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Convolutional Code Spread Multicarrier System

with Complementary Sequences

Chao ZHANG

School of Aerospace, Tsinghua Univ.,

Beijing, 100084, P. R. China

zhangchao@tsinghua.edu.cn

Dong CHENG

School of Electrical Engineering,

Xi’an Jiaotong University,

Xi’an, P. R. China

Yonghua ZHANG

School of Electrical Engineering,

Xi’an Jiaotong University,

Xi’an, P. R. China

Abstract—A novel convolutional code spread scheme for multi-

carrier system with high diversity gain is proposed in this paper.

This scheme, named by Convolutional Code Spread Multicarrier

(CCSM), convolutionally spreads the data symbols onto several

consecutive subcarriers with complementary sequences. Com-

pared with the traditional OFDM and OFDM-CDM systems,

CCSM enjoys high BER performance with equivalent spectrum

efficiency in multipath Rayleigh fading channels. Moreover, the

special structure of convolutional spreader in CCSM makes the

system suitable for high order modulation, which undoubtedly

endows the CCSM with the attractable capability of high trans-

mission capacity.

I. INTRODUCTION

Orthogonal Frequency Division Multiplex (OFDM) has

been considered as an efficient transmission scheme in mul-

tipath environment, especially for frequency selective fading

channels. OFDM and OFDM based transmission schemes,

such as OFDMA and MIMO-OFDM etc., have already been

proposed in some protocols or standards, e.g., DVB, IEEE

802.11a and IEEE802.16e etc. [1] [2]. In order to handle

the issue of the frequency selective fading, channel coding

is always involved to correct errors caused by deep fading

in some subcarriers. However, as we known, channel coding

brings the redundancy and decreases the spectrum efficiency

[3].

Multi-Carrier Code Division Multiple Access (MC-CDMA),

proposed as the new multiple access schemes based on a

combination of code division and OFDM, can achieve a good

diversity in frequency domain and efficiently fight against

deep fading in the subcarriers [4]. Moreover, each user can

be assigned unique orthogonal spectrum spread sequence and

spread the user data in frequency domain. However, the fading

in the subcarriers destroys the orthogonality between the

sequences and results in Multiple Access Interference (MAI).

Several types of combination schemes have to be utilized to

mitigate MAI, e.g., Controlled Zero Forcing, Minimum Mean

Square Error Combining (MMSEC) etc. [1]. In addition, MC-

CDMA has low spectrum efficiency because the user data are

spread over all the subcarriers. For each user, the bandwidth

efficiency is only 1/N of the conventional OFDM, where N

is the number of subcarriers.

Other several improved OFDM schemes via sequence

spread multiplex in the frequency domain like MC-CDMA

have been proposed to achieve diversity gain in point-to-

point transmission, such as OFDM Code-Division Multiplex-

ing (OFDM-CDM) [5] and Multi-Carrier Convolutional Mul-

tiplexing (MCCM) [6] etc. OFDM-CDM can be considered as

a special MC-CDMA system with all the sequences assigned

to ONE user. Although OFDM-CDM can get diversity gain, it

still suffers the interference caused by loss of orthogonality

among the sequences when the system load is high [5].

MCCM uses the convolutional structure to spread the user data

in the frequency domain. However, the interference between

successive user data appears intolerable, which is not analyzed

in [6].

In this paper, a novel convolutional code spread scheme for

multicarrier system with high diversity gain, named by Convo-

lutional Code Spread Multicarrier (CCSM), is proposed. This

scheme makes use of the ideal correlation property of com-

plementary sequences and has no loss in spectrum efficiency

compared with conventional OFDM transmission. The rest of

this paper is organized as follows. In Sect.II, complementary

sequences and the corresponding ideal correlation property are

reviewed as the preliminary knowledge. In Sect.III, the system

model of CCSM is proposed and analyzed. Furthermore, the

performance of the CCSM system is evaluated in Sect.IV.

In Sect.V, the applications for CCSM are discussed and in

Sect.VI, the conclusion is drawn to highlight the contributions.

