1

SINR Analysis and Energy Allocation of Preambleand Training for Time Division CT with Range

ExtensionQiongjie Lin and Mary Ann Weitnauer

School of Electrical and Computer EngineeringGeorgia Institute of Technology, Atlanta, Georgia 30332-0250

Email: lin3@gatech.edu; maweit@gatech.edu

Abstract—In this paper, we investigate the packet decodingperformance degradation due to the joint effects of synchro-nization and channel estimation errors in orthogonal frequencydivision multiplexing (OFDM)-based time-division cooperatingtransmission (TDCT) for the purpose of range extension. Thesignal to interference and noise ratio (SINR) of maximal ratiocombining (MRC) with OFDM TDCT is analyzed in multipathfading channels. We demonstrate the performance impairmenton decoding for different TDCT implementation scenarios withconventional synchronization and different channel estimationschemes through the simulation of the practical system with com-plete synchronization, channel equalization and decoding pro-cesses. We show that when the total energy for synchronizationand channel estimation is limited because of the range extensionobjective, the way energy is distributed between preamble andtraining sequence plays an important role.

Index Terms—Cooperative transmission, range extension,OFDM synchronization, channel estimation

I. INTRODUCTION

In cooperative transmission (CT), multiple radios in a net-work transmit copies of the same message through differentlyfading multipath channels, and a receiver combines the copiesin the physical layer. The cooperatively transmitting nodesform a virtual array, from which the receiver can derivediversity and array gains [1]. These gains can be used toincrease reliability, reduce transmit power, or extend range.In particular, range extension can overcome shadowing andpath loss that would otherwise partition the network. CTrange extension (CTREX) can benefit many types of wirelessnetworks. For example, it can increase the two-hop coveragearea of a single access point [2]. CT can be performedconcurrently (CCT) or in different time slots; we call thetime-slotted version “time-division CT” (TDCT). Meanwhile,the orthogonal frequency division multiplexing (OFDM) is anefficient technique for mitigating the effects of delay spreadin multipath wireless channels. This paper then treats OFDM-based TDCT with the objective of range extension in themultipath fading channel.

For OFDM wireless reception, timing/frequency synchro-nization and channel state information (CSI) estimation arewell known to be big issues. Symbol timing offsets larger thanthe cyclic prefix (CP) will introduce inter-symbol-interference

The authors gratefully acknowledge support for this research from theNational Science Foundation under grant CNS-1017984.

(ISI), while carrier frequency offsets (CFOs) will introduceinter-carrier-interference (ICI). Moreover, the packet decodingperformance is also affected by the CSI estimation error,which results from both the additive noise and the presenceof residual synchronization error.

The bit error rate (BER) impairment caused by CFO isevaluated in [3] in AWGN channels. In [4], the authorsevaluate the BER performance degradation, conditioned on thegiven multipath channel realization, by exploiting the Gaussianapproximation of the ICI. However, [3],[4] consider only theperformance degradation in terms of synchronization errorassuming perfect CSI is available at the receiver. On theother hand, various CSI estimators and their correspondingperformance analyses have been studied in the literature. In[5],[6],[7], LS (Least Square) and LMMSE (Linear Mini-mum Mean Square Error) channel estimation algorithms arediscussed under the assumption of perfect synchronization.On this subject, Cheon and Hong proposed a BER analysisin Rayleigh fading channel, incorporating both the effectsof the CFO and the CSI estimation errors [8]. But theyassume the channel estimate and the channel estimation errorare uncorrelated when the interference power is small. Asindicated in [4], the method presented in [8] may overestimatethe BER in some simulations. In [9], the effects of channelestimation error are analyzed in the presence of CFO in thefrequency-selective Rayleigh fading channel. However, theeffect of synchronization error on received data is ignored inthe derivation of BER.

