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A Spectral Efficient Relay-Based Transmit Diversity

Technique for SC-FDE without Cyclic Prefix

Ui-Kun Kwon, Dae-Young Seol, Gi-Hong Im, Senior Member, IEEE, and Young-Doo Kim†,Member,IEEE

Department of Electronic and Electrical Engineering

Pohang University of Science and Technology (POSTECH)

Pohang, Kyungbuk, 790-784, South Korea

Email: {kuk2580, jeviens, igh}@postech.ac.kr

†Samsung Advanced Institute of Technology (SAIT)

Gyunggi, 449-712, South Korea

Email: young-doo.kim@samsung.com

Abstract—This paper proposes a relay-based transmit diver-

sity technique for single carrier frequency-domain equalization

(SC-FDE). The proposed system achieves spatial diversity over

fading channels in a distribution fashion without cyclic prefix

(CP), which increases spectral efficiency of conventional relay-

based systems. The destruction of channel cyclicity due to the lack

of CP is recovered at the input of relay and destination. In order

to obtain spatial diversity, the transmit sequence of relay is effi-

ciently generated in time domain, realizing space-frequency block

code (SFBC). Corresponding destination structure using FDE,

timing synchronization, and highly accurate channel estimation

for the proposed system are also presented. Simulation results

show that the proposed system approaches the lower bound of

full-CP system.

Index Terms - SC-FDE, CP, relay, SFBC.

I. INTRODUCTION

Single-carrier frequency-domain equalization (SC-FDE) has

similar performance and essentially the same overall complex-

ity as orthogonal frequency division multiplexing (OFDM) [1].

Recently, the SC-FDE has drawn a great attention as an alter-

native to the OFDM, especially in the uplink communications,

because its transmit structure at mobile equipment is very

simple and lower peak-to-average power ratio (PAPR) signif-

icantly benefits the transmitter in terms of power efficiency.

Transmit diversity is an effective technique to combat the

fading effect in mobile wireless communications. A simple

and powerful diversity technique using two transmit anten-

nas, called space-time block code (STBC), was proposed by

Alamouti, guaranteeing full spatial diversity and full rate over

frequency-flat channels [2]. Based on the Alamouti scheme,

Al-Dhahir presented a combination of STBC with SC-FDE for

frequency-selective channels [3]. The performance of STBC

SC-FDE, however, considerably deteriorates in a time-varying

mobile environment. In order to mitigate the fast fading

distortion caused by high-speed mobility, Jang et. al. proposed

space-frequency block code (SFBC) SC-FDE, but it introduces

3dB PAPR increase over two transmit antennas, and additional

computational complexity at the transmitter [4]. Further, em-

ploying multiple antennas at the cellular mobile devices might

be restricted, due to the limitation of size and complexity.

Cooperative diversity overcomes these problems without addi-

tional complexity of multiple antennas [5]. Multiple terminals

in the network cooperate to form a virtual antenna array

1st time slot

S

R

DS

R

D

2nd time slot

hSR

hRD

hSD

Fig. 1. Schematic of distributed SFBC (D-SFBC) protocol.

realizing spatial diversity in a distributed fashion, although

each of them is equipped with only one antenna. For practical

implementation of the cooperative network, Mheidat et. al. and

Seol et. al. extended conventional STBC and SFBC SC-FDEs

in a distributed fashion, so called distributed STBC (D-STBC)

and D-SFBC SC-FDEs, respectively [6], [7].

The diversity benefit of D-STBC/SFBC SC-FDEs comes at

the price of decreasing spectral efficiency by transmitting one

data block over two time slots. In addition, a cyclic prefix

(CP), which should be longer than the length of channel

impulse response (CIR), is appended at the head of each

data block. Since the use of CP much lowers the spectral

efficiency, several approaches have been proposed to cope

with the problem in the case of collocated antenna systems

[8]-[10]. In this paper, we first propose a spectral efficient D-

SFBC SC system without CP. The proposed system achieves

diversity gain without any increase of PAPR and computational

complexity at the mobile equipment. In order to obtain spatial

diversity, the transmit sequence of relay is efficiently generated

in time domain, realizing SFBC. The destruction of channel

cyclicity due to the lack of CP is recovered at the input of relay

and destination. Corresponding destination structure using

FDE, timing synchronization, and highly accurate channel

estimation for the proposed system are also presented.

