Codes for Optical CDMA Report with matlab code for simulation

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c c c .c  c c  -  c c     c  c c c ,c   c  c c c     c -c   c c c   c c   c ,c c -c - c c ,c c 
hc # c c  ,c c  c c  c    c 

 c c c  cc     c ,c c ,c c c !c c c c c

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c c c 
c c  

c cc
c c c c
c
ccc
cc c
c 
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c 
 ccc
c c 
 c c !"#c
cc cc$
 cc$c

c
c
c %
&

cc


c c c   c c

'c 
 cc cc
c
cc
c
 c c
c
cc

 cccccc
cc&cc c

 
c
cc  ccc c cc 
c
c c &c c  c c c  c c   c  c 'c $
 c c $c $($

c )
 c $$)c   
c c
c c 
c
c 

c  c &
c 
 c  
 c *+c ",c -c 
 c  c c c $
 c $  c  c $$cc
cc c
 c 

cc
 cc&c
c
c
c 
c  ccc cc$(

 c$$c"(c$$c c

"(c $$c c c &c c
c  
c  c 
c c . c % c  
c c

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c0cc
ccc
ccc
c

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2c
cc
cccc ccc
c 
c
c"(c$$cc
ccc 
c c 
(c c c c c "341

2c "34c 1c 5c 
 c c 4(c $$c &c 
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c & c 
c c   c & c c c &c 
c 
c c c c 6c 31

2c3c1c c

'cc%c
cc
c
c& cc& c cc
7c c"(c ccc&
c

& c c 
c
c c c& c8cc 
c
 c4(c$$cc c& c cc
7c c4(c c  cc
c
 cc c 
c
c &c cc c c
 c 
c8c c  cc  &c 
 
c c c (   
cc (  
c c4(c$$c c c c

c

c
c c

· c     
 c 
c c ሼͲǡͳሽȂc  c c  c ݊ c 9
 c 
c ‫ ݓ‬c c &c 
c c c

ሺ݊ǡ ‫ݓ‬ǡ ݇ሻc$
 c$  c c$$‫ ܥ‬ccͳ ൑ ݇ ൑ ‫ ݓ‬൑ ݊cc ࢔ି૚

෍ ࢞ሺ࢏ሻ࢟ሺ࢏ ْ ࢔ ࣎ሻ ൑ ࢑  ࢏ୀ૙

. cc
c c cሼ‫ݔ‬ǡ ‫ݕ‬ሽc
c‫ ܥ‬cc ‫ ݕ ് ݔ‬c cc ߬ ് Ͳc

c c c ْ࢔  c  c 

c   ݊ c  ݇ c c c %
c 

c

c:c c

8Ȱሺ݊ǡ ‫ݓ‬ǡ ݇ሻc  c c  c  
& c 

c c c ሺ݊ǡ ‫ݓ‬ǡ ݇ሻc $
 c

$  c c$$c c ·       c!4c;cc! 3c Arrays w
 À      
 each row of every ሺܹ array
n

ሻ code


s requ
red o have Hamm
ng we
gh = 1.

3c Arrays w
h À À     
  here each row of any ሺܹ

ሻ code
n


s requ
red o have Hamm
ng we
gh < 1.

3c Arrays w
h À       À 
! here each co umn of every ሺܹ code array
n

ሻ


s requ
red o have Hamm
ng we
gh = 1.

3c Arrays w
h À À     À 
! here each co umn of any ሺܹ

c

ሻ array
n


s requ
red o have Hamm
ng we
gh < 1.

c cc

O ÀÀÀ  The op
ca corre aor can be
mp emened as shown
n h
s f
gure.

