Page 1

Impact of the Channel Time-selectivity on BER

Performance of Broadband Analog Network Coding

with Two-slot Channel Estimation

Haris Gacanin†, Mika Salmela‡ and Fumiyuki Adachi§

†Motive Division, Alcatel-Lucent Bell N.V., Antwerp, Belgium

‡School of Science and Technology, Aalto University, Aalto, Finland

§Graduate School of Engineering, Tohoku University, Sendai, Japan

†Email: harisg@ieee.org

Abstract—Network coding at the physical layer (PNC) can be

used to improve the network capacity in a wireless channel.

Broadband analog network coding (ANC) was introduced as

a simpler implementation of PNC. The coherent detection and

self-information removal in ANC require accurate channel state

information (CSI). In this paper, we theoretically investigate

an impact of the channel time-selectivity on the bit error rate

(BER) performance of broadband ANC with practical channel

estimation (CE) scheme using orthogonal frequency division

multiplexing (OFDM). The achievable BER performance gains

due to the first and second order polynomial time-domain

channel interpolation are evaluated using derived close-form BER

expressions.

Index Terms—Analog network coding, channel estimation,

BER analysis, OFDM.

I. INTRODUCTION

Recently, a physical-layer network coding (PNC) [1], [2]

and analog network coding (ANC) [3]-[7] schemes have been

proposed to increase the network capacity of bi-directional

communication in a frequency-nonselective fading channel.

On the other hand, in broadband wireless communications

the channel is both time- and frequency-selective due to the

user mobility and multipath propagation, respectively. These

properties of the wireless channel render schemes in [1]-

[7] not applicable for wireless communications. Thus, in [8],

broadband ANC scheme was proposed for communication

over a multipath (i.e., frequency-selective) channel.

The user mobility is an important factor in wireless com-

munications that affects the bit error rate (BER) performance.

Thus, a robust channel estimation (CE) is required to tract

the fast fading variations. In broadband ANC scheme for

coherent detection and self-information removal accurate CE

is required. In [10], the BER performance of broadband ANC

with pilot-assisted CE has been presented, but the tracking

against the channel time-selectivity has not been investigated.

In this paper, we theoretically investigate the impact of

channel time-selectivity on the BER performance of broadband

ANC with two-slot pilot-assisted CE scheme in a frequency-

selective fading channel. We present a closed-form BER ex-

pression based on orthogonal frequency division multiplexing

(OFDM) radio access. We investigate the achievable BER

performance gains in a fast-fading channel obtained by polyno-

mial time-domain channel interpolation. Our analytical results

shown that in a higher Eb/N0 region the BER performance

gains using pilot-assisted CE with both the first and second-

order polynomial time-domain interpolation depend on the

mobile user velocity.

The remainder of this paper is organized as follows. In

Section II, we present the network model. The performance

analysis is presented in Section III. In Section IV, the numer-

ical results and discussions are presented. Finally, the paper is

concluded in Section V.

II. NETWORK MODEL

A bi-directional relay network with users U0and U1, who

are assumed to be out of each other’s transmission range,

and relay R is illustrated in Fig. 1. The transmission frame

structure is illustrated in Fig. 2. The communication between

two users in the mth block takes place during two slots; (i)

in the first slot (q = 0) the users simultaneously transmit

to the relay (ii) during the second slot (q = 1) the relay

broadcasts the received signals to both users using an amplify-

and-forward protocol. The figure shows that the kth frame

consists of M blocks, where the first block (m = 0) is used

for pilot-assisted CE.

A. Radio Access Scheme

The jth (j ∈ {0,1}) user’s mth block symbol sequence

{dj,m(n); n = 0 ∼ Nc− 1, m = 0 ∼ M − 1} is fed to an

Nc-point inverse fast Fourier transform (IFFT) to generate the

jth user’s time-domain OFDM signal in the mthe frame (i.e.,

sj,m(t)). Then, an Ng-sample guard interval (GI) is added and

the GI-added OFDM signal is transmitted over a time-varying

frequency-selective fading channel.