II. PRELIMINARY KNOWLEDGE

In [7], the set of complete complementary sequences was

proposed. The definition is briefly reviewed in the following.

Denoting the sequence Siin length of L as

Si = (s(i)

0,s(i)

1,··· ,s(i)

L−1)

(1)

then, the correlation function of sequence S0and S1is defined

as

Rs0,s1(τ) =

L−τ−1

?

n=0

s(0)

n s∗(1)

n+τ

(2)

where s∗

of complete complementary sequences is defined as follows.

M × M sequences with the length of L

nis the complex conjugate of sequence sn. The set

Si,j = {si,j,l} = {si,j,0,si,j,1,··· ,si,j,L−1}

(3)

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where 0 ≤ l ≤ L − 1 and |si,j,l| = 1 form a sequence set as

S

=[Si,j]M×M

=

S0,0

S1,0

...

SM−1,0

S0,1

S1,1

...

SM−1,1

···

···

...

···

S0,M−1

S1,M−1

...

SM−1,M−1

(4)

If the sum of correlation function satisfies

1

ML0

M−1

?

j=0

RSi,j,Sk,j(τ) = δ(τ)δ(i − k)

(5)

(0 ≤ i,j,k ≤ M − 1)

where δ(n) is the Dirac function

δ(n) =

?

1,(n = 0)

0,(n ?= 0)

(6)

Then, we call it the set of complete complementary sequences

with the dimension of M. Sequence Si,j is the jth sequence

in the ith group in the set.

When M

=2, the set of complete complementary

sequences degrades into complementary Golay pairs [8].

Example :

If M = 2, the sequence set S is obtained as

S =

?

S0,0

S1,0

S0,1

S1,1

?

(7)

and if L = 8, then

S0,0 = (1,1,−1,1,−1,−1,−1,1)

(8)

S0,1 = (1,−1,−1,−1,−1,1,−1,−1)

(9)

S1,0 = (−1,−1,1,−1,−1,−1,−1,1)

(10)

S1,1 = (−1,1,1,1,−1,1,−1,−1)

(11)

The Sum of Auto-Correlation Functions (SACF)

1 ?

j=0RS0,j,S0,j

Functionsand

1 ?

correlation property of complementary sequences.

theSum ofCross-Correlation(SCCF)

j=0RS0,j,S1,jare shown in Fig.1, which reveals the ideal

0

(a) SACF.

0

(b) SCCF.

Fig. 1.SACF and SCCF of complementary sequences.

Symbol

Mapper

Convolutional

Spreader

IFFT

Add

CP

D/A

Channel

A/D

Remove

CP

FFT

Matched

Filter

De-Mapper

User Data

Fig. 2.The system model of CCSM.

III. SYSTEM MODEL

The system model of CCSM system is shown in Fig.2.

In the transmitter side, at first, the user data are mapped to

the symbol of the constellation for modulation, e.g., QPSK or

8PSK constellation. Then, the mapped symbols pass through

the convolutional spreader, which convolutionally spreads the

data symbol on several subcarriers with complementary se-

quences to achieve high diversity gain. After that, Inverse

Fast Fourier Transform (IFFT) is adopted like OFDM. With

padding the Cyclic Prefix (CP), the resultant CCSM signal is

sent out.

In the receiver side, the received signal is processed with

removing CP and Fast Fourier Transform (FFT). The special

designed Matched Filter for CCSM can pick up the data

symbol from the convolutionally spread sequences. Finally,

the user data is restored with De-Mapper module.

The Convolutional Spreader with complementary sequences

and special Matched Filter are two very important components

in CCSM system, which reflects the major significant contri-

bution of CCSM.

In CCSM, without loss of generality, the complementary

sequences with M = 2 are adopted, i.e., Golay complementary

sequences. The structure of the Convolutional Spreader in the

transmitter side is illustrated in Fig.3.

time slot

time slot

DDD

?

????

?

( )

x l

( )

a k

0T

DDD

?

????

?

( )

x l

( )

a k

1T

1(1)

C

1(0)

C

2( 1)

C L?

