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IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 10, OCTOBER 20111029

Analysis of Channel Estimation Error in Physical Layer Network Coding

Keyvan Yasami, Abolfazl Razi, and Ali Abedi, Senior Member, IEEE

Abstract—This article investigates the effect of erroneous

channel estimation on performance of physical layer network

coding over fading channels. In this scenario, the relay maps the

superimposed noisy modulated data, received from the two end

terminals, to network coded combination of the source packets.

We consider channel estimation error to be Gaussian distributed

and formulate the network coding error by the distance between

real and estimated points in the channel coefficients plane. Using

this model, we present a statistical lower bound on variance

of estimation error that can be tolerated by the relay terminal

without imposing a network coding error on the system.

Index Terms—Channel estimation error, physical layer net-

work coding, cooperative relaying, network throughput.

I. INTRODUCTION

S

interest is due to considerable gains that can be achieved by

application of network coding in wireless relaying scenarios.

Physical layer network coding makes use of inherent additive

nature of electromagnetic waves, arriving simultaneously at a

relay, to further improve network coding and increase network

throughput [2], [3].

Denoise-and-forward (DNF), a relaying scheme in which

there is no need to decode the received signal at the relay, is

introduced in [4], and later expanded in [5] to present adaptive

network coding and modulation design. However the authors

assume that relay terminal can accurately estimate the channel

coefficients. The robustness of the DNF physical layer network

coding scheme with respect to channel estimation error is

studied in [6]. Channel estimation error is considered for both

links and performance of this scheme, in terms of end-to-

end network throughput, is evaluated for different values of

estimation error. However, no analytical results are provided

to calculate this performance degradation.

In this article, Gaussian error in channel estimation is con-

sidered, and a novel analytical method to determine a bound

on tolerable error and hence performance of the network is

proposed. This paper compliments our previous work [6] with

an analytical lower bound on tolerable channel estimation

error to avoid majority of network coding errors. The results

presented in this article may be used as a design criterion for

practical systems utilizing physical layer network coding.

INCE network coding was introduced [1], wireless net-

work coding has received a lot of attention. This increased

Manuscript received February 11, 2011. The associate editor coordinating

the review of this letter and approving it for publication was I. Maric.

This work is financially sponsored by the National Aeronautics and Space

Administration (NASA) grant number EP-11-05-5404438.

The authors are with the Department of Electrical and Computer Engineer-

ing, University of Maine, Orono, ME, 04469 USA (e-mail: {keyvan.yasami,

abolfazl.razi, ali.abedi}@maine.edu).

Digital Object Identifier 10.1109/LCOMM.2011.082011.110301

The rest of this paper is organized as follows. Section II

describes the physical layer network coding and channel esti-

mation parameters. Section III presents problem formulation

and the proposed solution. Numerical and simulation results

are presented in section IV. Section V concludes this paper.

II. SYSTEM MODEL

Consider a simple bidirectional relaying scenario, where

terminal A and B have traffic to send to each other, and

terminal R acts as a relay [5]. Terminal R performs maximum-

likelihood (ML) detection, and quantizes the received signal

from simultaneous transmission of data from A and B, i.e.

푌푅 = 퐻퐴푋퐴+ 퐻퐵푋퐵+ 푍푅, using a denoising mapper

퐶 and a constellation mapper, ℳ, where 퐻퐴 and 퐻퐵 are

the channel coefficients from terminals A and B to relay R,

respectively; 푍푅∼ 푁(0,휎2

symbols from terminals A and B.

The network coded data can be written as:

푛); 푋퐴and 푋퐵are the modulated

푆푅= 퐶(ˆ푆퐴,ˆ푆퐵),

(1)

whereˆ푆퐴, andˆ푆퐵 are ML estimates of digital data from A

and B, namely 푆퐴 and 푆퐵. Denoising maps are optimized

by maximizing the minimum Euclidean distance between all

transmitted signal pairs (푆퐴,푆퐵) and their estimates (ˆ푆퐴,ˆ푆퐵)

[5].

The network code selection only depends on the channel

amplitude ratio, 훾, and phase difference, 휃, i.e, [6]:

퐶 = 푓(훾,휃).

(2)

Note that 퐻퐵/퐻퐴 = 훾 푒푥푝(푗휃). It has been shown that

erroneous channel estimation may lead to selection of a sub-

optimum code, since the distance profile used for network

code selection at the relay is based on the estimates of the

channel coefficients and not the actual values [6]. Moreover,

it has been shown that this non-optimal code selection may

not lead to a network error if estimation errors are in a certain

range. In this case, the network performance will not degrade

drastically. In section III, we formulate impact of imperfect

channel estimation on physical layer network coding to get a

lower bound on this error range.

