canada
June, 2013
1

Motivation
System Model
Simulation
Summary
Outline
1 Motivation
Cooperative Communications
2 System Model
3 Two-Symbol Detection
Time-Series Model
Non-Coherent Detection
BER Performance Analysis
4 Multiple-Symbol Detection
5 Simulation
Illustrative Results
6 Summary
2

Motivation
System Model
Simulation
Summary
Users help each other
Leverage coverage problems
Coverage extension
Relay
Shadow
Base Station
Relays
3

Motivation
System Model
Simulation
Summary
Dual-Hop Relaying
Source sends, Relay listen
Relay re-broadcasts its received signal
Source
h1
h2
Destination
Relay
4

Motivation
System Model
Simulation
Summary
Relay Protocols
Decode-and-Forward
Amplify-and-Forward (AF), (ﬁgure taken from reference 1)
Simplicity of relay function in Amplify-and-Forward relaying
1
A. Nosratinia, T. E. Hunter, A. Hedayat, ”Cooperative communication in wireless networks,”
Communications Magazine, IEEE , vol.42, no.10, pp.74,80, Oct. 2004
5

Motivation
System Model
Simulation
Summary
Detection
Coherent detection
Channel information required
Channel estimation: training symbols
Challenges: Estimation of SR channel, Mobility of users
Non-coherent detection
Diﬀerential modulations and demodulations
No channel estimation required
3 dB performance loss between coherent and non-coherent
detection in slow-fading channels
For fast-fading channels there would be higher loss that needs
to be examined!
6

Motivation
System Model
Simulation
Summary
Differential Dual-Hop Relaying (D-DH)
Rayleigh flat-fading channels, hi [k] ∼ CN(0, 1), i = 1, 2 at
time index k
Auto-correlation between two channel coefficients, n symbols
apart, E{hi [k]h∗
i [k + n]} = J0(2πfi n)
Transmission process is divided into two phases
h1[k] h2[k]
Source
Relay
Destination
7

Motivation
System Model
Simulation
Summary
Diﬀerential Dual-Hop Relaying: Phase I
Information bits convert to M-PSK symbols: v[k] ∈ V,
V = {ej2π(m−1)/M , m = 1, . . . , M}.
Diﬀerential encoding: s[k] = v[k]s[k − 1], s[0] = 1
h1[k]
Source
Relay
Destination
Received signal at Relay:
x[k] =
√
P0h1s[k] + w1[k], w1[k] ∼ CN(0, N0)
8

Motivation
System Model
Simulation
Summary
Diﬀerential Dual-Hop Relaying: Phase II
Relay multiplies received signal with A and forwards
h2[k]
Source
Relay
Destination
Received signal at Destination:
y[k] = A P0h[k]s[k] + w[k]
• Cascaded channel: h[k] = h1[k]h2[k]
• Equivalent noise: w[k] = Ah2[k]w1[k] + w2[k]
9

Motivation
System Model
Simulation
Summary
Time-Series Model
Channel Variation Over Time
Common assumption: slow-fading, hi [k] ≈ hi [k − 1]
Rayleigh fading, hi [k] ∼ CN(0, 1)
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
f
D
T
s
=.001
f
D
T
s
=.01
f
D
T
s
=.03
Amplitude
time index, k
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
f
D
T
s
=.001
fD
Ts
=.01
fD
Ts
=.03
time index, k
Auto-Correlation
10

Motivation
System Model
Simulation
Summary
Time-Series Model
Channel Time-Series Models
Direct channel:
hi [k] = αi hi [k − 1] + 1 − α2
i ei [k], i = 1, 2
αi = J0(2πfi n) auto-correlation
ei ∼ CN(0, 1), independent of hi [k − 1]
Cascaded channel:
h[k] = αh[k − 1] + 1 − α2h2[k − 1]e1[k]
α = α1α2: auto-correlation of cascaded channel
e1 ∼ CN(0, 1), independent of h[k − 1]
11

Motivation
System Model
Simulation
Summary
Time-Series Model
Two-Symbol Diﬀerential Detection
y[k] = αv[k]y[k − 1] + n[k]
n[k] = w[k]−αv[k]w[k−1]+ 1 − α2A P0h2[k − 1]s[k]e1[k]
Detection
ˆv[k] = arg min
v[k]∈V
|y[k] − v[k]y[k − 1]|2
(1)
12

Motivation
System Model
Simulation
Summary
Time-Series Model
Bit Error Rate
Pb(E) =
1
4π
π
−π
g(θ)J(θ)dθ (2)
J(θ) = b3(θ) 1 + (b1 − b2(θ))eb2(θ)
E1(b2(θ)) (3)
b1, b2(θ), b3(θ) depend on system parameters and channels
auto-correlation, E1(x) exponential integral function.
Error Floor
lim
(P0/N0)→∞
Pb(E) =
1
4π
π
−π
g(θ)
1 − α2
α2q(θ) + 1 − α2
dθ (4)
13

Motivation
System Model
Simulation
Summary
To overcome error ﬂoor
Take N received symbols: y = [ y[1], y[2], . . . , y[N] ]t
y = A P0diag{s}diag{h2}h1 + w (5)
where s = [ s[1], · · · , s[N] ]t
, h2 = [ h2[1], · · · , h2[N] ]t
,
h1 = [ h1[1], · · · , h1[N] ]t
and w = [ w[1], · · · , w[N] ]t
.
ML detection:
ˆs = arg max
s∈CN
E
h2
1
πN det{Ry}
exp −yH
R−1
y y . (6)
Ry covariance matrix of y, depends on h2
14

Motivation
System Model
Simulation
Summary
Replace Ry by Ry = E
h2
{Ry}
ˆs = arg min
s∈CN
yH
R
−1
y y = arg min
s∈CN
Us 2
(7)
where U = (LHdiag{y})∗ and L is obtained by the Cholesky
decomposition of C−1 = LLH, C = A2P0Rh + (1 + A2)N0IN.
Rh = toeplitz{ϕ1(0)ϕ2(0), . . . , ϕ1(N − 1)ϕ2(N − 1)}.
Solve by sphere decoding with low complexity
15

Motivation
System Model
Simulation
Summary
Simulation Setup
Correlated channels h1[k], h2[k] ∼ CN(0, 1)
Normalized Doppler frequencies f1, f2
Three simulation cases:
f1 f1 Channels status
Case I .001 .001 both slow-fading
Case II .01 .001 SR fast-fading
Case III .02 .01 both fast-fading
Ampliﬁcation factor: A = P1/(P0 + N0)
Power allocation: P0 = P1
16

Motivation
System Model
Simulation
Summary
- BER in diﬀerent cases using DBPSK
10 15 20 25 30 35 40 45 50 55 60
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Simulation, N=2
Analysis, N=2
MSDSD, N=10, Case II
MSDSD, N=10, Case III
P0/N0 (dB)
BER
Case I
Case II
Case III
Error Floor
17

Motivation
System Model
Simulation
Summary
BER in diﬀerent cases using DQPSK
10 15 20 25 30 35 40 45 50 55 60
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Simulation, N=2
Analysis, N=2
MSDSD, N=10, Case II
MSDSD, N=10, Case III
P0/N0 (dB)
BER
Case I
Case II
Case III
Error Floor
18

Motivation
System Model
Simulation
Summary
Summary
Diﬀerential dual-hop transmission in time-varying channels
Two-symbol non-coherent detection
• Channel time-series model
• Bit-error-rate analysis
• Error ﬂoor in fast fading channels
Multiple-symbol detection
Thank You!
19

Differential Dual-Hop Relaying over Time-Varying Rayleigh-Fading Channels

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