To the best of our knowledge there are few existing workson analyzing the joint impact of synchronization and channelestimation errors on the BER of TDCT in terms of rangeextension, or equivalently, at very low SNR. The BER perfor-mance of multiple-input and multiple-output (MIMO) OFDMsystem is analyzed with CFO and CSI estimation errors in[10], where the CFO and CSI estimation errors are modeledas independent zero-mean RVs. In [11], the effect of CFOsestimation error on CSI estimation is discussed, however,the mean square error (MSE) of the channel estimator forSISO/MIMO-OFDM system is addressed instead of BER.Some recent works [12], [13] consider the OFDM channelestimation problem in the presence of CFO and phase noisewith focus on the development of new channel estimationalgorithms and corresponding BER analyses are missing.

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Fig. 1. Illustration of two-hop TDCT system. Fig. 2. Illustration of the structure of relay nodes and destination.

In this paper, we consider the joint effects of synchroniza-tion and channel estimation errors on TDCT for the purposeof range extension. We analyze the post-combiner signal toinference and noise ratio (SINR) to evaluate the joint impactof residual CFOs and channel estimation errors on decodingnumerically. The BER of coded OFDM TDCT in the multipathfading channel is then investigated through the simulationof the entire practical system with complete synchronizationand CSI estimation processes. The conventional SISO OFDMsynchronization [14] scheme is applied on each relay link. Achannel estimator that averages channel estimates over severalconsecutive subcarriers is also considered. Through the MonteCarlo simulation of the whole OFDM TDCT system, weobserve that the CFO and channel estimation errors jointlyaffect the SINR performance, and the performance degradationfactors could be modeled as co-link interference and noise,which is proportional to number of cooperating relays, K andinter-link interference and noise, which is proportional to K2.

The paper is organized as follows. Section II describes thesystem model considered for this work. Section III presents theSINR analysis for post-combiner OFDM signals in terms ofrange extension and energy limitation. In Section IV, we showthe simulation results of BER of coded BPSK OFDM signalsfor various TDCT scenarios with different synchronization andchannel estimation schemes. Finally, Section V concludes thepaper.

II. SYSTEM MODEL

We consider a half-duplex time-division cooperative com-munication system with one source node, S, a relay clusterof K cooperating relay nodes {R1, R2, ..., RK}, and a des-tination node, D, as shown in Fig.1. Either the decode-and-forward (DF) or amplify-and-forward (AF) relaying schemecould be adopted for the relay nodes. We assume directcommunication between source and destination is not avail-able. There are two phases of transmission to achieve thecommunication between source and destination. In the firstphase, the source node, S, broadcasts the message to potentialrelay nodes. All the relay nodes that correctly decode theheader (AF) or the packet (DF) from the source node willparticipate in the second phase, keeping the same offsets fortransmission that they learned in reception.

In this work, we focus on the decoding performance at thereceiver during the second phase, assuming the original datafrom the source node has been decoded successfully and isready to be sent at the relay cluster. The output OFDM symbolof the kth relay is given by the N point complex modulationsequence

xki =1√N

N−1∑n=0

Xnej2πin/N , (1)

where, Xn ∈ {Xpn, X

tn, X

dn} is the modulated symbol in

the frequency domain, where Xpn denotes the preamble for

synchronization, Xtn denotes the training sequence for CSI

estimation, while Xdn denotes the data symbol; i and n are

the time and subcarrier indices, respectively; N is number ofsubcarriers of one OFDM symbol.

Since the OFDM system is not sensitive to the timing offset,we assume the timing error is smaller than the CP length. Letthe normalized frequency offset error be denoted εk = ∆fkTs,where ∆fk is the residual CFO of the kth link after the CFOcompensation and Ts is the sample duration. Denoting thephase offset θk0 for the kth link, the received OFDM symbolduring the kth time slot at the destination can be expressed by

rki =ejθ

k0

√N

N−1∑n=0

HknXne

j2πi(εk+n)/N + zki , (2)

where Hkn is the channel response of the nth subcarrier of the

kth relay link, and zki ∼ CN (0, σ2Z) is an additive Gaussian

noise. With perfect symbol timing, the received data from thekth relay in the frequency domain at the receiver after FFTbecomes

Rkn =1

N

N−1∑i=0

rki e−j2πin/N + Zkn = Hk

nXn + Ikn + Zkn, (3)

where Zkn is a frequency domain additive Gaussian noise,and Hk

n denotes the distorted channel response, which canbe written as

Hkn =

Hknsin(πεk)

Nsin(πεk/N)ej(πε

k(N−1)/N+θk0 ), (4)

3

and the ICI due to the residual CFO is

Ikn =

N−1∑m=0,m6=n

HkmXm

sin(πεk)ej(πεk(N−1)/N+θk0 )

Nsin(π(m− n+ εk)/N)e

−jπ(m−n)N .

