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arXiv:1111.7069v1 [cs.IT] 30 Nov 2011

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Differential Modulation for Bi-directional

Relaying with Analog Network Coding

Lingyang Song, Yonghui Li, Anpeng Huang, Bingli Jiao,

and Athanasios V. Vasilakos

Abstract

In this paper, we propose an analog network coding scheme with differential modulation (ANC-

DM) using amplify-and-forward protocol for bidirectional relay networks when neither the source

nodes nor the relay knows the channel state information (CSI). The performance of the proposed

ANC-DM scheme is analyzed and a simple asymptotic bit error rate (BER) expression is derived.

The analytical results are verified through simulations. It is shown that the BER performance of

the proposed differential scheme is about 3 dB away from that of the coherent detection scheme.

To improve the system performance, the optimum power allocation between the sources and the

relay is determined based on the simplified BER. Simulation results indicate that the proposed

differential scheme with optimum power allocation yields 1-2 dB performance improvement over

an equal power allocation scheme.

Index Terms

Differential modulation, bi-directional relaying, analog network coding, amplify-and-forward

protocol

I. INTRODUCTION

Bi-directional relay communication has attracted considerable interest recently [1]–[6],

and various bi-directional relay protocols for wireless systems have been proposed [1]–[3].

Copyright (c) 2010 IEEE. Personal use of this material is permitted. However, permission to use this material for any

other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org.

This work was partially supported by the National Natural Science Foundation of China under Grant number 60972009

and 60811130529.

LingyangSong,BingliJiao andAnpengHuangarewithPekingUniversity,China(e-mail:

lingyang.song@pku.edu.cn,jiaobl@pku.edu.cn,hapku@pku.edu.cn).

Yonghui Li is with University of Sydney, Australia (e-mail: lyh@ieee.org).

Athanasios V. Vasilakosis is with University of Western Macedonia, Greece (e-mail: vasilako@ath.forthnet.gr).

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In [1] [2], the conventional network coding scheme is applied to the bi-directional relay

network. Two source nodes transmit to the relay, separately. The relay decodes the received

signals, performs binary network coding, and then broadcasts network coded symbols back

to both source nodes. However, this scheme may cause irreducible error floor due to the

detection errors which occur at the relay node.

In [3]–[5], an amplify and forward based network coding scheme, referred to as the analog

network coding, was proposed. In this scheme, both source nodes transmit at the same time so

that the relay receives a superimposed signal. The relay then amplifies the received signal and

broadcasts it to both source nodes. Analog network coding is particularly useful in wireless

networks as the wireless channel acts as a natural implementation of network coding by

summing the wireless signals over the air.

Almost all existing works in bi-directional relay communications using analog network

coding assume that the sources and the destination have perfect knowledge of channel state

information (CSI) for all transmission links. As a result, coherent detection can be readily

employed at either sources or relay, or both [1]–[5]. In some scenarios, e.g. the slow fading

environment, the CSI is likely to be acquired by the use of pilot symbols. However, when

the channel coefficients vary fast, channel estimation may become difficult. In addition,

the channel estimation increases computational complexity in the relay node and reduces

the data rate. Moreover, it would be difficult for the destination to acquire the source-to-

relay channel perfectly through pilot signal forwarding without noise amplification. Hence,

differential modulation without need of any CSI would be a practical solution.

In a differential bi-directional relay network, each source receives a superposition of

differentially encoded signals from the other source, and it has no knowledge of CSI of both

channels. All these problems present a great challenge for designing differential modulation

schemes in two-way relay channels. In [6], differential receivers for differential two-way

relaying were presented using analog network coding. These non-coherent schemes were

realized by averaging the channel coefficients of the substraction of adjacent received signals.

However, the approaches may result in more than 3 dB performance loss compared to the

coherent schemes due to instantaneous detection errors.

