On ZCZ sequences and its Application to MC-DS-CDMA

June 23, 2005

SUK HOON NOH

Coding and Information Theory LabYONSEI University

Coding and Information Theory Lab 2/22

Contents

Introduction- Definition of ZCZ sequences, Sequences, Correlation and Interferences

ZCZ Sequences- Proposed Ternary ZCZ sequences

MC-DS-CDMA using ZCZ sequences- System Model, BER Analysis- Simulation Results

Cyclic Extension of ZCZ sequences- Signal Analysis on Perfect Elimination of Interferences

Conclusions

Coding and Information Theory Lab 3/22

ZCZ Sequences : GO(N, M, Z0)

ZCZ / Generalized Orthogonal Sequences : GO(N, M, Z0)

( N : Sequence length, M : family size, Z0 : ZCZ length)- are defined as Sequences with Zero Correlation Zone (ZCZ) Z0

Where

- Well Known bound for binary ZCZ sequences MZ0 N/2

for ternary ZCZ sequences MZ0 N (accurately, MZ0 N+Z0 -1)

s r, allfor ,0for 0,

sr 0,for ,0sr 0,for ,

)(1

0

)*()(,

N

no

sn

rnsr

Z

Naa

Orthogonal

1( ) ( )

0

1 1, : Complex numberN

r rn n

na a

N

Coding and Information Theory Lab 4/22

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

Phase

Cro

ssco

rrel

atio

n

Crosscorrelation of GO(32, 4, 4)

-15 -10 -5 0 5 10 15-5

0

5

10

15

20

25

30

35

Phase

Aut

ocor

rela

tion

Autocorrelation of GO(32, 4, 4)

Fan’s ZCZ Sequences GO(32, 4, 4) (N=32, M=4, Zo=4)

(1)

(2)

(3)

(4)

,

{ }{ }{ }{ }

{ 0000 32 0000

n

n

n

n

r r

aaaa

,

}{ 0000 0 0000 }r s

ZCZ

ZCZ

In-phase

Coding and Information Theory Lab 5/22

Sequences, Correlation & Interferences

Interferences and Spreading Sequences - Autocorrelation of a sequences MPI- Crosscorrelation between sequences MAI, MUI

Periodic and Aperiodic Correlation - No ISI, Synch. among different symbols Full correlation of spreading signals

Only Periodic correlation of sequences - ISI, Asynch. among different symbols Partial correlation of spreading signals

Aperiodic correlation of sequences

Partial Interference

Spreading Sequences Design Issues- Sequences families with impulsive ideal autocorrelation and zero crosscorrelation

between sequences are IMPOSSIBLE to design : by Well known Welch’bound

Coding and Information Theory Lab 6/22

Definition of Correlation

Definition 1 Periodic AutoCorrelation Function (ACF) & CrossCorrealtionFunction (CCF)

Definition 2 Aperiodic ACF & CCF

1( ) *( )

,0

ACF, if ( )

CCF, if r s

Nr s

A A n nn

r sa a

r s

1( ) *( )

,0

ACF, if ( )

CCF, if r s

N mr s

A A n n mn

r sR m a a

r s

a 1a 0a 1Na

0a

0a 1Na

0a 1Na ma

1N ma

Coding and Information Theory Lab 7/22

Proposed Ternary ZCZ Sequences

Let any row of matrix F(n) be ZCZ sequences with GO(Nn, Mn, Zn) based on complementary sequences [*] [*] X.M. Deng and P.Z. Fan, “Spreading Sequences sets with Zero Correlation Zone,”Electronics Letters,

Vol.36, No.11, 25th May 2000

Theorem 1 Any row of matrix F’(n) made from padding Nk Zs of Zeros with length L as below is ZCZ sequences with (Nn+2NkL, Mn, Zn+L)

1,1 1,2 1,2

2,1 2,2 2,2( )

,1 ,2 ,

k

k

n n n k

n n nN

n n nNn

n n nM M M N

F F F

F F FF

F F F

1,1 1,2 1,2

2,1 2,2 2,2( )

,1 ,2 ,

'

k

k

n n n k

n n nN

n n nNn

n n nM M M N

F Z F Z F Z

F Z F Z F ZF

F Z F Z F Z

Coding and Information Theory Lab 8/22

Proposed Ternary ZCZ Sequences

EX) Binary ZCZ sequences GO(8, 4, 1)

EX) Proposed Ternary ZCZ sequencesGO(16, 4, 3)

144

143

142

141

134

133

132

131

124

123

122

121

114

113

112

111

)1(

FFFFFFFFFFFFFFFF

F

00000000000000000000000000000000

144

143

142

141

134

133

132

131

124

123

122

121

114

113

112

111

)1(

ZFZFZFZFZFZFZFZFZFZFZFZFZFZFZFZF

F

Coding and Information Theory Lab 9/22

EX) Cha’s Ternary ZCZ GO(16, 6, 1)