(5)

A. Synchronization

As we can see from Fig.2, synchronization is the firstoperation on the received baseband signal at the destination.The objective of OFDM synchronization is finding the start ofpacket (SOP) and the CFO. In this paper, we perform a SISOOFDM synchronization scheme for each link of TDCT.

S&C method[14]: This very popular synchronizationmethod for SISO OFDM link was proposed by Schmidl andCox. The timing and CFO estimations are done based on twoOFDM preamble symbols. For TDCT, the synchronizationbased on the S&C method is applied on each relay linkindependently during different time slots.

B. Channel estimation

As shown in Fig.2, the CSI is estimated in the frequencydomain after CFO compensation, removing CP, and FFT. Inthis paper, we compare two channel estimation methods.

1) LS estimator: the estimate of Hkn based on the training

sequence Xtn for the LS estimator is

HkLS,n =

RknXtn

= Hkn + ηkn, (6)

where Rkn is shown in Eq.(3), and ηkn =Ikn+ZknXtn

denotes thechannel estimation error of the nth subcarrier.

2) Improved LS (ILS) estimator: By taking advantageof the similarity of channel response among D consecutivesubcarriers, we may improve the channel estimation accuracyby simply performing a moving average over subcarriers witha window of length D.

HkILS,n =

1

D

D/2∑d=−D/2

HkLS,n+d (7)

III. IMPACT OF RESIDUAL CFOS AND CHANNELESTIMATION ERROR

A. SINR analysis

The received signal after MRC at the destination can bewritten as

Xn =

K∑k=1

(Hkn)†Rkn

=

K∑k=1

{(Hkn + ηkn)†Hk

nXdn + (Hk

n)†(Ikn + Zkn)}

=

K∑k=1

|Hkn|2Xd

n +

K∑k=1

{(ηkn)†HknX

dn + (Hk

n)†(Ikn + Zkn)},

(8)

where ′′†′′ means conjugate operation.

To evaluate the performance in terms of distance d (thedistance between relay cluster and the destination), the channelgain for each subcarrier is modeled as

E(|Hkn(d)|2) = σ2

0 d−β , (9)

where d = d/d0 is the normalized distance, and σ20 =

E{|Hkn(d0)|2} = 1.

Assuming that the channel response is stationary during onepacket duration, the average SINR for nth subcarrier at thedestination is

SINRn =σ2XE{

∑Kk

∑Kl |Hk

n|2|H ln|2}∑K

k

(CLINk +

∑Kl,l 6=k ILINk,l

) , (10)

and

CLINk = E{σ2X |Hk

n|2|ηkn|2 + |Hkn|2(|Ikn|2 + |Zkn|2)},

ILINk,l = E{Hkn(H l

n)†(Ikn)†I ln}, (11)

where, CLINk is the co-link interference and noise generatedwithin the kth link, while ILINk,l is the inter-link interferenceintroduced between different links k and l.