II. PROTOCOL AND SYSTEM MODEL OF THE PROPOSED

CP-LESS D-SFBC SC

D-SFBC systems require two time slots for the transmis-

sion of one information block as depicted in Fig. 1. The

source communicates with the relay during the first time

slot, while the destination does not receive direct signal from

the source. Then, both the source and relay communicate

with the destination in the second time slot to complete

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

978-1-4244-2075-9/08/$25.00 ©2008 IEEE

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IFFT

P/S

Channel Cyclicity

Reconstruction

Normalization

S/P

FFT

Conjugate

. . .

. . .

. . . . . .

_

_

. . .

Virtual Block

(a)

Conjugate &

Time Reverse

N/2-

Circular

Shift

(b)

Channel Cyclicity

Reconstruction

S/P

FFT

SFBC

Combining

. . . . . .

FDE

IFFT . . .

P/S

. . . . . .

(c)

_

_

Fig. 2.

block. (c) Block diagram of destination.

Relay and destination structures of CP-less D-SFBC SC-FDE. (a) Conceptual block diagram of relay. (b) Efficient implementation of relay virtual

the transmission of SFBC signal. In conventional distributed

systems [6], [7], CP is appended at the beginning of each

block, which is a redundant data transmission. The CP is used

to eliminate the interblock interference (IBI) and transform

the linear convolution channel into the circular convolution

channel. Due to the circular convolution, SC receiver can do a

simple one-tap frequency domain equalization for frequency-

selective multipath channels [4], [7]. Note that in the protocol

of D-SFBC, each link among the source, relay and destination

is used for only one of the two time slots, i.e., when the link

between the source and relay is busy, the other links are idle

in the first time slot, and vice versa in the second time slot

(see Fig. 1). In the proposed system, we do not use CP, since

there is no IBI as far as the length of CIR is no longer than the

length of one data block. The destruction of channel cyclicity

due to the lack of CP is recovered at the relay and destination.

We consider all underlying links experience frequency se-

lective fading. The CIR for the transmitting node (A)-to-

receiving node (B) is given by hAB=[hAB(0),hAB(1),...,

hAB(LAB)]T, where LAB denotes the CIR length. In this

paper, subscripts S, R, and D stand for the source, relay,

and destination nodes, respectively. In the first time slot, the

sequence of source, which is equal to the QAM or PSK

modulated information block x = [x(0),x(1),...,x(N −1)]T,

i.e., x1

length. Unless LSR is longer than N, the received signal at

the relay is given by

?

where ¯ x1

complex additive white Gaussian noise (AWGN) vector with

each entry having a zero-mean and variance of N0/2 per

S=x, is transmitted to the relay, where N is the block

rR=

ESRHSR¯ x1

S+ nR

(1)

Sis the x1

Sappended by LSR zeros, and nR is a

dimension. EAB represents the average energy available at

the receiving node B, and includes path loss and shadowing

effects in A → B link for simplicity. HSRis the (N+LSR)×

(N +LSR) lower triangular Toeplitz channel matrix with the

first column equal to hSRappended by N −1 zeros. Note that

unlike the conventional cyclic prefixed systems, HSR is not

the circulant matrix. Fig. 2 (a) shows the conceptual block

diagram of relay, realizing SFBC. Since there is no IBI in

the received signal at the relay, we can effectively reconstruct

the cyclicity of channel matrix HSRby adding the last LSR

elements of rRto the first LSRelements of rRas follows

?rR(n) + rR(N+n),

Then, the reconstructed signal in (2) can be rewritten in the

matrix form as follows

?

Here, H?SR is an N×N circulant matrix with entries

[H?SR]k,l = hSR((k − l) mod N). Thus, (3) has exactly

the same form as the conventional cyclic prefixed systems,

except that the power of noise n?

is increased as much as twice. r?R is normalized as ˜ rR ?

r?R/

N

the discrete Fourier transform (DFT) of ˜ rR,˜RR, is processed

like below

?

where XR is the transmit signal of the relay in frequency

domain, and (·)∗denotes the complex-conjugate operation.

Changing the signs of odd components and permutating the

r?

R(n)=

0 ≤ n < LSR,

LSR≤ n ≤ N−1.

rR(n),

(2)

r?R=

ESRH?SRx1

S+ n?R.

(3)

R[n] for n=0,...,LSR−1

?N0to ensure unit average energy, and

?

ESR+?1+LSR

XR(2l)

XR(2l + 1)

?

=

?−˜R∗

R(2l + 1)

˜R∗

R(2l)

?

, l = 0,1,...,N

2− 1

(4)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

Page 3

even and odd components can be conducted by multiplying

the following matrices, S and P, respectively.

?

where Ik×k is the k×k identity matrix, and ⊗ denotes the

Kronecker product. Then, the transmit signal of the relay can

be represented as

S=I N

2×N

2⊗

1

0

0

−1

?