Comparison of bounds to Cha’s ternary ZCZ- Cha’s ternary sequences bounds : MZ03N/4 - Proposed ternary sequences bounds : MZ0N, if L

Proposed Ternary ZCZ Sequences

1

2

3

4

5

6

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

MMM

FMMM

Coding and Information Theory Lab 10/22

MC-DS-CDMA

Various propagation environments - Diverse delay spread ( indoor, open rural, suburban, urban )

; distributed over [0.1us, 3us]

- More flexible for diverse channel More capacity and Better performance

MC-DS-CDMA Higher degree of freedom by using Time-Frequency spreading

More flexible for the diverse environments

Optimum diversity gain through path (time) diversity and frequency diversity

Total spreading factor : SFTotal = SFtime x SFfrequency

# of TD spreading sequences : Max. Mt user groups

# of FD spreading sequences : Max. Mf users per group Max. Mt x Mf users

Coding and Information Theory Lab 11/22

MC-DS-CDMA System Model

Transmitter Block

)(taq

)(taq

]1[rc )cos( 1,1 tw

)cos( ,1 twcN

]1[rc )cos( 1, twU

][ cr Nc

)cos( , twcNU][ cr Nc

)(tbk S/P

1,kb

Ukb ,

Copier1st

Copier Uth

Tb

bk1[i]

Tc

Ck[M]

ak[1]

Ck[j]

ak[N]

FDSpreading

TDSpreading

Coding and Information Theory Lab 12/22

MC-DS-CDMA System Model

Proposed Receiver Block based on Rake Receiver

)(tr

)cos( 1,1 tw)( 1,11tas 1,11

bT

dt0

bT

dt0

[1]tc

)cos( ,1 twcN [ ]t cc N

1,1sZ

cNsZ ,1

1,l̂b

)( ,11 Is ta I,11

)cos( 1, twU

)( 1,cUNs ta 1,cUN

bT

dt0

bT

dt0

[1]tc

)cos( , twcNU [ ]t cc N

cNsUZ ,

1,sUZ

Ulb ,ˆ

)( , IUNs cta IUN c ,

Comb.

Coding and Information Theory Lab 13/22

BER Analysis

BER of Receiver based on Rake receiver over multipath non-fading channel

BER of Receiver based on Rake receiver over multipath Rayleigh fading channel (MRC in time domain & EGC in frequecy domain)

2 2

2

[ ] 2 var[ ] 2

c be

o c b o

E z N EP Q Q Qz N N E N

2

0 00

1

2( | ) ( ) ( )2( 1)

where ( ) exp ( )( 1)!

c

c

be

f c b c

N L

N Lc

L EP P error f d Q f dM N E N N L

f UN L

Coding and Information Theory Lab 14/22

BER Analysis

BER of Receiver based on Rake receiver over multipath Rayleigh fading channel (MRC in time domain & ORC in frequecy domain)

1 2

1 2

1 2 1 20 0 0

1 2 1 20 0 00

fold1

11

( | ) ( ) ( ) ( )

2 ( ) ( ) ( )

1 where ( ) exp( 1)!

N c cc

N c cc c

c

i

e n N N

bN NN

N nnc

L

ii L

i

P P error f f f d d d

EQ f f f d d dNN

f UL

( ) for 1, 2,i ci N

Coding and Information Theory Lab 15/22

BER Analysis

BER of Receiver based on Rake receiver over multipath Nakagkmi-mfading channel (MRC in time domain & EGC in frequecy domain)

BER of Receiver based on Rake receiver over multipath Nakagkmi-mfading channel (MRC in time domain & ORC in frequecy domain)

2

0 00

1

2( | ) ( ) ( )2( 1)

where ( ) exp ( )( )

c

c

be

f c b c

N Lm

N Lmc

mL EP P error f d Q f dM mN E N N mL

f UmN Lm m

1 2

1 2

1 2 1 20 0 0

1 2 1 20 0 00

fold1

11 1

( | ) ( ) ( ) ( )

2 ( ) ( ) ( )

where ( ) exp( )

N c cc

N c cc c

c

n

e n N N

bN NN

N nnc

Lm

n nn Lm

P P error f f f d d d

EQ f f f d d dNN

fmLm m

( ) for 1, 2,n cU i N

Coding and Information Theory Lab 16/22

ZCZ sequences in Simulation

Binary ZCZ GO(32, 4, 4)

Ternary ZCZ GO(32, 4, 4)