We assume the links are statically identical: σ2H

=

E{|Hkn|2}; σ2

HH= E{Hk

n(H ln)†} 6= 0, (k 6= l)(because the

channel estimation error is affected by ICI (Eq.(5)), whichare correlated among different links); σ2

H= E{|Hk

n|2};σ2HH

= E{Hkn(H l

n)†} = 0, (k 6= l); σ2I = E{|Ikn|2} is the

averaged power of co-link ICI; σ2II = E{Ikn(I ln)†}, (k 6= l)

is the averaged power of inter-link ICI; σ2η = E{|ηkn|2}

is the averaged power of co-link channel estimation error ;σ2ηη = E{ηkn(ηln)†}, (k 6= l) is the averaged power of inter-link

channel estimation error; σ2X = E{(Xn)†Xn} is the averaged

power of source signal, and σ2X

= E{(Xn)†Xn} is the powerof received signal after MRC. Then, the averaged SINR canbe approximated as

SINRa =K2σ2

Xσ4H

σ2IN

=K2

σ2IN

, (12)

where

σ2IN = K

(σ2Xσ

2Hσ

2η + σ2

H(σ2I + σ2

Z) + (K − 1)(σ2HH

σ2II))

= Kσ2CLIN +K(K − 1)σ2

ILIN ,

σ2IN =

σ2IN

σ2Xσ

4H

= Kσ2CLIN +K(K − 1)σ2

ILIN , (13)

where, σ2CLIN = σ2

Xσ2Hσ2η + σ2

H(σ2I + σ2

Z) is the averagedco-link interference and noise, while σ2

ILIN = (σ2HH

σ2II) is

the averaged inter-link interference and noise.The key to approximating the average SINR after MRC

is the normalized interference and noise power σ2IN , which

consists of the normalized power for CLIN and ILIN,

1

SINRa=σ2CLIN

K+

(K − 1)σ2ILIN

K. (14)

As we will show later, the σ2CLIN , which is orders of

magnitude higher than σ2ILIN , dominates the SINR.

4

B. Energy distribution optimization

As we can see from Eq.(12), the effective SINR at desti-nation is compromised due to interference introduced by bothresidual CFOs and channel estimation error. For the purposeof range extension, we aim to reach longer transmissiondistance without increasing the energy on each relay link.Therefore, when the given energy for synchronization andchannel estimation is limited, how to distribute the total energyto achieve the optimal decoding performance at the destinationcould be modeled as an optimization problem,

minimizew1

BER(K,E, d)

subject to wi ∈ [0, 1];

2∑i=1

wi = 1,

Es = E ∗ w1; Ec = E ∗ w2

(15)

where, K is the number of cooperating relays; E =E(∑p |Xp|2 +

∑t |Xt|2) = cE0 is total energy for synchro-

nization and channel estimation, which is multiple times ofthe energy of single OFDM data symbol as E0 = E{|Xd|2};Es and Ec are the energy used for synchronization andchannel estimation respectively. d is the distance betweenthe relay cluster and the destination. w1 is the weight forsynchronization, while w2 is the weight for channel estimation.

IV. SIMULATION

The joint effects on decoding performance are very com-plicated without knowing the statistics of the synchronizationand channel estimation errors for all K relay links. To avoidthe unrealistic evaluation based on assumptions of ideal errormodels, we simulate the whole TDCT system shown in Fig.2,for three cases.

Cases: S1: S&C’s method for synchronization, and LSestimator for channel estimation; S2: S&C’s method forsynchronization, and ILS estimator (with D=8) for channelestimation; I: ideal case with perfect synchronization and CSIavailable at destination.

The system parameters used are illustrated in Table I.TABLE I

SYSTEM PARAMETERS

Channel Frequency selective fadingSampling rate, 1/Ts 10 MHzRMS delay spread 5 × 10−8s

Modulation BPSKFFT length, N 128

Used subcarriers 112CP length, Ng 8

Normalized CFO betweenrelay cluster and destination 2.2Average SNR@d0, γ(d0) 20dB

Interleaving and convolutional coding Rc = 1/2

A. Synchronization performance

Since OFDM is not sensitive to timing error, we assumeperfect timing and simulate the CFO estimation of S&C’smethod. The mean and mean square error (MSE) of CFO

estimation error ε, MSEε are simulated as

ε =1

(M − 1)K

N∑m=1

K∑k=1

|εkm|, (16)

MSEε =1

(M − 1)K

N∑m=1

K∑k=1

(εkm)2, (17)

where, εkm is the residual CFO for mth link during the nthtrial after CFO estimation and compensation.