The propagation channel is characterized by the mth frame

impulse response given by hk

τl), where L denotes the number of paths, hk

the path gain between the relay R and jth user Uj at slot

q during the mth block of kth frame, δ(·) denotes the delta

function and τl denotes the time delay of the lth path. We

q,j,m(τ) =?L−1

l=0hk

q,j,m(l) denotes

q,j,m(l)δ(τ −

978-1-4244-8331-0/11/$26.00 ©2011 IEEE

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R

1

U

0

U

First slot Second slot

Coverage

of U0

Coverage

of U1

Coverage

of R

h0,0(τ)

h0,1(τ)

h1,0(τ)

h1,1(τ)

Fig. 1. Bi-directional relay network using ANC.

assume that the GI is assumed to be longer than the maximum

channel time delay. The channel gain at the nth subcarrier

is represented by Hk

we consider the mth block transmission and without loss of

generality the frame index k can be omitted in the following.

First time slot: By assuming perfect time and frequency

synchronization at the relay, during the first time slot (TS0),

the signals from both of the users T0and T1are transmitted

to the relay terminal R. The received signal at the relay is

amplified and broadcasted over a frequency-selective fading

channel. For the sake of the analysis we normalize the transmit

signal by a factor β, which is the square root of its noise

variance.

Second time slot: During the second slot, by assuming

perfect time and frequency synchronization, the received signal

at the jth user Ujafter FFT can be expressed as

q,j,m(n) = FFT[hk

q,j,m(τ)]. Henceforth

Rj,m(n) =

√P

β

[√Pd0,m(n)H0,0,m(n)

+√Pd1,m(n)H0,1,m(n) + Nr,m(n)]H1,j,m(n) + Nj,m(n), (1)

for n = 0 ∼ Nc− 1, where Nj,m(n) is the zero-mean noise

having variance N0/Ts due to the AWGN. The jth user Uj

removes its self-information as

˜Rj,m(n) = Rj,m(n) −P

for n = 0 ∼ Nc− 1. The decision variables are given by

ˆdj,m(n) =˜Rj,m(n)wj,m(n)

βdj,m(n)H0,j,m(n)H1,j,m(n)

(2)

(3)

for n = 0 ∼ Nc− 1, where wj,m(n) denotes the equalization

weight [8].

B. Channel Estimation Scheme

The frame structure is illustrate in Fig. 2 with the kth frame

divided into one pilot block and M − 1 data blocks. The

channel estimates are obtained from the pilot signal, which

is transmitted in the first block (i.e., m = 0) of the each

frame. Blocks are divided into two stages, corresponding to

U0

U1

U1

U0

U1

U1

RR

kth frame (M blocks)

kth frame (M blocks)

……

Pilot (m=0)Pilot (m=0) Data (m=1)Data (m=1)

U0

U1

U1

RR

U0

U1

U1

RR

Data (m=M-1) Data (m=M-1) Pilot (m=0) Pilot (m=0)

RR

……

blockblock

Ts

Ts

(k+1)th frame(k+1)th frame

tt

U0

U0

U0

U0

Fig. 2. Frame structure.

the first and second time slot, TS0 and TS1, respectively,

each consisting of Nc+ Ngsamples (i.e., duration of Ts).

In the first time slot TS0, the users U0 and U1, re-

spectively transmit their pilot signals, p0(t) and p1(t) =

p0((t−Δ)modNc), where Δ denotes the time shift [10]. The

relay estimates the channel gains and in the second time slot

TS1 broadcasts its pilot signal p0(t) to both users. Finally,

the users estimate the corresponding channel gains using the

broadcasted pilot signal in the first block. The estimated

CSIs obtained from the pilot block are used in detecting the

following M − 1 data blocks within the kth frame. For more

details please refer to [10].

III. PERFORMANCE ANALYSIS

We first present the CE error model and then, we develop a

closed-form BER expression with practical pilot-assisted CE

scheme presented in [10], where the BER with perfect CSI is

only presented as a reference.

A. CE Error Model

The estimated channel gains for the mth block within the

kth frame can be represented as

¯Hk

q,j,m(n) = Hk

q,j,m(n) + ?k

q,j,m(n)

(4)

for n = 0 ∼ Nc−1, where ?k

error. We model the channel estimation error ? as a zero-mean

complex Gaussian random variable with the variance given by

σ2

The decision variables after coherent detection in the mth

OFDM data block given by (3) can be expressed as

ˆdj,m(n) = Xj,m(n)Y∗

q,j,m(n) is the channel estimation

e.

j,m(n),

(5)

for n = 0 ∼ Nc− 1, where

⎧

⎪

⎪

⎪

Note that ¯ c represents the logical negation (i.e., NOT) of

c. In the above expressions, we assume that for the given

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Xj,m(n) =P

βd¯j,m(n)H0,¯j,m(n)H1,j,m(n)

√P

βH1,j,m(n)Nr,m(n)+Nj,m(n)

+P

−P

−P

+?0,j,0(n)H1,j,0(n)+?0,j,0(n)?1,j,0(n),

= H0,¯j,m(n)H1,j,m(n) + H0,¯j,0(n)?1,j,0(n)

+?0,¯j,0(n)H1,j,0(n)+?0,¯j,0(n)?1,j,0(n).