In this paper, we propose an analog network coding scheme with differential modula-

tion (ANC-DM) using amplify-and-forward protocol for bidirectional relay networks so that

the CSI is not required at both sources and the relay. The performance of the proposed

ANC-DM scheme is analyzed and a simple asymptotic bit error rate (BER) expression is

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derived. The analytical results are verified through simulations. They show that the proposed

differential scheme is about 3 dB away compared to the coherent detection scheme. To

improve the system performance, the optimum power allocation between the sources and

the relay is determined based on the provided simplified BER. Simulation results show that

optimum power allocation yields 1-2 dB performance improvement over an equal power

allocation scheme.

Note that unlike [6], the destination realizes differential detection by subtracting away its

own contribution in the received signals based on the power estimation. Besides, our scheme

requires linear complexity, while the detectors in [6] are much complex and non-linear which

may calculate the modified Bessel function of second kind. Note also that the application of

differential modulation for the bi-directional relaying in [1] [2] with digital network coding

is relatively straightforward as the received signals at the relay from the sources can be

decoded separately due to the use of orthogonal transmissions. However, for amplify-and-

forward relaying [3]–[5], the relay receives a superposition of differentially encoded signals

from the sources which makes the final detection at the source difficult.

The rest of the paper is organized as follows: In Section II, we describe the proposed

differential modulation scheme. Theoretical analysis is given in Section III. Section IV

presents the optimal transmit power allocation between the sources and the relay. Simulation

results are provided in Section V. In Section VI, we draw the main conclusions.

Notation: Boldface lower-case letters denote vectors, (·)∗stands for complex conjugate,

(·)Trepresents transpose, E is used for expectation, Var represents variance, ?x?2= xHx,

and R(·) denotes real part.

II. DIFFERENTIAL MODULATION FOR BIDIRECTIONAL RELAY NETWORKS

A. Differential Encoding

We consider a three-node bi-directional relay network consisting of two source nodes,

denoted by S1and S2, and one relay node, denoted by R. All nodes are equipped with one

antenna and operate in a half-duplex way so that the complete transmission can be divided into

two phases, as shown in Fig. 1. In the first phase, both source nodes simultaneously send the

differentially encoded information to the relay, and in the second phase, the relay broadcasts

the combined signals to both sources. Let c1(t)∈A denote the symbol to be transmitted by

the source S1at the time t, where A represents a unity power M-PSK constellation set. In

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the differential modulation bi-directional relay system, the signal s1(t) sent by the source S1

is given by

s1(t) = s1(t − 1)c1(t),c1(t)∈A

(1)

Similarly, the signal transmitted by S2at the time t is given by

s2(t) = s2(t − 1)c2(t),c2(t)∈A

(2)

B. Differential Decoding

In the bi-directional relayed transmission, the source nodes first broadcast the information

to the relay. For simplicity, we assume that the fading coefficients are constant over one

frame of length L, and change independently from one frame to another. The received signal

in the relay at time t can be expressed as

yr(t) =√p1h1s1(t) +√p2h2s2(t) + nr(t).

(3)

where p1 and p2 represent the transmit power at S1 and S2 respectively, h1 and h2 are

the Rayleigh fading coefficients with zero mean and unit variance between S1and R, and

between S2and R, respectively, nr(t) denotes zero mean complex Gaussian random variable

with two sided power spectral density of N0/2 per dimension, and we furthermore assume

S1, S2, and R have the same noise variance.

In the second phase, the relay R amplifies yr(t) by a factor β and then broadcasts its

conjugate, denoted by y∗

r(t), to both S1and S2. The corresponding signal received by S1at

time t, denoted by y1(t), can be written as

y1(t) = β√prh1y∗

r(t) + n1(t)

= µs∗

1(t) + νs∗

2(t) + w1(t),

(4)

where β = (p1|h1|2+ p2|h2|2+ N0)−1

β√p1pr|h1|2, ν ? β√p2prh1h∗

ANC schemes [3]–[5], y∗

2, prrepresents the transmit power by the relay, µ ?

2, and w1(t) ? β√prh1n∗

r(t)+n1(t). Note that unlike traditional

r(t) is transmitted from the relay, which obviously yields µ > 0.

The reason of doing this is to make the receiver easily estimate µ and then subtract µs∗

1(t)

in (4) for differential detection.