For the simulation, E = 2E0, and w1 = w2 = 0.5. Theresulting BER for coded BPSK, in terms of normalized rangeextension distance dk/d0, is shown in Fig.3.

1 2 3 4 5 6 7 8

0.5

1

1.5

2

Normalized distance d/d0

mean o

f C

FO

estim

ation e

rror

K=1,s1

K=1,s2

K=2,s1

K=2,s2

K=4,s1

K=4,s2

K=8,s1

K=8,s2

(a) Mean

1 2 3 4 5 6 7 8

10−2

100

Normalized distance d/d0

MS

E o

f C

FO

estim

ation e

rror

K=1,s1

K=1,s2

K=2,s1

K=2,s2

K=4,s1

K=4,s2

K=8,s1

K=8,s2

(b) Variance

Fig. 3. Statistics of CFO estimation error, when w1 = w2 = 0.5, E = 2E0.

According to Fig.3, since the synchronization scheme isthe same for both cases S1 and S2, their performance curvesoverlap with each other. The synchronization is done on eachSISO link independently, therefore performance statistics areidentical for different numbers of cooperating relays, K. Theperformance of mean and MSE of CFO estimation errordegrades as distance.

B. Channel estimation performanceThe performance of channel estimation is measured using

the normalized mean square error (MSE)

MSEH = E{ 1

KN

K∑k=1

N−1∑n=0

|Hkn − Hk

n|2

|Hkn|2

}, (18)

where, the distorted channel, Hkn , is computed as Eq.(4).

1 2 3 4 5 6 7 8

10−1

100

101

Normalized distance d/d0

MS

E o

f C

SI estim

ation

K=1,s1

K=1,s2

K=2,s1

K=2,s2

K=4,s1

K=4,s2

K=8,s1

K=8,s2

Fig. 4. MSE of channel estimation, when w1 = w2 = 0.5, E = 2E0.

As we can see from Fig.4, the proposed ILS estimatoroutperforms the conventional LS estimator with lower MSEbecause of the benefit from averaging.

5

C. SINR estimation

The key to estimating the average SINR in Eq.(12) isto get the statistics of the averaged and normalized co-linkinterference and inter-link interference as shown in Eq.(13).

The power of the co-link and inter-link interferences intro-duced by residual CFO and CSI estimation error are averagedamong K relay links through N subcarriers as

σ2y = E{

( 1

KN

N−1∑n=0

K∑k=1

(ykn)†ykn)},

σ2yy = E{

( 1

(K(K − 1))N

N−1∑n=0

K∑k=1

K∑l,l 6=k

(ykn)†yln)}, (19)

where, y ∈ {I, η} is the dummy symbol; Ikn and ηkn arecomputed as Eq.(5) and Eq.(6), respectively. The expectationis computed through sample mean during the simulation.

The power of the normalized co-link interference and noise,σ2CLIN , and the normalized inter-link interference and noise,σ2ILIN , are estimated as

σ2CLIN =

σ2Xσ

2Hσ2η + σ2

H(σ2I + σ2

Z)

σ2Xσ

4H

,

σ2ILIN =

σ2HH

σ2II

σ2Xσ

4H

. (20)

0 2 4 6 810

−10

10−5

100

105

Normalized Distance d/d0

Norm

aliz

ed inte

rfere

nce a

nd

nois

e p

er

link

s1 CLIN

s1 ILIN

s2 CLIN

s2 ILIN

*The color denotes different K, which is same as Fig.4.

Fig. 5. Normalized interference and noise

1 2 3 4 5 6 7 8

−60

−50

−40

−30

−20

−10

0

10

20

Normalized distance d/d0

SIN

R d

B

K=1 SINR

K=1 SINR approx

K=1 SNR

K=2 SINR

K=2 SINR approx

K=2 SNR

K=4 SINR

K=4 SINR approx

K=4 SNR

K=8 SINR

K=8 SINR approx

K=8 SNR

Fig. 6. SINR vs distance for different TDCT cases for S1

In Fig.5, we observe that the ILIN is several orders ofmagnitude weaker than the CLIN. That’s because the linkchannel gains are independent and both the synchronizationand channel estimation are done independently for each relayand destination link. When we improve the CSI estimation,both the CLIN and ILIN are improved as S2 with ILSoutperforms S1.

