Coding and Information Theory Lab YONSEI...
Transcript of Coding and Information Theory Lab YONSEI...
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