¬c

   
  

 ^ ^ 

  3c ef 4 x 4 mar
x represens a 2-D OO. 3c 4 recangu ar b ocks represens op
ca corre aor. 3c 9
gh s
de
s rece
ved oupu 
me a
gned Each recangu ar box represens an op
ca f
er
mp emened us
ng a f
ber-5ragg gra
ng or an op
ca m
cro-resonaor. Th
s f
er ref ecs
gh of he wave engh shown a ongs
de he box and a ows
gh of a oher wave enghs o pass hrough. F
ers p aced furher a ong he ref ec
on pah w
suffer an
ncreased de ay and
n h
s manner, he p acemen of he f
ers can be ad6used o br
ng he pu ses of a he d
fferen wave enghs
n he des
red code mar
x
no 
me a
gnmen a he oupu of he corre aor. 

c

c
c c

O 8   · "# $8O
 . c c
c c c  c c ܹǡ ܶǡ ‫ݓ‬ǡ ݇ǡ c c ߔ ሺܹ‫ܶݔ‬ǡ ‫ݓ‬ǡ ݇ሻ c  c c  c  
& c



c ccሺܹǡ ܶǡ ‫ݓ‬ǡ ݇ሻ cc4(c$$c c c
c 
 cc 
 c 
c

4(c$$ccc c
ccc c"(c$$c
c
c4"c c
c c c O 8     ' c ‫ ܥ‬c
c c ሺܹ‫ܶݔ‬ǡ ‫ݓ‬ǡ ݇ሻc 4(c $$c c &c
 
 c c  (
c 
c c c

  c
c ‫ ܥ‬c cc cc c
c c
 cc
 cc cc

 c cc ܹ‫ ܶݔ‬cc c cc "(c
 c c  c ܹܶ c-c c  c 
c  c c c ܹܶǡ ‫ݓ‬ǡ ݇ሻc c 
7c ?ܶȁ‫ܥ‬ȁc -
c &
c  c c  c  c & c c c
c c c& c c4(c$$cc
c
c c
c
c4"c c

-c cc=  c5 c c
c4(c$$c

c!;!0# c ࢃࢀିሺ࢒ି૚ሻ ࢃ ࢃࢀି૚ ǥ ǥ ǥ ቔ ࢎቕ ࢝ ࢝ି૚ ࢝ିሺ࢒ି૚ሻ

5 ሺࢃ࢞ࢀǡ ࢝ǡ ࢑ሻ ൑ ඌ ቔ

ሺௐ்ି௟ሻሺ௪ି௞ሻ

c F ൌ ‹ ቀܹܶ െ ݈ǡ ቔሺ௪ି௟ሻ మ

ିሺௐ்ି௟ሻሺ௞ି௟ሻ

ǥ ቕඐ ǣ ൌ ࡶ࡯ ሺࢃ࢞ࢀǡ ࢝ǡ ࢑ሻǤ

ቕቁǤ݈ c
c
 cͳ ൑ ݈ ൑ ݇ െ ͳǡc

ccሺ‫ ݓ‬െ ݈ ሻଶ ൐ ሺܹܶ െ ݈ ሻሺ݇ െ ݈ ሻǤc c

c

cc c

"

 ñ
 ݊ ൌ ܹܶ 

 ‫݅ܬ‬ǡ ݅ ‫  א‬ሼ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ሽ

 
 ሺܹܶǡ ‫ݓ‬ǡ ݇ሻPP  
 





 





 
 ሺܹܶǡ ‫ݓ‬ǡ ݇ሻPP  ‫ ࢏ࡶ܅‬ሺ‫܂܅‬ǡ ‫ܟ‬ǡ ‫ܓ‬ሻ ൑ ࡶ࢏ ሺ‫܂ܠ܅‬ǡ ‫ܟ‬ǡ ‫ܓ‬ሻ ൑ ‫ ࢏ࡶ܅‬ሺ‫܂܅‬ǡ ‫ܟ‬ǡ ‫ܓ‬ሻ ൅ ሺ‫ ܅‬െ ૚ሻ



 ‫݅ܬ‬ሺܹܶǡ ‫ݓ‬ǡ ݇ሻ 




 
    PP 


 

       ñ 







 PP 




     
   