+

βdj,m(n)H0,j,m(n)H1,j,m(n)

βdj,m(n)H0,j,0(n)H1,j,0(n)

βdj,m(n)H0,j,0(n)?1,j,0(n)

Yj,m(n)

(6)

Page 3

{Hq,j,m(n)} both Xj,m(n) and Yj,m(n) are zero mean com-

plex Gaussian random variables. Thus, the jth user’s BER

within the mth frame can be represented as [12]

j,m(n)] < 0] =1

P4b,m= P[Re[Xj,m(n)Y∗

2

?

1 −

μ

?2 − μ2

?

(7)

,

where P[a] and μ, respectively, denote the probability of a

and the normalized covariance given as

Re[gxy]

?gxxgyy− Im[gxy]2,

with gxx = E[|Xj,m(n)|2], gyy = E[|Yj,m(n)|2], gxy =

E[Xj,m(n)Y∗

B. Second-order Moment Functions

In this subsection, we evaluate the impact of imperfect CSI

with practical pilot-assisted CE scheme for broadband ANC

[10] briefly explained in the previous section with different

interpolation techniques.

1) First-order interpolation: The 1storder interpolated

channel gain¯Hk

¯Hk

M

[Hk

+m

for n = 0 ∼ Nc− 1. Thus, the second-moment covariance

functions for the first-order interpolation can be expressed as

⎧

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⎪

⎪

⎪

⎪

⎪

⎪

⎪

where J0(·) is the zeroth order Bessel function of first kind

and fDis the maximum Doppler shift. σ2

channel estimation error ?q,j,m(n) and σ2

noise power due to AWGN with 1/Tsbeing data symbol rate.

We note here that we assume the Jakes fading model, where

incoming rays constituting each propagation path arrive at a

user with uniformly distributed angles with the correlation

given by E[Hk

We also consider the block fading, where the fading gains

remain constant during one time slot and vary slot-by-slot.

μ =

(8)

j,m(n)]

q,j,m(n) at the mth block is obtained as

q,j,m(n) =M−m

q,j,0(n) + ?k

q,j,0(n) + ?k+1

q,j,0(n)]

q,j,0(n)]

M[Hk+1

(9)

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gxx = 2P2

−2P2

+(M

+P2

+4P2

+4P2

+4P2

= 2P2

gyy =?(M−m

+4(M−m

+4(M−m

s

β2 + (Ps

?(M−m

?(M−m

β2(M−m

β2(M−m

β2 + (Ps

β2+ 1)2σ2

m)J0(2πfDTsm)

m)J0(2πfDTs(M − m))?2

β2

β2(M−m

m)(M

m)2(M

β2+ 1)2σ2

m)2?2(1 + 2σ2

m)(M

m)2(M

gxy =Ps

β

+(M

=Ps

βA2,

n

s

β2

s

m)2+ (M

m)3(M

m)2?2(1 + 2σ2

m)3(1+2σ2

m)2J2

n+P2

e)2

s

m)(1+2σ2

e)J0(2πfDTsM)

e)J0(2πfDTsM)

0(2πfDTsM)

β2A1+P2

s

s

sss

β2gyy,

m)2+ (M

m)3(M

m)3(1 + 2σ2

m)2J2

?(M−m

e)2

+4(M−m

m)(1 + 2σ2

e)J0(2πfDTsM)

e)J0(2πfDTsM)

0(2πfDTsM),

m)J0(2πfDTsm)

m)J0(2πfDTs(M−m))?2

(10)

eis the variance of

n= N0/Ts is the

q,j,m(n)Hk?

q?,j,m(n)] = J0(2πfDTs(k−k?) [11].

