Differential Amplify-and-Forward Relaying in Time-Varying Rayleigh Fading Channels

Diﬀerential Amplify-and-Forward Relaying in
Time-Varying Rayleigh Fading Channels
M. R. Avendi and Ha H. Nguyen
Department of Electrical & Computer Engineering
University of Saskatchewan
Saskatoon, SK, Canada, S7N5A9
m.avendi@usask.ca, ha.nguyen@usask.ca
April, 2013
M. R. Avendi and Ha H. Nguyen ( Department of Electrical & Computer Engineering University of Saskatchewan Saskatoon, SK, CanadDiﬀerential Amplify-and-Forward Relaying in Time-Varying Rayleigh Fading ChannelsApril, 2013 1 / 18

Motivation
Demand for high date rate and better quality are increasing
Main obstacle to achieve better performance is channel fading
Solution is to use diversity techniques
Spatial diversity using multiple antennas gives better spectral
eﬃciency
Multiple antennas cannot be implemented in many applications such
as mobile units due to lack of space
Proposed solution is cooperative diversity [1]

Cooperative Communications
Cooperative communication utilizes other users to construct a virtual
antenna array
Relay protocols [2]
– Decode-and-Forward: Decodes the received signal from Source and
re-transmits to Destination
– Amplify-and-Forward (AF): Ampliﬁes the received signal from
Source and forwards to Destination
Source Destination
Relay
Direct channel
Cascaded channel

Amplify-and-Forward (AF) Relaying
Simplicity of relay function in AF relaying makes it attractive
Coherent detection: requires channel state information of all links
Channel estimation is a challenge specially in time-varying
environment
Diﬀerential modulation and non-coherent detection can be used to
avoid channel estimation [3, 4]
In slow-fading channels, 3 dB performance loss exists between
coherent and non-coherent detection
For fast-fading channels there would be higher loss that needs to be
examined!

System Model: Differential AF (D-AF) Relay Network
All links are Rayleigh flat-fading denoted by hij[k] ∼ CN(0, 1),
ij ∈ {sd, sr, rd} at time index k
Auto-correlation between two channel coefficients, n symbols apart,
E{hij[k]h∗
ij[k + n]} = J0(2πfijn)
Transmission process is divided into two phases
hsr[k] hrd[k]
hsd[k]
Source
Relay
Destination

System Model: 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
Received signal at Relay:
ysd [k] =
√
P0hsds[k] + wsd[k], wsd[k] ∼ CN(0, 1)
Received signal at Destination:
ysr[k] =
√
P0hsr[k]s[k] + wsr[k], wsr[k] ∼ CN(0, 1)

System Model: Phase II
Relay ampliﬁes the received signal with A and forwards
Received signal at Destination: yrd[k] = A
√
P0h[k]s[k] + w[k]
– Cascaded channel: h[k] = hsr[k]hrd[k]
– Equivalent noise: w[k] = Ahrd[k]wsr[k] + wrd[k]
Given hrd[k], w[k] ∼ CN(0, σ2), σ2 = 1 + A2|hrd[k]|2
Received SNR: ρ = A2P0|hrd[k]|2/(1 + A2|hrd[k]|2)

Time-Series Models
Slow-fading: hsd[k] ≈ hsd[k − 1], h[k] ≈ h[k − 1]
Time-varying channels:
Rayleigh channels: hij[k] = αijhij[k − 1] + 1 − α2
ijeij[k]
ij ∈ {sd, sr, rd}
αij = J0(2πfijn) auto-correlation
eij ∼ CN(0, 1) independent of hij[k − 1]
Cascaded channel: h[k] = αh[k − 1] +
√
1 − α2hrd[k − 1]esr[k]
α = αsrαrd is the auto-correlation of the cascaded channel and
obtained by multiplying the auto-correlations of SR and RD channels.

Detection
ysd[k] = αsdv[k]ysd[k − 1] + nsd[k]
nsd[k] = wsd[k] − αsdv[k]wsd[k − 1] + 1 − α2
sd
√
P0s[k]esd[k],
nsd[k] ∼ CN(0, σ2
nsd
), σ2
nsd
= 1 + α2
sd + (1 − α2
sd)P0
yrd[k] = αv[k]yrd[k − 1] + nrd[k]
nrd[k] = w[k] − αv[k]w[k − 1] + 1 − α2A P0hrd[k − 1]s[k]esr[k]
– Given hrd[k]: nrd[k] ∼ CN(0, σ2
nrd
),
σ2
nrd
= σ2(1 + α2 + (1 − α2)ρ)

Combining
Maximum Ratio Combining (MRC) method
ζ = b0y∗
sd[k − 1]ysd[k] + b1y∗
rd[k − 1]yrd[k]
Optimum weights: bopt
0 = αsd/σ2
nsd
, bopt
1 = α/σ2
nrd
Conventional (CDD) weights: bcdd
0 = 1/2, bcdd
1 = 1/2(1 + A2)
Proposed weights:
b0 = αsd/[1 + α2
sd + (1 − α2
sd)P0]
b1 = α/[(1 + α2
)(1 + A2
) + (1 − α2
)A2
P0]
The output of the combiner can be used for non-coherent detection
by minimizing below over all M-PSK symbols
ˆv[k] = arg min
v[k]∈V
|ζ − v[k]|2.