As the relay has no CSI, the normalization factor β has to be obtained indirectly. We may

rewrite the received signals in (3) in a vector format, given by

yr=√p1h1s1+√p2h2s2+ nr,

(5)

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where yr= [yr(1),...,yr(L)]T, s1 = [s1(1),...,s1(L)]T, s2 = [s2(1),...,s2(L)]T, and

nr= [nr(1),...,nr(L)]T. To estimate the average receive power, we multiply the received

signals by its Hermitian transpose as

?yr?2= p1|h1|2sH

1s1+ p2|h2|2sH

+ 2√p1R{h1nH

2s2+ 2√p1p2R{h∗

rs1} + 2√p2R{h2nH

1h2sH

1s2}

rs2} + nH

rnr

(6)

By taking the expectation of (6), β can be then approximated at high SNR by

?

where E{sH

0.

β =

E{yH

ryr}

L

≈

?

?yr?2

L

,

(7)

1s1} = E{sH

2s2} = L, E{nH

rnr} = LN0, and E{sH

1s2} = E{nH

rs1} = E{nH

rs2} =

Similarly, the received signal at S2can be calculated as

y2(t) = β√prh2y∗

r(t) + n2(t),

(8)

As S1and S2are mathematically symmetrical, as shown in (4) and (8), for simplicity, we

in the next only discuss the decoding as well as the corresponding theoretical analysis for

signals received by S1.

Recalling the differential encoding process in (2), (4) can be further written as

y1(t) = µs∗

1(t) + νs∗

2(t) + w1(t),

= µs∗

1(t) + νs∗

2(t − 1)c∗

2(t) + w1(t),

(9)

Obviously, since s1(t) is known, to decode c2(t) in (9), µs∗

1(t) should be removed. To do

this, we have to first estimate µ, where µ > 0.

Similar to (10), the received signal in (4) can be expressed in a vector form as follows

y1= µs1+ νs2+ w1,

(10)

where y1= [y1(1),...,y1(L)]Tand w1 = [w1(1),...,w1(L)]T. At high SNR, we may

approximately obtain

µ2+ |ν|2≈ yH

1y1/L.

(11)

Since the source node S1can retrieve its own information s1(t − 1) and c1(t), based on (1)

and (9), we have the following transformation

? y1(t) ? c∗

1(t)y1(t − 1) − y1(t)

= νs∗

2(t − 1)(c1(t) − c2(t))∗+ ? w1(t),

(12)

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where ? w1(t) ? c∗

1(t)w1(t − 1) + w1(t). Then, |ν|2can be approximately calculated in a

similar way as (11)

|ν|2≈

? yH

1? y1

LE[|s2(t − 1)|2]E[|c1(t) − c2(t)|2],

(13)

where ? y1= [? y1(1),..., ? y1(L − 1)]T, E[|s2(t − 1)|2] = 1 , and the calculation of E[|c1(t) −

where ∆ ?

L

−

to the noise effect, and thus we set µ ≈ 0 instead. The estimation method given in (14) is

evaluated in Fig. 2.

c2(t)|2] is given in Section-II-C. As µ is positive, by combining (7), (11) and (13), we have

µ ≈

√∆, ∆ > 0

0,∆ ≤ 0

(14)

yH

1y1

? yH

1? y1

LE[|s2(t−1)|2|c1(t)−c2(t)|2]. Note that in low SNR, ∆ could be negative due

By subtracting µs∗

1(t), (9) can be further written as

y′

1(t) ? y1(t) − µs1(t)

= νs∗

2(t − 1)c∗

1(t − 1) − w1(t − 1))c∗

2(t) + w1(t)

= (y′

2(t) + w1(t),

(15)

Finally, the following linear decoder can be used to recover c2(t)

? c2(t) = arg max

c2(t)∈ARe?y′

1(t)y′

1

∗(t − 1)c2(t)?.

(16)

And c1(t) can be differentially decoded in a similar way by the source S2. Note that in

comparison to traditional differential modulation [7], the extra complexity comes from the

µ estimation in (14), which is linear and only comprises a few number of additions and

multiplications. But the receiver in [6] requires complicated computations, such as the zeroth-

order modified Bessel function of the second kind.