, P=I N

2×N

2⊗

?

0

1

1

0

?

(5)

xR= WHPS{W˜ rR}∗

=

ESR+?1+LSR

where (·)Hdenotes the conjugate transpose, W is a DFT

matrix, and n??

Using DFT properties, xR can be also efficiently obtained

in time domain without the fast Fourier transform (FFT) and

inverse FFT (IFFT) operations as follows [7]

xR= rc(n)×j sin(2πn

N

where rc(n) = ˜ r∗

The efficient implementation of relay virtual block is depicted

in Fig. 2 (b).

In the second time slot, the source and relay transmit

x2

Lmax=max(LSD,LRD) is longer than N, the received signal

at the destination is given by

?

where ¯ xR, ¯ x2

tively, and nD is a complex AWGN vector with each entry

having a zero-mean and variance of N0/2 per dimension.

HRD and HSD are the (N+Lmax) × (N+Lmax) lower

triangular Toeplitz channel matrixes with the first column

equal to hRDand hSDappended by zeros, respectively.

?

ESR

N

?N0

WHPS{WH?

SRx1

S}∗+n??

R(6)

R=WHPS{Wn?R}∗/

?

ESR+?1+LSR

N

?N0.

) + rc(n−N

2)N×cos(2πn

N

)

(7)

R(−n)N, and (n)N represents n (mod N).

S, remaining the same as x1

S, and xR, respectively. Unless

rD =

ERDHRD¯ xR+

?

ESDHSD¯ x2

S+ nD

(8)

Sare xR, x2

Sappended by Lmaxzeros, respec-

III. DESTINATION STRUCTURE OF THE PROPOSED

CP-LESS D-SFBC SC USING FDE

Fig. 2 (c) shows the block diagram of destination. The

cyclicity of HRDand HSDin (8) is effectively reconstructed

as in (2) by adding the last Lmaxelements of rDto the first

Lmaxelements of rD. Then, the signal at the destination can

be written as follows

?

where n?

power, and H?RDand H?SDare N×N circulant matrixes with

entries [H?RD]k,l= hRD((k − l) mod N) and [H?SD]k,l=

hSD((k − l) mod N), respectively. We can rewrite (9) as

follows

?

N

+

ESDH?

ERDH?

r?

D=

ERDH?

RDxR+

?

ESDH?

SDx2

S+ n?

D

(9)

Dis the AWGN vector with a partially increased

r?

D=

ERDESR

ESR+?1+LSR

?N0

H?

RDWHPS{WH?

SRx}∗

?

SDx +

?

RDn??

R+ n?

D.

(10)

By omitting

term of (10) can be written as

?

ERDESR/?ESR+?1+LSR

RDWHPS{WH?

N

?N0

SR{Wx}∗

SRS{Wx}∗(11)

SRWH) are

?, the first

H?

SRx}∗=WHΛRDPSΛ∗

=WHΛRDPΛ∗

where ΛRD(= WH?

N×N diagonal matrices. If we assume that the channel

frequency responses (CFRs) between adjacent subcarriers are

approximately constant, i.e., ΛSR(2k)∼= ΛSR(2k+1), where

ΛSR(k) is the kth diagonal element of ΛSR, (11) can be given

as

RDWH) and ΛSR(= WH?

WHΛRDPΛ∗

SRS{Wx}∗∼= WHΛRDΛ∗

By substituting (12) into (10), r?

SRPS{Wx}∗. (12)

Dcan be rewritten as

r?

D= γ1WHΛEQPS{Wx}∗+ γ2H?

where γ1=

n=√ERDH?

is the equivalent CFR for the S→R→D link. Then, by

performing FFT on r?

follows

SDx + n

?N0), γ2=√ESD,

(13)

?

ERDESR/(ESR+?1+LSR

N

RDn??

R+n?

D, and ΛEQ=ΛRDΛ∗

SR. Here, ΛEQ

D, R?

D(=Wr?D) can be obtained as

R?

D= γ1ΛEQPSX∗+ γ2ΛSDX + N

(14)

where ΛSD=WH?

ing complex-conjugate operation on odd elements, we can

divide (14) into even and odd components, and change it into

the matrix form as

?

∼=

γ1Λ∗

γ2Λ∗

? Λ?

SDWH, X=Wx, and N=Wn. After tak-

R?

k?

R?

D(2k+1)

?γ2ΛSD(2k) −γ1ΛEQ(2k)

kX?

D(2k)

R?∗

?

EQ(2k)

k+ N?