In Fig. 6, the simulation of approximated SINR in Eq.(12)is compared with the SINR, computed as E{K2σ2

X |Hkn|

2}E{|X|2−K2σ2

X |Hkn|2},

and the SNR = σ20 d−β γ without any interference introduced

by CFO and CSI estimation errors.According to the simulation result in Fig.6, the numerical

analysis for SINR in Section III is justified as the approximatedSINR curves match the SNIR curves very well. The result forcase S2 is similar as S1 with all the curves shifted to the lefta little bit since the interference is lower than S1. And wecan see that the SNR and SINR separate long distance. That’sbecause the when the SNR is low at the long distance, theperformance of both synchronization and channel estimationbecome worse, and the interference introduced to the systemdominate the SINR in stead of noise.

D. BER performance

The BER performance of coded BPSK for different scenar-ios are simulated through Monte Carlo simulation as shownin Fig.7.

1 2 3 4 5 6 7 810

−6

10−4

10−2

100

normalized Distance d/d0

BE

R

K=1,s1

K=1,s2

K=1,I

K=2,s1

K=2,s2

K=2,I

K=4,s1

K=4,s2

K=4,I

K=8,s1

K=8,s2

K=8,I

Fig. 7. BER vs distance for different TDCT cases, when w1 = w2 =0.5, E = 2E0

According to simulation result in Fig.7, it is expected thatS2 outperforms S1 because of the improved channel estimator.

Brute-force search for energy optimization: For theenergy distribution optimization problem in Eq.(15), due totime consumption of the simulation, we targeted the ideal BERaround 10−3. The ideal energy distribution weights were in-vestigated through Brute-force search for limited energy case,where E = 2E0. Based on the simulation result in Fig.7, forK = 1, 2, 4, 8 to achieve the BER of 10−3 under ideal case, weset the normalized extended distance d/d0 = 1, 1.7, 2.4, 3.2for S1, and d/d0 = 1.5, 2.2, 3.2, 4.2 for S2, respectively. TheBER as a function of w1 is shown in Fig.8.

As shown in Fig.8, for K=1, 2, 4, 8 in S1 the optimal weightfor synchronization w1 is around 0.3, 0.6, 0.7, 0.7, and is 0.7,0.8, 0.8, 0.8 in S2, respectively. The the optimal weights are

6

0.2 0.4 0.6 0.8 110

−4

10−3

10−2

10−1

Weight for synchronization: w1

BE

R

K=1

K=2

K=4

K=8

(a) S1

0.2 0.4 0.6 0.8 110

−3

10−2

10−1

Weight for synchronization: w1

BE

R

K=1

K=2

K=4

K=8

(b) S2

Fig. 8. BER vs energy weights for different TDCT cases

very close for different K when the corresponding optimalBER is similar. That’s because the errors and offsets areidentically distributed. When the CSI estimation performanceis improved, energy is shifted to the preamble; that’s why theoptimal weights for synchronization in S2 are bigger than S1.

V. CONCLUSION

In this paper, the joint effects of synchronization andchannel estimation errors are investigated for TDCT in termsof range extension distance. The post-combiner SINR atthe destination has been analyzed theoretically and verifiedthrough simulation. The performance degradation caused byco-link interference and noise, which is proportional to numberof cooperating relays, K, is much larger than the performancedegradation introduced by inter-link interference and noise,which is proportional to K2 when the synchronization andchannel estimation are done independently. Since the channelestimation error is affected by the synchronization error, whenthe total energy is limited for the purpose of range extension,the way to distribute the energy for the synchronization andchannel estimation matters, and can be formed as an optimiza-tion problem (the effective scheme to find the optimal weightsis not the focus of this paper). The optimal energy distributiondepending on the performance of synchronization and channelestimation schemes varies for different system settings.

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