By ignoring the second order term, the corresponding SNR of the proposed differential

detection scheme can be written as

γd≈

|ν|2

2Var{w1(t)}

β2p2pr|h1|2|h2|2

2(β2prN0|h1|2+ N0)

ψ2ψr|h1|2|h2|2

2((ψ1+ ψr)|h1|2+ ψ2|h2|2+ 1),

≈

≈

(17)

where Var{w1(t)} = β2prN0|h1|2+ N0, ψ1? p1/N0, ψ2? p2/N0, and ψr? pr/N0.

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Alternatively, if coherent detection is used, under the assumption of h1, h2and N0available

at S1, after subtracting µs1(t)∗from y1(t) in (4), the corresponding SNR can be calculated

as

γc?

|ν|2

Var{w1(t)}

(ψ1+ ψr)|h1|2+ ψ2|h2|2+ 1.

=

ψ2ψr|h1|2|h2|2

(18)

By comparing (17) and (18), we can easily obtain

γd≈γc

2

(19)

which clearly indicates that the differential detector in (16) suffers around 3 dB performance

loss compared to the coherent scheme.

C. The Calculation of E[|c1(t) − c2(t)|2] in (13)

From (13), the average power of c1(t) − c2(t) needs to be calculated. When M-PSK

constellations are applied, the number of symbols produced in the new constellation by

c1(t) − c2(t) is finite. Hence, it is easy to derive the average power of the new constellation

sets. Note that the value of c1(t)−c2(t) can be equal to zero, which may affect the estimation

accuracy in (13).

In order to overcome this problem, we may properly choose a rotation angle for the symbol

modulated in source S2by c2(t)e−jθ, ensuring that c1(t) − c2(t) in (13) is nonzero. For a

M-PSK constellation, the effective rotation angle is in the interval [−π/M,π/M] from the

symmetry of symbols. For a regular and symmetrical constellation, the rotation angle may

be simply set as θ = π/M. Similar approach may be used to generate the rotation angle for

other types of constellations.

Here, we give two examples on how to compute the average symbol power:

1. Supposing BPSK constellation {−1,1} is used, we have c1(t) − c2(t) ∈ {−2,0,2}.

Hence, the calculation of average power in the new set is straightforward.

2. Supposing S1uses the BPSK set {−1,1}, by constellation rotation, S2can use {−j,j}.

And we can get c1(t)−c2(t) ∈ {−1−j,−1+j,1−j,1+j}. Hence, it is also easy to derive

the average power in the new constellation set.

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III. PERFORMANCE ANALYSIS

For simplicity, we in this section analyze the BER performance using BPSK for the

proposed ANC-DM scheme, and we assume p1 = p2 = ps and ps = λpr, where λ > 0,

and thus, ψ1= ψ2= ψs= λψr. (17) can be rewritten as

γd≈

ψsψ′

r|h1|2|h2|2

2(1 + λ)(ψ′r|h1|2+ ψs|h2|2+ 1),

(20)

where ψ′

r? (1 + λ)ψr.

Let X = γd, and the BER for BPSK modulation can generally be expressed as

BER = E[Q(2X)] =

1

2√π

?∞

0

exp(−x)FX(x)

√x

dx,

(21)

where Q(·) is the Gaussian-Q function, FX(x) is the cumulative distribution function (CDF)

of X. The right side of the equation can be readily obtained by integration by parts. The

above expression is useful as it allows us to obtain the BER directly in terms of the CDF of

X.

By using a general result from [8], the BER in (21) can be approximated in the high SNR

regime by considering a first order expansion of the CDF of X. Specifically, if the first order

expansion of the CDF of X can be written in the form

FX(x) =

αxN+1

N+1(N + 1)λ

+ o(xN+1+ε),ε > 0,

(22)

where λ represents the average transmit SNR. At high SNR, the asymptotic BER is given

by [8]

BER =

αΓ(N +3

2√πλ

2)

N+1(N + 1)

+ o?(λ)−(N+1)?.

?

(23)

The PDF of X can be obtained with the help of [9]

PX(x) =8(1 + λ)2xexp(−2(1 + λ)(ψ−1

s

+ ψ′−1

r )x)

ψsψ′r

ψs+ ψ′

?ψsψ′r

+2K0

r

×K1

?