2) Second order interpolation: The 2ndorder interpolated

channel gain¯Hk

q,j,m(n) at the mth block is obtained as

¯Hk

q,j,m(n) =(M−m)(m−2M)

+m(2M−m)

+m(m−M)

2M2

[Hk

q,j,0(n) + ?k

q,j,0(n) + ?k+1

[Hk+2

q,j,0(n)]

M

[Hk+1

q,j,0(n) + ?k+2

q,j,0(n)]

q,j,0(n)]

2M2

(11)

for n = 0 ∼ Nc− 1. Thus, the second-moment covariance

functions for the second-order interpolation are given by

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functions for practical pilot-assisted CE with both the first

and second order interpolation we obtain the normalized

covariance as

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gxx= 2P2

×J0(2πfDTsm)J0(2πfDTs(M − m))

+P2

β2M4

−2P2

×J0(2πfDTs(2M − m)) −(m−M)2(2M−m)2

×J2

−m2(m−M)2

= 2P2

gyy=?((M−m)4(2M−m)4

8M8

+m2(m−M)2(2M−m)4

2M8

+m4(m−M)2(2M−m)2

M8

×J2

+?m2(m−M)3(2M−m)3

+?2m4(m−M)(2M−m)3

M8

+m2(m−M)3(2M−m)3

2M8

×J0(2πfDTsM) −?m(m−M)4(2M−m)3

M8

×J0(2πfDTs2M),

gxy=(m−M)2(2M−m)2

4M4

×J2

×J2

×J0(2πfDTsm)J0(2πfDTs(M − m))

−Psm(m−M)2(2M−m)

×J0(2πfDTsm)J0(2πfDTs(2M − m))

+m2(m−M)(2M−m)

M4

×J0(2πfDTs(2M − m))

=Ps

βA2.

s

β2 + (Ps

β2+ 1)2σ2

n+ 2m(m−M)(2M−m)2

M4

P2

β2

s

sm(m−M)2(2M−m)

sm2(m−M)(2M−m)

β2M4

J0(2πfDTsm)J0(2πfDTs(2M − m))

J0(2πfDTs(M − m))

2M4

P2

s

β2J2

0(2πfDTs(2M − m)) +P2

β2+ 1)2σ2

P2

β2

s

0(2πfDTsm) − 2m2(2M−m)2

2M4

β2 + (Ps

M4

0(2πfDTs(M − m))

s

β2gyy

β2gyy,

P2

β2J2

s

s

n+P2

s

β2A1+P2

s

16M8

+m4(2M−m)4

M8

+m4(m−M)2(2M−m)2

2M8

?(1 + 2σ2

+m4(m−M)4

16M8

+m2(m−M)4(2M−m)2

e)2+?m2(m−M)2(2M−m)4

M8

J2

−m3(m−M)3(2M−m)2

0(2πfDTs2M)

−m(m−M)3(2M−m)4

−2m3(m−M)(2M−m)4

−m3(m−M)3(2M−m)2

4M8

+m3(m−M)2(2M−m)3

M8

?

?

− 2m3(m−M)2(2M−m)3

4M8

0(2πfDTsM) +m2(m−M)4(2M−m)2

0(2πfDTs2M)

M8

M8

×J2

0(2πfDTsM)J2

M8

2M8

+m4(m−M)3(2M−m)

2M8

2M8

?(1 + 2σ2

?(1 + 2σ2

e)

+m3(m−M)4(2M−m)

4M8

e)

Ps

βJ2

0(2πfDTsm) +Psm2(2M−m)2

Ps

β

βM4

0(2πfDTs(M − m)) +m2(m−M)2

0(2πfDTs(2M − m)) −m(m−M)(2M−m)2

4M4

M4

Ps

β

2βM4

Ps

βJ0(2πfDTs(M − m))

(12)

3) BER Evaluation: Using the second moment covariance

μ =

A2

?

(2+ (Es

2N0)−1+ (Es

2N0)−2+ A1+ gyy)gyy

.

(13)

Page 4

TABLE I

NUMERICAL SIMULATION PARAMETERS.

Tx

Data modulation

Block size

GI

L-path block Rayleigh fading channel

Equalization

QPSK

Nc = 256

Ng = 32

Channel

RxMRC

Thus, the BER performance for broadband ANC with first and

second order interpolation schemes is finally derived as

P4b,m=

?

The average BER expression for the OFDM frame is finally

calculated by averaging the M − 1 data blocks as

M−1

?

Next a closed-form BER for broadband ANC with perfect

knowledge of CSI is given as a reference.

1

2

1 −

A2

?

(4+2(

Es

2N0)−1+2(

Es

2N0)−2+2A1+2gyy)gyy−A2

2

?

.

(14)

P4b=

m=1

P4b,m.