2(2)

C L?

1( 2)

C L?

1(1)

C L?

2(1)

C

2(0)

C

(1)

a kL

??

( 1)

a kL

??

Fig. 3. The structure of convolutional spreader of CCSM.

The input data symbols go through an FIR filter weighted

by the elements of the complementary sequences. Specifically,

in Eq.7, the user data are processed by S0,j,j = 0,1, and

S1,j,j = 0,1, separately. Take S0,0and S0,1as example in the

following analysis. For easy description, denote S0,0and S0,1

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Page 3

as sequences C1 = {c1(i)} and C2 = {c2(i)} respectively,

where 0 ≤ i ≤ L − 1. There are two FIR filters, one for time

slot T0 and one for T1. The period of time slot is the time

period for one OFDM symbol, which is denoted by TS. In

the time slot T0, the data symbol a(k) is input and spread by

C1. In the time slot T1, the same data symbol a(k) is input

and spread by C2. Since i = 0,1, the Convolutional Spreader

at most consists of four FIR filters, the output of FIR filter

weighted by Si,0,i = 0,1, will be added up together and sent

out in the time slot T0. Similarly, the output of FIR filter

weighted by Si,1,i = 0,1, will be added up together and sent

out in the time slot T1. Meanwhile, the time slot T0and T1

run alternatively.

0T

0f

1f

2f

1

Nf

?

0f

1f

2f

1

Nf

?

1( 1)

C a NL

??

1(2)

C a

1(1)

C a

1(0)

C a

2(1)

C a NL

??

2(2)

C a

2(1)

C a

2(0)

C a

10

s

TTT

??

)(

1

LNaC

?

)(

2

LNaC

?

Fig. 4.The illustration of sequence spread in the frequency domain.

The illustration of sequence spread in the frequency domain

is illustrated in Fig.4. Similar to a pipeline structure [9], the

sequences C1and C2are multiplied by data symbol a(k) and

transmitted on L consecutive subcarriers in the time slot T0

and T1respectively. The multiplied sequence for a(k+1) has

one chip-shift compared with the multiplied sequence for a(k)

in the frequency domain, i.e., one subcarrier shift.

FFT

Weighted

by

id

?

( )

r t

( )

dr t

( )

R t

1

C

2

C

( )

s

D T

Matched Filer

Fig. 5.The structure of the matched filter for CCSM.

The structure of Matched Filter for CCSM in the receiver

side is illustrated in Fig.5. The outputs of FFT are weighted

by di,0 ≤ i ≤ N − 1, and pass through Matched Filter,

where N indicates the number of utilized subcarries. Matched

Filter consists the matched filter for C1and the matched filter

for C2, which are indicated by¯C1and¯C2respectively. The

output of the matched filter¯C1 is delayed with one OFDM

symbol period TS and added up together with the output of

the matched filter of¯C2.

IV. PERFORMANCE EVALUATION

A. Numerical Analysis

According to Sect.III, without loss of generality, take S0,0

and S0,1as example. Also, use C1and C2to denote S0,0and

S0,1. Assume the N −L+1 symbols of user data, denoted by

a(l) and 0 ≤ l ≤ N − L, will be spread by C1and C2. The

resultant data will be transmitted on N subcarriers. Moreover,

the magnitude and the phase of channel impulse response for

the ith subcarrier is represented as ρiand θ(j)

where θ(1)

i

is for the time slot of C1and θ(2)

slot of C2. Therefore, the received signal is denoted by r(t)

and

i, j = 1 or j = 2,

is for the time

i

r(t)=

N−L

?

·Rect(2Ts,Ts,0)]

N−L

?

·Rect(2Ts,Ts,Ts)]

+n(t)

l=0

l+L−1

?

i=l

[ρic1(i − l)a(l)cos(ωct + 2πit/Ts+ θ(1)

i )

+

l=0

l+L−1

?

i=l

[ρic2(i − l)a(l)cos(ωct + 2πit/Ts+ θ(2)

i )

(12)

where θ(j)

phase of the transmitted signal in the time slot T0and T1. In

the above equation, Rect(a,b,c) denotes the unit amplitude

rectangular function with time period a, the duty ratio is b/a

and the initial phase is c.

i

= ϕi+ θ(j)+ ωcτi,j = 1,2, θ(j)is the initial

rd(t)=

2

Ts

?