4(1 + λ)x

?ψsψ′r

U (2(1 + λ)x),

?

?

4(1 + λ)x

?ψsψ′r

??

(24)

where K0(·) and K1(·) are the zeroth-order and first-order modified Bessel functions of

the second kind, respectively, and U(·) is the unit step function. Note that the exact BER,

which is complicated in computation, does not have a closed-form solution. However, at high

SNR, when z approaches zeros, the K1(z) function converges to 1/z [10], and the value of

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the K0(z) function is comparatively small, which could be ignored for asymptotic analysis.

Hence, PX(x) in (24) can be approximated as

PX(x) ≈2(1 + λ)(ψs+ ψ′

r)exp(−2(1 + λ)(ψ−1

ψsψ′r

s

+ ψ′−1

r )x)

.

(25)

For the ANC-DM, the CDF of destination SNR γdcan be approximated as

FX(x) ≈ 1 − exp?−2(1 + λ)(ψ−1

≈ 2(1 + λ)(ψ−1

s

+ ψ′−1

r )x?

s

+ ψ′−1

r

)x + o(x1+ε).

(26)

Finally, comparing (26) with (22) and (23), the asymptotic BER of ANC-DM at high SNR

can be approximated as

BER ≈2(1 + λ)Γ(3

=(1 + λ)(ψ−1

2)

2√π

(ψ−1

s

+ ψ′−1

r )

s

2

+ ψ′−1

r )

.

(27)

IV. TRANSMIT POWER ALLOCATION

In this section, we discuss how to allocate power to both sources and the relay subject to

total transmission power constraint. It can be seen from (27) that the asymptotic BER of the

proposed differential modulation scheme depends non-linearly upon psand pr. Hence, when

the total transmit power is fixed, 2ps+ pr= p, the power allocation problem over Rayleigh

channels can be formulated to minimize the asymptotic BER at high SNR in (27)

minBER

s.t. 2ps+ pr= p (0<ps<p, 0<pr<p),

(28)

where we assume p1= p2= psand ps= λpr.

The power allocation problem is to find pssuch that the BER in (27) is minimized subject

to the power constraint by solving the following optimization problem

L(ps) = BER + ξ(2ps+ pr− p),

(29)

where ξ is a positive Lagrange multiplier. The necessary condition for the optimality is found

by setting the derivatives of the Lagrangian in (29) with respect to psand prequal to zero,

respectively. Reusing the power constraint, we can calculate at high SNR that

ps=p

4,

pr=p

2,

(30)

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which indicates the power allocated in the relay should be equal to the total transmit power

at both sources in order to compensate the energy used to broadcast combined information

in one time slot.

V. SIMULATION RESULTS

In this section, we provide simulation results for the proposed ANC-DM scheme. We

also include corresponding coherent detection results for comparison. All simulations are

performed for a BPSK modulation over the Rayleigh fading channels. The frame length is

L = 100. For simplicity, we assume that 2ps+pr= p = 3, and S1, S2and R have the same

noise variance N0. The SNR ψscan be then calculated as ψs= ps/N0.

Fig. 2 shows the simulated BER performance for differential and coherent ANC schemes in

bi-directional relaying without using constellation rotation. Equal transmit power allocation

is applied: ps = pr = p/3. It can be seen that the differential scheme suffers around 3

dB performance loss compared to the coherent ANC scheme, which has been validated by

(19). We also include the Genie-aided result by assuming that µ is perfectly known by the

source such that traditional differential decoding can be performed. It shows from the results

that there is almost no performance loss using the estimation method in (14) which clearly

justifies the robustness of the proposed differential decoder.

It is worthwhile mentioning that, in [6], it uses similar Genie-aided result as a benchmark

as well. The major difference is that the detectors in [6] have much inferior performance than

the Genie-aided result, and it has about 6dB performance loss in comparison to the coherent

detection scheme. However, our proposed detection algorithm has comparable performance

with the genie-aided result, and only has 3dB performance loss than the coherent detection

results. This clearly indicates that our proposed method outperforms the differential detectors

in [6]. The main performance loss in [6] is due to that uncoherent detection approach is

employed by statistically averaging off the impact of channel fading coefficients ignoring

the instantaneous channel state information. But, our method utilizes differential detection

relying on the operation with the previous received signals which could be more adaptive to

variation of the channels.