(15)

C. BER with perfect CSI

In the case of perfect knowledge of CSI the second moments

are given by gxx= P2/β2+ (P/β2+ 1)2σ2

gxy= P/β. Thus, the average BER is obtained by

P4b=1

2

1+ 2(Es

n, gyy = 1 and

?

1 −

1

?

2N0)−1+ 2(Es

2N0)−2

?

.

(16)

Next we present the numerical results and discussions based

on the analysis presented in the this section.

IV. NUMERICAL RESULTS

The numerical simulation parameters are shown in Table

I. We assume ideal coherent QPSK modulation/demodulation

with Nc= 256 and GI length of Ng= 32. The propagation

channel is an L-path Rayleigh fading channel, where the path

gains {hq,l,j,m;l = 0 ∼ L − 1} are zero-mean independent

complex variables with E[|hq,l,j,m|2] = 1/L. The maximum

time delay of the channel is assumed to be less than the guard

interval and that all paths are independent of each other. fDTs

denotes the normalized Doppler frequency, where 1/Tsis the

transmission symbol rate (fDTs = 10−3corresponds to a

mobile terminal speed of approximately 83 km/h for a trans-

mission data rate of 100 Msymbols/s and a carrier frequency

of 5GHz). Distance-dependent path loss and shadowing loss

are not considered.

Figure 3 shows the BER performance of broadband ANC

using pilot-assisted CE with the first-order interpolation as a

function of Eb/N0with fDTsas a parameter with σ2

and M = 16. The tracking ability is noticeable improved for

pilot-assisted CE with the first-order interpolation in compar-

ison to the case without interpolation (i.e., 0thorder). The

figure shows that pilot-assisted CE scheme with the first-order

e= 10−3

1.E-03

1.E-02

1.E-01

1.E+00

5 101520 25 30

Average Eb/N0 (dB)

fDTs= 0.02

fDTs= 0.01

fDTs= 0.001

0th order interpolation

1st order interpolation

QPSK

σe2= 10-3

M= 16

Average BER

Fig. 3.BER performance using 1storder interpolation.

1.E-03

1.E-02

1.E-01

1.E+00

5 10 15 20 2530

Average Eb/N0 (dB)

fDTs= 0.02

fDTs= 0.01

fDTs= 0.001

0th order interpolation

2nd order interpolation

QPSK

σe2= 10-3

M= 16

Average BER

Fig. 4. BER performance using 2ndorder interpolation.

interpolation slightly improves the BER perfomance for the

mobile user veocity of about 80 km/h (corresponds to the

normalized Doppler frequency fDTs= 10−3). On the other

hand, for the mobile user velocity of about 800 km/h (corre-

sponds to the normalized Doppler frequency fDTs= 10−2)

the significant BER performance improvement is observed in

comparison with the CE case where interpolation is not used.

However, for a higher mobile user velocity (corresponding to

the normalized Doppler frequency fDTs= 10−1) a BER floor

is observed due to a fast fading variation that cannot be tracked

Page 5

by the first-order interpolation.

Figure 4 shows the BER performance of broadband ANC

using pilot-assisted CE with the second-order interpolation as

a function of Eb/N0 with fDTs as a parameter. The values

of σ2

user speed of about 80 km/h (i.e., normalized Doppler fre-

quency fDTs= 10−3) the second-order interpolation slightly

improves the BER performance in comparison with the CE

case when interpolation is not used (i.e., 0thorder). The larger

BER performance gain with the second-order interpolation can

be observed for the mobile user velocity of about 800 km/h

(i.e., normalized Doppler frequency fDTs= 10−2), while for

the higher velocity (corresponding to the normalized Doppler

frequency fDTs = 10−1) the BER performance severely

degrades since the channel fluctuations are too fast and they

cannot be encountered by the polynomial interpolation.

eand M are same as in Fig. 3. For the the mobile

V. CONCLUSION

In this paper, we presented the closed-form BER expressions

for bi-directional ANC with practical pilot-assisted CE scheme

in a frequency-selective fading channel. As a physical layer

access we consider OFDM radio. We evaluate the impact

of the first and second order time domain interpolation on

the practical pilot-assisted CE scheme on the achievable BER

performance of bi-directional ANC with OFDM. It was shown

that the BER performance gains in a higher Eb/N0 region

for both the first and second-order polynomial time-domain

channel interpolation depend on a mobile user velocity. Further

performance improvements can be obtained by using a higher

order interpolation techniques, but their analysis may become

very difficult if not impossible to track.

ACKNOWLEDGMENT

This work was supported in part by 2010 KDDI Research

Grant Program.

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