Error Performance
Let v1, v2 ∈ V, v1 is sent and v2 is detected
PEP: Ps(E12) = Ps(v1 → v2), dmin = v1 − v2
Ps(E12) =
1
π
π/2
0
I1(θ)
1 + 1
2 sin2
θ
γsd|dmin|2
dθ
γsd = α2
sdP0/(2P0(1 − α2
sd) + 4)
I1(θ) = ǫ1(θ) 1 + (β1 − β2(θ))eβ2(θ)E1(β2(θ))
ǫ1(θ) = 4(1−α2)A2P0+8A2
1
sin2(θ)
α2A2P0|dmin|2+4(1−α2)A2P0+8A2
β1 = 4/[2(1 − α2)A2P0 + 4A2]
β2(θ) = 8/[ 1
sin2
(θ)
α2A2P0|dmin|2 + 4(1 − α2)A2P0 + 8A2]

Error Floor
Performance is related to the auto-correlations of the channels
For fast-fading channels at high transmit power, eﬀective SNR in
each path hits a wall as: lim
P0→∞
γsd = α2
sd/(2(1 − α2
sd)) and
lim
P0→∞
E[γrd] = α2/(2(1 − α2))
Consequently an error ﬂoor exists at high transmit power as:
lim
P0→∞
Ps(E12) =
1
2
−
α2
sd(1 − α2)
2(α2
sd − α2)
α2
sd|dmin|2
α2
sd|dmin|2 + 4(1 − α2
sd)
+
α2(1 − α2
sd)
2(α2
sd − α2)
α2|dmin|2
α2|dmin|2 + 4(1 − α2)

Simulation
Three simulation scenarios:
fsd fsr frd
Scenario I .001 .001 .001
Scenario II .01 .01 .001
Scenario III .05 .05 .01
Ampliﬁcation factor: A = P1/(P0 + 1)
Equal power allocation: P0 = P1 = P/2

Simulation Results
BER of D-AF relaying in three scenarios using DBPSK
0 5 10 15 20 25 30 35 40 45 50 55
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Simulation CDD
Simulation TVD
Error Floor Optimum
P (dB)
BER
Scenario I
Scenario II
Scenario III
Lower bound

Simulation Results
BER of D-AF relaying in three scenarios using DQPSK
0 5 10 15 20 25 30 35 40 45 50 55
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Simulation CDD
Simulation TVD
Error Floor Optimum
P (dB)
BER
Scenario I
Scenario II
Scenario III
Lower bound

Summary
In this paper:
Diﬀerential Amplify-and-Forward (D-AF) relaying for a three-node
network was presented
A new time-series model for the cascaded channel was given
New combining weights for MRC method were proposed
Performance analysis of the system in time-varying channels was
provided
The existence of an error ﬂoor and SNR walls in fast-fading channels
were shown
This work is extended to multi-node relay networks in [5]

References I
1- A. Sendonaris and E. Erkip and B. Aazhang
“ User cooperation diversity. Part I. System description.”
IEEE Trans. on Wireless Comm. Nov. 2003
2- J.N. Laneman and D.N.C. Tse and G.W. Wornell
“ Cooperative diversity in wireless networks: Eﬃcient protocols and
outage behavior.”
IEEE Trans. on Info. Theory Dec. 2004
3-T. Himsoon and W. Su and K.J.R. Liu
“Diﬀerential Transmission for Amplify-and-Forward Cooperative
Communications.”
Signal Processing Letter Sept. 2005

References II
4-Z. Fang and L. Li and X. Bao and Z. Wang
“ Generalized Diﬀerential Modulation for Amplify-and-Forward
Wireless Relay Networks.”
IEEE Trans. on Vehi. Tech. July. 2009
5-M. R. Avendi and H. H. Nguyen
“Performance of Diﬀerential Amplify-and-Forward Relaying in
Multi-Node Wireless Communications.”
To appear in IEEE Tran. on Vehi. Techno. April 2013

Differential Amplify-and-Forward Relaying in Time-Varying Rayleigh Fading Channels

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