2Ts

[r(t)dicos(ωct + 2πit/Ts+ θ(1)

iRect(2Ts,Ts,0)

+θ(2)

?

+c2(i − l)Rect(2Ts,Ts,Ts))] + η

i Rect(2Ts,Ts,Ts))dt] + η

N−L

a(l)[ρidi(c1(i − l)Rect(2Ts,Ts,0)

=

l=0

l+L−1

?

i=l

(13)

where η is the noise item

η=

2

Ts

?

2Ts

[n(t)dicos(ωct + 2πit/Ts+ θ(1)

i Rect(2Ts,Ts,0)

+θ(2)

i Rect(2Ts,Ts,Ts))]dt

(14)

Then, the single rd(t) passes through matched filter and we

can get the detection variable of the user data. Without loss of

generality, the detection variable for user data a(k) is denoted

by zk, and

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Page 4

zk

=

k+L−1

?

N−L

?

+η

S + I + η

i=k

ρidi(c2

1(i − k) + c2

2(i − k))a(k)

+

l=0

l?=k

k+L−1

?

i=k

ρidi(c1(i − l)c∗

1(i − k)+c2(i − l)c∗

2(i − k))a(l)

=

(15)

where S indicates the signal of user data a(k), I indicates the

interference from other user data. It is easy to get SIN R as

follows

SIN R =

PS

PI+ Pη

(16)

Where PSis the power of the user data a(k)

PS =2Eb

TbL(

k+L−1

?

i=k

ρidi)2

(17)

PIis the power of the interference from other user data

PI =

Eb

2TbL(

N−L

?

l=0

l?=k

k+L−1

?

i=k

ρidi(c1(i − l)c∗

1(i − k)

+c2(i − l)c∗

2(i − k)))2

(18)

Pηis the power of noise

Pη =2N0

Tb

k+L−1

?

i=k

d2

i

(19)

In the above equations, Tb = 2TS. Gaussian approximation

can be employed to evaluate the BER performance

Pe = Q(√SIN R)

(20)

where “Q” represents Q function [3]. Specifically, whichever

EGC,MRC or MMSEC combiner is applied, diis the function

of ρi, the

?∞

·dρkdρk+1···dρk+L−1

where Pe(ρk,ρk+1,··· ,ρk+L−1) is the bit error probability

conditioned on ρi, f(ρk,ρk+1,··· ,ρk+L−1) is the joint p.d.f.

of ρi. Assuming ρk,ρk+1,··· ,ρk+L−1 distribute indepen-

dently with Rayleigh distribution, the following equation is

obtained.

Pe

=

0

Pe(ρk,ρk+1,··· ,ρk+L−1)f(ρk,ρk+1,··· ,ρk+L−1)

(21)

f(ρk,ρk+1,··· ,ρk+L−1) = f(ρk)f(ρk+1)···f(ρk+L−1)

and

(22)

f(ρ) =

ρ

σ2exp(−ρ2

2σ2),0 ≤ ρ ≤ ∞

(23)

where σ2is the variance of ρ. If MMSEC is applied, then

dj=

ρ2

ρj

j+N0.

B. Spectrum Efficiency

If only S0,0 and S0,1 are employed in CCSM, due to

the same data transmitted in the time slot T0 and T1, the

spectrum efficiency is half of that of the conventional OFDM.

However, if S1,0and S1,1are involved, the spectrum efficiency

is restored. Therefore, we confirm the maximum spectrum

efficiency of CCSM equals to that of the conventional OFDM,

which consolidates the significance of CCSM, i.e., achieving

diversity gain without loss of spectrum efficiency.

C. BER Performance

According to the analysis in Sect.IV-A, the BER perfor-

mance can be evaluated in AWGN channel as well as multi-

path Rayleigh fading channels. The simulation parameters are

shown in Table I.

