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

MotivationSystem Model

Two-Symbol DetectionMultiple-Symbol Detection

SimulationSummary

Differential Dual-Hop Relaying over Time-Varying

Rayleigh-Fading Channels

M. R. Avendi and Ha H. Nguyen

Department of Electrical & Computer EngineeringUniversity of Saskatchewan

Canada

June, 2013

1



SimulationSummary

Outline

1 MotivationCooperative Communications

2 System Model

3 Two-Symbol DetectionTime-Series ModelNon-Coherent DetectionBER Performance Analysis

4 Multiple-Symbol DetectionMultiple-Symbol Detection

5 SimulationIllustrative Results

6 Summary

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SimulationSummary

Cooperative Communications


Users help each otherLeverage coverage problems

Coverage extension

RelayShadow

Base Station

Relays

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SimulationSummary


Dual-Hop Relaying

Source sends, Relay listen

Relay re-broadcasts its received signal

Source

h1h2

Destination

Relay

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SimulationSummary


Relay Protocols

Decode-and-Forward

Amplify-and-Forward (AF), (figure taken from reference 1)

Simplicity of relay function in Amplify-and-Forward relaying1A. Nosratinia, T. E. Hunter, A. Hedayat, ”Cooperative communication in wireless networks,”

Communications Magazine, IEEE , vol.42, no.10, pp.74,80, Oct. 2004

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SimulationSummary


Detection

Coherent detection

Channel information requiredChannel estimation: training symbolsChallenges: Estimation of SR channel, Mobility of users

Non-coherent detection

Differential modulations and demodulationsNo channel estimation required3 dB performance loss between coherent and non-coherentdetection in slow-fading channelsFor fast-fading channels there would be higher loss that needsto be examined!

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SimulationSummary

Differential Dual-Hop Relaying (D-DH)

Rayleigh flat-fading channels, hi [k] ∼ CN (0, 1), i = 1, 2 attime index k

Auto-correlation between two channel coefficients, n symbolsapart, E{hi [k]h∗i [k + n]} = J0(2πfin)

Transmission process is divided into two phases

h1[k]h2[k]

Source

Relay

Destination

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SimulationSummary

Differential Dual-Hop Relaying: 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

h1[k]

Source

Relay

Destination

Received signal at Relay:x [k] =

√P0h1s[k] + w1[k], w1[k] ∼ CN (0,N0)

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SimulationSummary

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]

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SimulationSummary

Time-Series ModelNon-Coherent DetectionBER Performance Analysis

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 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

fD

Ts=.001

fD

Ts=.01

fD

Ts=.03

Amplitude

time index, k0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

fD

Ts=.001

fD

Ts=.01

fD

Ts=.03

time index, k

Auto-Correlation

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SimulationSummary


Channel Time-Series Models

Direct channel:

hi [k] = αihi [k − 1] +√

1− α2i ei [k], i = 1, 2

αi = J0(2πfin) auto-correlationei ∼ 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 channele1 ∼ CN (0, 1), independent of h[k − 1]

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SimulationSummary


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

v [k] = arg minv [k]∈V

|y [k]− v [k]y [k − 1]|2 (1)

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SimulationSummary


BER Performance Analysis

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 channelsauto-correlation, E1(x) exponential integral function.

Error Floor

lim(P0/N0)→∞

Pb(E ) =1

4π

π∫

−π

g(θ)1− α2

α2q(θ) + 1− α2dθ (4)

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SimulationSummary

Multiple-Symbol Detection


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:

s = arg maxs∈CN

{

Eh2

{1

πNdet{Ry}exp

(

−yHR−1y y

)}}

. (6)

Ry covariance matrix of y, depends on h2

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SimulationSummary


Replace Ry by Ry = Eh2{Ry}

s = arg mins∈CN

{

yHR−1y y

}

= arg mins∈CN

{‖Us‖2

}(7)

where U = (LHdiag{y})∗ and L is obtained by the Choleskydecomposition 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

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SimulationSummary

Illustrative Results

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

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SimulationSummary



- BER in different cases using DBPSK

10 15 20 25 30 35 40 45 50 55 6010

−5

10−4

10−3

10−2

10−1

100

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

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SimulationSummary



BER in different cases using DQPSK

10 15 20 25 30 35 40 45 50 55 6010

−5

10−4

10−3

10−2

10−1

100

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

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SimulationSummary

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!

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

Engineering

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