F

00000000 0000000000000000 0000000000000000 0000000000000000 00000000

F

Coding and Information Theory Lab 17/22

Simulation Result I

MC-DS-CDMA using ZCZ sequence over AWGN and multipath non-fading

channel

0 2 4 6 8 10 12 14 16 18 2010

-5

10-4

10-3

10-2

10-1

100 MC-DS-CDMA QPSK BER Performance for 4(4x1) Users

Eb/Io

BE

R

Walsh over AWGNB ZCZ over AWGNT ZCZ over AWGNWalsh over 2-path CHB ZCZ over 2-path CHT ZCZ over 2-path CH

0 2 4 6 8 10 12 14 16 18 2010

-5

10-4

10-3

10-2

10-1

100

MC-DS-CDMA QPSK BER Performance for 32(4x8) Users

Eb/Io

BE

RWalshBinary ZCZTernary ZCZAWGN

Coding and Information Theory Lab 18/22

Simulation Result II

MC-DS-CDMA using ZCZ sequence for 32 users over multipath Rayleighfading channel

0 2 4 6 8 10 12 14 16 18 2010

-2

10-1

100

MRC & EGC MC-DS-CDMA QPSK BER Performance for 32 Users over 2-Path Rayleigh CH

Eb/Io

BE

R

WalshWalsh & PN IWalsh & PN IIBinary ZCZTernary ZCZ

0 2 4 6 8 10 12 14 16 18 2010

-5

10-4

10-3

10-2

10-1

100

MRC & ORC MC-DS-CDMA QPSK BER Performance for 32 Users over 2-Path Rayleigh CH

Eb/Io

BE

R

WalshWalsh & PN IWalsh & PN IIBinary ZCZTernary ZCZ

Coding and Information Theory Lab 19/22

Cyclic Extension of ZCZ sequences

Nonzero Partial Correlation in Decision value for nth subcarrier- Asynch., different value between consecutive symbols Aperiodic correlation- Causing some interferences called Partial Interferences in MC-DS-CDMA receiver

Nonzero Partial Correlation

,

,, , ,

, ,

( 1) (0) ( 1) (0), , , ,

, ,

(0) ( 1), ,

( ) ( ) ( )

( )

( )( 1) ( 1) ( ) ( ) ,

( )

( 1) ( 1)

b n j

n j

s q q s s q q s

s q q s

T

k n i q n i s n j

n i n j

b b ck A A k A A k A A k A A

c c

n i n j

bk A A k A A

c

b t a t a t dt

ifT T Tb R N m b R m b R N m b R mNT NTifT b R m b R N mNT

(0) ( 1), ,

, ,

( ) ( ) ( )

, , ( 1)

s q q s

b ck A A k A A

c

n i n jbc c

c c

T T b R m b R N mNT

TN m mT m TT T

Coding and Information Theory Lab 20/22

Cyclic Extension of ZCZ sequences

Cyclic Extension of ZCZ sequences- A=(a1a2… aN-1) be ZCZ sequence with length N, ZCZ=Z- is defined as follows if maximum delay among multipaths is d.

- The concatenation of cyclically repeated sequence with length +1 to the front and the end of original ZCZ sequence ( +1 Z )

- Practically concatenation of the cyclically repeated version with length +1 of original spreading symbol to the front and the end of the original spreading symbol similar to Cyclic Prefix in OFDM

- But may be some difficult in practical implementation.- Increased overhead : 0 2( +1), low spectral efficiency : 1/N 1/{N+2( +1)/Tc}

c

dT

0a 1Na 1Na 1Na 0a a

Coding and Information Theory Lab 21/22

Cyclic Extension of ZCZ sequences

Resulted Partial Correlation function after Cyclic Extension of ZCZ sequences

- Resulted BER after Cyclic Extension of ZCZ Sequences

,

,, , ,

(0) (0), , , ,

(0) (0), , , ,

,

( ) ( ) ( )

( )( 1) ( ) , if

( )( 1) ( ) , if

0

Since (

b n j

n j

s q s q

s q s q

q s

T

k n i q n i s n j

b b ck A A k A A n i n j

c c

b b ck A A k A A n i n j

c c

A A

b t a t a t dt

T T Tb m b mNT NT

T T Tb m b mNT NT

n

1

( ) ( )

0

,,

) 0, for 0 , 1

, , max( ), ( 1)

Nq s

k k nk

n dbn d c c

c c

a a n Z Z

TN m mT m TT T

0

2Q bl

EPeN

Coding and Information Theory Lab 22/22

Concluding Remarks

Proposed ternary ZCZ Sequences- Flexible ZCZ length- Not optimal but better bound than Cha’s ternary ZCZ sequences

MC-DS-CDMA using ZCZ sequences- Effective elimination of interferences inside ZCZ

Cyclic Extension of ZCZ sequences- Theoretically Perfect elimination of all interferences if every MPD is inside ZCZ- Same BER performance as AWGN channel even in multipath non-fading channel- Increased overhead, low spectral efficiency