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

Differential Amplify-and-Forward Relaying inTime-Varying Rayleigh Fading Channels

M. R. Avendi and Ha H. Nguyen

Department of Electrical & Computer EngineeringUniversity of Saskatchewan

Saskatoon, SK, Canada, [email protected], [email protected]

April, 2013

M. R. Avendi and Ha H. Nguyen ( Department of Electrical & Computer Engineering University of Saskatchewan Saskatoon, SK, Canada,Differential 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 spectralefficiency

Multiple antennas cannot be implemented in many applications suchas mobile units due to lack of space

Proposed solution is cooperative diversity [1]


Cooperative Communications

Cooperative communication utilizes other users to construct a virtualantenna array

Relay protocols [2]– Decode-and-Forward: Decodes the received signal from Source andre-transmits to Destination– Amplify-and-Forward (AF): Amplifies the received signal fromSource 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-varyingenvironment

Differential modulation and non-coherent detection can be used toavoid channel estimation [3, 4]

In slow-fading channels, 3 dB performance loss exists betweencoherent and non-coherent detection

For fast-fading channels there would be higher loss that needs to beexamined!


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 = {e j2π(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]|2Received 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− α2ijeij[k ]

ij ∈ {sd, sr, rd}αij = J0(2πfijn) auto-correlationeij ∼ 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 andobtained 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− α2sd

√P0s[k]esd[k],

nsd[k] ∼ CN (0, σ2nsd

), σ2nsd

= 1 + α2sd + (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, σ2nrd

),σ2nrd

= 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: bopt0 = αsd/σ2nsd

, bopt1 = α/σ2nrd

Conventional (CDD) weights: bcdd0 = 1/2, bcdd1 = 1/2(1 + A2)

Proposed weights:

b0 = αsd/[1 + α2sd + (1− α2

sd)P0]

b1 = α/[(1 + α2)(1 + A2) + (1− α2)A2P0]

The output of the combiner can be used for non-coherent detectionby minimizing below over all M-PSK symbolsv̂ [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 + 12 sin2 θ

γsd|dmin|2dθ

γsd = α2sdP0/(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

1sin2(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 ineach path hits a wall as: lim

P0→∞γsd = α2

sd/(2(1 − α2sd)) and

limP0→∞

E [γrd] = α2/(2(1 − α2))

Consequently an error floor exists at high transmit power as:

limP0→∞

Ps(E12) =1

2− α2

sd(1− α2)

2(α2sd − α2)

√

α2sd|dmin|2

α2sd|dmin|2 + 4(1− α2

sd)

+α2(1− α2

sd)

2(α2sd − α2)

√

α2|dmin|2α2|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

100

Simulation CDDSimulation 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 5510

−6

10−5

10−4

10−3

10−2

10−1

100

Simulation CDDSimulation 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-nodenetwork 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 wasprovided

The existence of an error floor and SNR walls in fast-fading channelswere 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 andoutage 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 CooperativeCommunications.”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-ForwardWireless Relay Networks.”IEEE Trans. on Vehi. Tech. July. 2009

5-M. R. Avendi and H. H. Nguyen“Performance of Differential Amplify-and-Forward Relaying inMulti-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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