TABLE I

SIMULATION PARAMETERS.

The length of Complementary Sequences

The total number of subcarries

The number of utilized subcarries

The time period of OFDM symbol

Guard interval in time domain

Modulation type

Transmission rate

Combination scheme

L = 4

Nc = 64

N = 52

Ts = 4µs

Gl= 0.8µs

BPSK

R = 13Mbps

MMSEC

In the simulation, three typical systems are studied, i.e.,

conventional OFDM, OFDM-CDM, and the proposed CCSM

system.

Fig.6 shows the simulation result of AWGN Channel.

The three systems in AWGN channel have equivalent BER

performance, because there is no diversity gain in AWGN

channel for OFDM-CDM as well as CCSM system. Under this

circumstance, the performances of OFDM-CDM and CCSM

both degrade into the performance of conventional OFDM.

02468 10

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0(dB)

BER

conventional OFDM

OFDM−CDM

CCSM

Fig. 6.BER performance in AWGN channel.

Fig.7 shows the simulation result of multipath Rayleigh

fading channel. In the figure, the 4-fold diversity of conven-

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

tional OFDM is also illustrated for reference. The conven-

tional OFDM has the worst performance. The OFDM-CDM

outperforms the conventional OFDM, but is inferior to CCSM.

CCSM has the best performance among the three systems.

The 4-fold diversity indicates the bound of the diversity gain.

However, in order to achieve diversity gain without the loss of

spectrum efficiency for the conventional OFDM system, other

technologies must be employed, such as the space diversity.

05101520 25 303540

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0(dB)

BER

conventional OFDM

OFDM−CDM

CCSM

4−fold diversity

Fig. 7.BER performance in multipath Rayleigh fading channel.

Fig.8 shows the BER performance for CCSM systems with

different length (L) of complementary sequences. From the

figure, it is easy to get the concept that the longer complemen-

tary sequences yield higher diversity gain, i.e., the CCSM with

longer complementary sequences has better BER performance.

05 1015 2025 3035 40

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0(dB)

BER

L = 2

L = 4

L = 8

L = 16

Fig. 8.BER performance with different L.

V. APPLICATION ANALYSIS

In Sect.IV-B, the equivalence of the spectrum efficiency of

CCSM and the conventional OFDM has been revealed. In

practical scenarios, as we known, the conventional OFDM

scheme has guard interval at the two terminal parts of the

frequency band. For instance, 802.16d has 256 subcarriers,

but only 200 subcarriers are used for data and pilots. Other

56 subcarriers carry no data and are assigned for the guard

interval in the frequency domain. Also, in the simulation in

Sect.IV, only 52 subcarriers within 64 subcarriers are utilized.

If CCSM employs the complementary sequences in length of

4, it is reasonable to borrow 3 subcarriers from the guard

interval, so that the CCSM has exactly the same transmission

rate and spectrum efficiency with the conventional OFDM.

If the length of complementary sequences is not so long,

the occupation of the guard interval is not so severe, and

the performance of CCSM hardly degrades caused by the

interference from the adjacent frequency band.

In addition, benefited from the special FIR structure of con-

volutional spreader, the CCSM can be applied with different

modulation schemes, especially the high order modulation,

e.g., 8PSK etc. Also, different types of complementary se-

quences can be adopted. The details can be referred to [10].

VI. CONCLUSION

Convolutional Code Spread Multiplex (CCSM) scheme em-

ploys convolutional spreader with complementary sequences to

get high diversity gain. Compared with the traditional OFDM

and OFDM-CDM, CCSM has no loss of spectrum efficiency,

but better BER performance in multipath Rayleigh fading

channel. Moreover, due to the special structure of the inner

Convolutional Spreader, CCSM can adequately utilize high

order modulation to further increase the transmission rate and

spectrum efficiency of the system.

ACKNOWLEDGMENTS

This work is supported by National Basic Research Program

of China (2007CB310601) and NSFC (60532070).

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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.