III. PROPOSED ANALYTIC SOLUTION

In this section, we assume Gaussian channel estimation

error for both links 퐴 and 퐵, i.e.,

ˆ퐻퐴= 퐻퐴+ 푍1,

ˆ퐻퐵= 퐻퐵+ 푍2,

(3)

where 푍1and 푍2are independent Gaussian random variables,

푍1,푍2∼ 푁(0,휎2) . Relay node will use this erroneous data

to perform network coding utilizing (2). The distance between

1089-7798/11$25.00 © 2011 IEEE

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1030IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 10, OCTOBER 2011

Fig. 1.

network code, 퐶푖. For instance, note the symmetry between regions associated

with codes 퐶0and 퐶1and between regions associated with codes 퐶2to 퐶9.

Network Coding Map [5]. Each region is associated with a unique

the Channel point 푃1 = (퐻퐴,퐻퐵) and the estimated point

푃2= (ˆ퐻퐴,ˆ퐻퐵) in 퐻퐴− 퐻퐵 plane can be calculated using

(3) as,

√

Furthermore the network coding map consists of discrete

region in the 훾 − 휃 plane according to (2). These regions,

푅푖, correspond to different network codes. Each region, 푅푖,

may be represented by a statistical center, 푃(푥푐,푦푐), with

coordinates defined in 훾 cos휃 − 훾 sin휃 plane as,

푥푐= 피[푥∣푅푖],

where 푥 = 훾 cos휃 and 푦 = 훾 sin휃, and 피[.] denotes

expectation function.

The shortest distance between this point and any neighbor-

ing region is considered as an error threshold, 푇푖, for each

region, 푅푖. If the points 푃1and 푃2are in different regions of

the network coding map, then there will be a network error due

to channel estimation error. At any region in 훾 cos휃 −훾 sin휃

plane, if the distance between the points 푃1and 푃2is greater

than the corresponding threshold, i.e., 푑푃1푃2≥ 푇푖, or in terms

of squred distance,

푑2

푑푃1푃2=푍12+ 푍22.

(4)

푦푐= 피[푦∣푅푖],

(5)

푃1푃2≥ 푇2

푖,

(6)

a network coding error will happen. To consider all points

and regions, expected value of squared distance, 푑2

weighted threshold of all regions should be considered, i.e.,

푃1푃2, and a

피[푑2

푃1푃2] ≥ 푇2,

(7)

where expectation is over the Gaussian random variables 푍1

and 푍2and 푇2is the total weighted squared threshold defined

as,

푇2=

∑

푖

푃푅푖푇2

푖.

(8)

where 푃푅푖is the probability of region 푅푖. Substituting 푑푃1푃2

from (4) into (7), we have,

휎2≥ 푇2/2,

(9)

which is lower bound on variance of estimation error, 휎2, for

which a network coding error will happen. Note that we have

TABLE I

PROBABILITY OF REGIONS IN NETWORK CODING MAP OF FIG. 1 FOR

RICIAN CHANNEL EXAMPLE (K-FACTOR = 10푑퐵)

Region (푅푖)

푅0 & 푅1

푅2 to 푅9

Probability (푃푅푖)

0.2600

0.0600

considered all the regions in calculating (9) to account for

their probabilities.

Numerical solutions and simulations for this lower bound,

under specific channel realizations, are presented in the next

section.

IV. NUMERICAL RESULTS

In this section, a numerical value for the lower bound of

(9) is derived. The proposed analytic solution in section III

works regardless of the modulation scheme used at terminals

A and B. However we assume QPSK modulation in this

section for simplicity of our proposed solution. Although the

proposed method may be applied to various channel models,

in this article we used Rician model as an example. Probability

distribution function (pdf) for both channels, 퐻퐴and 퐻퐵, is

given by,

푓(ℎ∣푣,휎푟) =

ℎ

휎2

푟

푒푥푝(−(ℎ2+ 푣2)

2휎2

푟

)퐼0(ℎ푣

휎2

푟

),

(10)

where 퐼0(.) is modified zeroth order Bessel function of first

kind, ℎ is either 퐻퐴 or 퐻퐵, with Rician K-factor (dB) =

10푙표푔10(푣2/2휎2

For case of QPSK modulation at both terminals 퐴 and 퐵

and closest-neighbor clustering method the network code map

(2) is presented in Fig. 1 [5]. As shown in this figure, 10

network codes are obtained in this case. The joint pdf of

훾 cos휃 and 훾 sin휃 can be calculated using (10),

푟).

푓푋,푌(푥,푦) =

∫∞

where 푥 = 훾 cos휃, 푦 = 훾 sin휃, and 훾2= 푥2+ 푦2. (10) and

(11) are used to generate numerical results in the following

manner. Integrating (11) over each region, 푅푖, determines

region probabilities. Table I demonstrates these probabilities

for Rician channels with K-factor = 10푑퐵. Note that region

푅0and 푅1and regions 푅2to 푅9are symmetric as depicted in

Fig. 1. Fig. 2 depicts regions 푅0and 푅4with their neighboring

borders. 푅0 and 푅4 have 2 and 4 discrete sections, 푆푖푗,

respectively. The corresponding statistical centers, (푥푐,푦푐),

shortest distance to neighboring regions, 푇푖푗, and section

probabilities, 푃푆푖푗, are given for all sections of each region in

Table II.