SD(2k)

??

X(2k)

X∗(2k+1)

?

+

?

N(2k)

N∗(2k+1)

?

k.

(15)

Due to the central limit theorem with a relatively large number

of block length N, N(k) can be assumed to have a complex

Gaussian distribution, and the variance of N(k) per dimension

is given by

?

N

σ2

N(k)=

ERD|ΛRD(k)|2?1+LSR

From (15) and (16), we can now derive an SFBC combining

receiver under the minimum mean squared error (MMSE)

criterion, which is

?˜Λ(k)

˜Λ(k) ? |γ1ΛEQ(2k)|2+ |γ2ΛSD(2k)|2

ˆX?

SNRI2×2)−1Yk=

N

?

ESR+?1+LSR

?N0

+

?

1+Lmax

N

??

N0

2. (16)

Yk? Λ?H

kR?

k=

0

0

˜Λ(k)

?

X?

k+ Λ?H

kN?

k

k? (Λ?H

kΛ?

k+

1

?

ˆ X(2k)

ˆ X∗(2k+1)

?

. (17)

Note that decisions of the SC using FDE are made in the

time domain, although channel equalization is performed in

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

Page 4

the frequency domain. Therefore, the estimates of information

blocks can be obtained as

ˆ x = WHˆX

(18)

whereˆX = [ˆ X(0),ˆ X(1),...,ˆ X(N−1)]T.

IV. TIMING SYNCHRONIZATION AND CHANNEL

ESTIMATION

Even if the source and relay transmit signals simulta-

neously, different delays from the source and relay result

in an asynchronous arrival at the destination. Thus, timing

synchronization between the source and relay has been one

of the key challenges to implement a distributed network.

To cope with this problem, the CPs with increased length

L = {max(LRD,LSD)+timing offset}, yet decreasing the

spectral efficiency, would be required in conventional dis-

tributed cyclic prefixed systems to guarantee the cyclicity of

channel matrixes. Further, the effect of timing offset turns

into a cyclic shift of the received signal with delay, which

corresponds to a phase rotation in frequency domain. Without

loss of generality, we assume that the signal from relay is

received with time delay Toffset. Then, the timing offset should

be compensated through a phase-rotated CFR like below

?

In the CP-less D-SFBC sytem, the destruction of channel

cyclicity due to the timing offset can be easily reconstructed

with the proposed channel cyclicity reconstruction method.

The phase rotated CFR can be obtained with the following

proposed channel estimation method.

We estimate the CFR by using a Chu sequence [11] as

a training sequence, which has constant amplitude in both

frequency and time domains. This property precludes the

PAPR increase of the transmit signal. The nth element of a

length-N Chu sequence is given by

?

where r is relatively prime to N. Note in (17) that, at the

destination of proposed system, only the ΛEQand ΛSDare

required instead of each ΛSR, ΛRD, and ΛSDfor the SFBC

combining and channel equalization. For the estimation of

ΛEQ, the source transmits a training sequence cEQ in the

first time slot. After the channel cyclicity reconstruction, the

received signal at the relay is given by

?

as in (3), and c?

tain the CFR ΛEQ(=ΛRDΛ∗

conjugation and time-reversal operations are taken on ˜ c?

Λ?

EQ(k) =

e−j2πk

NToffsetΛRD(k)

?

Λ∗

SR(k).

(19)

cN(n) =

ejπrn2/N

ejπrn(n+1)/N

,

,

for even N

for odd N

,

(20)

c?

R=

ESRH?

SRcEQ+ n?

R

(21)

Ris normalized, yielding ˜ c?

SR) at the destination, complex

R. In order to ob-

Ras

cRD(n) = ˜ c?∗

R(−n)N,

(22)

which corresponds to complex conjugate operation in the

frequency domain. Then, the relay transmits cRDin the second

468101214 1618

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

ESD/N0(dB)

Bit Error Rate

SISO SC−FDE with CP (Perfect CSI)

D−SFBC SC−FDE with CP (Perfect CSI)

D−SFBC SC−FDE w/o CP (Perfect CSI)

CP−less D−SFBC SC−FDE with CCR (Perfect CSI)

CP−less D−SFBC SC−FDE with CCR (Estimated CSI)

CSI: Channel State Information

CCR: Channel Cyclicity Reconstruction

Fig. 3.

ESR/N0= 20dB).

BER performance of CP-less D-SFBC SC-FDE (N=1024, QPSK,

time slot. After the channel cyclicity reconstruction, the DFT

of received signal at the destination is given by

C?