In Fig. 3, we compare the analytical and simulated BER performance of the proposed

differential modulation scheme. Equal transmit power allocation is also applied by setting

ps= pr= p/3. From the figure, it can be observed that at high SNR, the analytical BER

derived by (27) converged to the simulated result, which justifies the validation of (27).

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In Fig. 4, we examine the BER performance of the proposed differential modulation

protocol with power allocation by setting ps= p/4 and pr= p/2 subject to the total power

constraint. From Fig. 4, it can be observed that with optimal power allocation, the proposed

scheme obtains about 2 dB performance gain in comparison with the equal power allocation

scheme at high SNR. We also compare the result by another power setting: ps= 0.4p and

pr= 0.2p, and inferior result can be again observed compared to the optimal power allocation.

In Fig. 5, we plot the BER curves in terms of λ = ps/prdefined in (20) using different

noise variance N0 with the asymptotic BER constraint in (27). With the power constraint

2ps+ pr= p, the SNR can be therefore calculated by ψs=

λp

(2λ+1)N0, where p = 3. It shows

that best performance is obtained when λ = 0.5. In other words, ps= p/4 and pr= p/2 is

the optimal power setting between the sources and the relay, which further verify the power

allocation strategy in (30). From the figure, we can also see that the asymptotic BER is very

close to the simulated results for various SNR values, and they result in the same power

allocation solution of ps= p/4 and pr= p/2.

In Fig 6, we examine the BER results of the proposed differential modulation scheme

without and with using constellation rotation, where the signal constellation used by S1is

rotated by π/2 relative to that by S2. It can be observed that the new result has very similar

with the curve without rotating constellations. This indicates that using constellation rotation

may not give system any gains given large frame length.

Fig. 7 shows the average normalized mean square error (MSE) of µ estimation in (14) as a

function of the SNR, where the average normalized MSE is calculated as

1

N

?N

k=1(µ(k)−? µ(k))2

E[µ(k)]

and ? µ(k) is the estimate of µ(k). It can be observed that the estimation is quite accurate,

particularly at high SNR.

VI. CONCLUSIONS

In this paper, we have proposed a simple differential modulation scheme for bi-directional

relay communications using analog network coding when neither sources nor the relay has

access to channel state information. Simulation results indicate that there exist about 3 dB

loss compared to the coherent detection scheme. Analytical BER is derived to validate the

proposed method. In addition, based on the asymptotic BER at high SNR, an optimal power

allocation between the sources and the relay was derived to enhance the system performance.

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Kyoto, Aug. 2007.

[6] T. Cui F. Gao and C. Tellambura,”Physical layer differential network coding for two-way relay channels”, Proc. IEEE

GlobeCom, New Orleans, CA, 2008.

[7] J. G. Proakis, Digital communications, 4th ed. New York: McGraw-Hill, 2001.

[8] Z. Wang and G.B. Giannakis, ”A simple and general parameterization quantifying performance in fading channels,”

IEEE Trans. Commun., vol.51, no.8, pp.1389-1398, Aug. 2003.

[9] M. O. Hasna and M. S. Alouini, ”End-to-end performance of transmission systems with relays over Rayleigh-fading

channels,” IEEE Trans Wireless Commun., vol. 2, pp. 1126-1131, Nov. 2003.

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S1:

Differential

Encoding

S2:

Differential

Encoding

S1:

Differential

Decoding

S2:

Differential

Decoding

Fig. 1.Block diagram of the proposed ANC-DM scheme.

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048 12162024 28323640

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

ψs [dB]

BER

Differential

Genie−Aided Differential

Coherent

Fig. 2.Simulated BER performance by differential and coherent detections, where ps = pr = p/3.

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−3

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ψs [dB]

BER

Simulated

Analytical

Fig. 3.Analytical and Simulated BER performance by the proposed differential scheme, where ps = pr = p/3