By symmetry of the distributions and regions as shown in

Fig. 1, 푅1has same thresholds as 푅0, and rest of regions have

same thresholds as 푅4. Weighted threshold for each region can

be defined as,

푇푖=

푗

0

푤3

휎4

푟

푒푥푝[(훾2+ 1)푤2+ 2푣2

2휎2

푟

]퐼0(푤푣

휎2

푟

)퐼0(푤푣훾2

휎2

푟

)d푤, (11)

∑

푃푆푖푗푇푖푗.

(12)

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YASAMI et al.: ANALYSIS OF CHANNEL ESTIMATION ERROR IN PHYSICAL LAYER NETWORK CODING 1031

Fig. 2.

statistical centers (푥푐,푦푐). Sections and borders are given for any fading

channel model, statistical centers are given for Rician channel example with

K-factor = 10푑퐵.

Regions 푅0 and 푅4 with corresponding sections, borders, and

Using (8), (12), values of 푃푅푖, 푃푆푖푗, and 푇푖푗, from Tables I

and II, we have, 푇2≈ 0.1, and from (9),

휎 ≥ 0.22.

End-to-end network throughput in terms of number of de-

livered packets for different values of 휎 (standard deviation)

are shown in Fig. 3. Note that packet error rate is used to

evaluate the network throughput, since in practice, we cannot

know which bit location has failed. We have applied cyclic

redundancy check codes to packets to check whether one

packet failed or succeeded.

In this simulation set, 퐻퐴 and 퐻퐵 are considered to be

Rician distributed with previously mentioned parameters (K-

factor = 10푑퐵). Packets of length 256 symbols are considered.

It is assumed that channel coefficients are constant during

transmission of each packet. Signal to Noise Ratio (SNR) is

defined as (피[∣퐻퐴∣2] +피[∣퐻퐵∣2])/2휎2

of zero mean additive Gaussian noise of channels. Moreover,

to further concentrate on performance degradation due to

network coding error, perfect ML detection is assumed at both

end terminals, A and B, as well as relay, R.

Fig. 3 depicts end-to-end throughput for different values

of 휎 (standard deviation) together with pure XOR network

coding. In this figure, 휎 is increased by steps of size 0.04

from 0.14 to 0.30. A larger gap between curves corresponding

to 휎 values of 0.22 and 0.26 is seen. This larger gap is due

to network coding error, occurring for a larger quantity of

symbols in comparison with number of network coding errors

for smaller values of 휎. The value of 휎 corresponding to this

gap is between 0.22 and 0.26, which is in agreement with the

lower bound given by (13). Also note that the performance

curve for 휎 values greater than 0.22 are very close to the

performance curve of XOR network coding, i.e., for these

values of 휎, there is no advantage in using 10 denoising maps

at the relay than just the XOR network coding.

(13)

푛, where 휎2

푛is variance

V. CONCLUSIONS

This article presents a statistical lower bound on variance

(standard deviation) of channel estimation error to prevent net-

work coding error, in a two way relaying system with physical

layer network coding. Gaussian distributed error in estimation

TABLE II

STATISTICAL CENTERS, THRESHOLDS, AND SECTION PROBABILITIES OF

REGIONS 푅0AND 푅4FOR RICAN CHANNEL EXAMPLE (K-FACTOR =

10푑퐵)

Region

(푅푖)

Section

(푆푖푗)

Statistical

Center

(푥푐,푦푐)

(1.0090, 0)

(−1.0090,0)

(1.6479, 1.0658)

(0.6899, 1.0522)

(−0.4493, 0.6627)

(−0.4260, 0.2947)

Threshold (푇푖푗)

(Shortest

Distance)

0.4206

0.4206

0.2472

0.1899

0.1383

0.0740

Section

Probability

(푃푆푖푗)

0.1300

0.1300

0.0047

0.0253

0.0253

0.0047

푅0

푆01

푆02

푆41

푆42

푆43

푆44

푅4

101520 25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SNR (dB)

End−to−end Troughput (bps/Hz)

σ=0.14

σ=0.18

σ=0.22

σ=0.26

σ=0.30

XOR NetCoding

Fig. 3.

values of 휎 - Rician channel example with K-factor = 10푑퐵.

End-to-end Network throughput (bps/Hz) vs SNR (dB) for different

of channel coefficients is considered, and the network coding

error event is formulated. Using the expected value of distance

between real and estimated points in the channel coefficients

plane, thresholds and statistical centers for different regions

are defined as shown in Fig. 2. Utilizing region probabilities

and thresholds, a lower bound on variance of estimation error

is calculated. Although the proposed method may be applied to

various channel models, in this article we used Rician model

as an example. In case of Rician channels with K-factor =

10푑퐵, presented in Table II, the lower bound on standard

deviation is calculated to be 0.22. Simulation results verified

the analytically calculated lower bound on variance in terms

of network throughput degradation.

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