D= γ1ΛRDΛ∗

SRC∗

EQ+N = γ1ΛEQC∗

?={WcEQ}∗?, ΛEQ can be

EQ+N.

(23)

With the knowledge of C∗

estimated. ΛSD could be easily estimated in an additional

time slot, but it is inefficient in the sense of channel estimation

overhead.

We design two training sequences with a length-N, which

are orthogonal in frequency domain, in order to estimate the

CFRs, ΛEQand ΛSD, at the same time. The sequences are

given as

EQ

cEQ(n) = cN/2(n)N/2, cSD(n) = ej2πn

N cEQ(n).

(24)

Since cEQ is a simple repetition of length-N/2 Chu se-

quences and cSD is a phase rotated version of the cEQ, the

sequences inherit the desirable unit PAPR property. And those

sequences are orthogonal in the frequency domain as follows

CEQ=[CN/2(0),0,CN/2(1),0,...,CN/2(N/2−1),0],

CSD=[0,CN/2(0),0,CN/2(1),...,0,CN/2(N/2−1)],

where CN/2 is the DFT of cN/2. The nulled CFRs are

reconstructed by using the interpolation algorithm described

in [12]. It is noted that the phase rotation due to the timing

offset in (19) is reflected in the estimated CFRs.

(25)

V. SIMULATION RESULTS

The bit-error rate (BER) performance of the proposed CP-

less D-SFBC SC-FDE system is investigated through computer

simulations. We consider uncoded systems with the block

size of N=1024, QPSK constellation, and 5MHz bandwidth.

All underlying links experience frequency-selective channels,

where S→D and S→R links are modeled as the COST-207

typical urban (TU) channel [13], and R→D link is the 2-path

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

Page 5

05 10 1520 2530 35 40

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Timing Offset (Samples)

Bit Error Rate

D−SFBC SC−FDE with CP (w/o TOC, Perfect CSI)

D−SFBC SC−FDE with CP (with TOC, Perfect CSI)

CP−less D−SFBC SC−FDE with CCR (Perfect CSI)

CP−less D−SFBC SC−FDE with CCR (Estimated CSI)

CSI: Channel State Information

TOC: Timing Offset Compensation

CCR: Channel Cyclicity Reconstruction

Fig. 4.

offsets (N=1024, QPSK, ESR/N0= 20dB, ESD/N0= 18dB).

BER performance of CP-less D-SFBC SC-FDE for different timing

channel with a uniform power delay profile. We assume that

the S→D and R→D links are balanced, i.e., perfect power

control.

Fig. 3 shows the performance of CP-less D-SFBC SC-FDE.

Note that the performance of D-SFBC SC-FDE without CP

is even worse than that of the SC-FDE with single antenna

at the high SNR. Under the assumption of perfect channel

knowledge, it is observed that the proposed CP-less D-SFBC

SC-FDE with channel cyclicity reconstruction approaches the

cyclic prefixed D-SFBC SC-FDE with only 0.3dB gap at the

BER= 10−5. The CP-less D-SFBC SC-FDE with channel esti-

mation shows a BER degradation of about 0.6dB at BER=10−5

as compared with the case of perfect channel state information

(CSI). As such, the proposed system increases the spectral

efficiency by the amount of CP length, which depends on the

channel length. Fig. 4 shows the performance of CP-less D-

SFBC SC-FDE for different timing offsets. CPs with length

L=max(LSD,LRD) are used for the cyclic prefixed D-SFBC

SC-FDE in the second time slot. Without the timing offset

compensation mentioned in (19), the D-SFBC SC-FDE breaks

down with a small timing offset. Even though the timing offset

is compensated, the BER performance of D-SFBC SC-FDE

gets worse as the timing offset increases, due to the insufficient

CP. In contrast, the CP-less D-SFBC SC-FDE with channel

cyclicity reconstruction gives a reliable performance over the

whole range of timing offset, for both cases of perfect and

estimated CSIs.

VI. CONCLUSIONS

This paper proposes a D-SFBC SC-FDE without CP, which

increases the spectral efficiency of conventional distributed

systems. The destruction of channel cyclicity caused by the

lack of CP is effectively recovered at the relay and destina-

tion with only LSR and Lmax complex addition operations,

respectively. At the relay, the proposed system achieves spatial

diversity by realizing the SFBC in time domain. Timing

synchronization and highly accurate channel estimation for

the proposed system are also presented. Simulation results

show that the proposed CP-less D-SFBC SC-FDE approaches

the lower bound of full-CP system, and is robust against the

timing offset. It is noted that the proposed channel cyclicity

reconstruction method can also be applied to the distributed

STBC systems.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.