   

 


 ࢫ   ·       c 4(c $$c c &c  c c c c c c 
c cܹ ൌ ݂ ሺ‫ݐ‬ሻǡ Ͳ ൏ ‫ ݐ‬൏ ܶ െ ͳǡ Ͳ ൏

ܹ ൏ ܹ ൌ ͳǡc 
 c
c
 c   c c 
(‫ ݐ‬ൌ ݂ሺߣሻc

c 
c c  c

ܵఛ c c ܵఛ ሺȈǡ ݂ሺȈሻሻ c
c c ߬
c 
c 
c 
c c c ൫Ȉǡ ݂ ሺȈሻ൯c   c c


c%
ccc -
c


c ccc 
c
cc
c$$c
c:c  c c ݇ c

c

' c c  
ܵఛ ൫‫ݔ‬ǡ ݂ ሺ‫ ݔ‬ሻ൯ ൌ ൫‫ݔ‬ǡ ݂ ሺ‫ ݔ‬ሻ൯ c c %
c ݇ c  
c c c ݂ c c c ݃ c c


c݂ ് ݃ c ߬ ് Ͳc c

    88
 9c c c 
 c 
c c ò ሺ‫ ݍ‬ሻ c  c 4(c $$-c  c c 
 c


c c ò ሺ‫ݍ‬ሻ
c
c&c ܿሺ‫ݐ‬ሻc
c

௙ሺ௫ሻ

௚ሺ௫ሻ


c
 
 c 
 c 

>c

3c ݂ ݃ 





൑ ‫ ݐ‬ 3c ݂݃ 




 
 3c

௙ሺ௫ሻ

௚ሺ௫ሻ

് ܽǡ
 ܽ 


 

3c ݂  

c

cc c

Jܿሺ ሻ
s g
ven as ࢉሺ ሻ

ࢗ૛ ି૚ ࢗ ቊ ൒ ࢗ૛ ା૚

૚ ૛ ૜ ૝ ૞ ૟
 ൒ૠ

૛ ି૟ Τ

ࢗ

ૠ

The above cond

ons guaraneed ha each func
on
s couned on y once. Nሺ௫ሻ


  
         
  

 
 


௚ሺ௫ሻ

ܿ‫ݏ ݋‬

 Two

ܿ ܿ e emens Õ   Õ  are equa prov
ded here ex
s an e emen  such haÕ    Õ .
݀
݀

"

    #ሺ ሻ   

   ሺ ሻ ߁  

 








 



ሺ ሻ    
 

!

   !
 
  



  !
   
 

 ߁ ௜ ሾ 
  ሾ 
   !   ሺ ሻ   ሺ ሻ  I fo ows ha he e emens of

ሺ“ሻ

can be arranged so as o form an orb
 of

s
e (q + >   ‫ ۾‬૚ ሺ‫ܙ‬ሻ  ܑ Õ  ܑ  ૙ ૚ !  ‫ܙ‬Ǣ Õ  ‫ ۾‬૚ ሺ‫ܙ‬ሻ ቅ  

 

À  À



 



À
Þ  ࢗ ૚ ࢗÀ À e " # " $, and #% func
ons

Nሺఒሻ

௚ሺఒሻ

some subse of ሺ$ሻ of s
e#. Here we cons
der ra
ona

mapp
ng wave engh
no 
me. Assoc
ae o each 
me s o , he h e emen of

a cyc
c represena
on of



ሺ ሺ$ሻሻ. e us def
ne>

Where, g
ven a ra
ona func
on, he operaor ܰ d
v
des ou he common facors beween numeraor and denom
naor and
n add

on, sca es he wo so as o make he numeraor mon
c [9].

c

cc c

ons
der
ng # and
are re a
ve y pr
me, and



& ,

s obv
ous ha he operaor ܰresu s


n some ra
ona func
on wh
ch sa
sf
es he cond

ons of sec
on 5. We need o d
scard a ra
ona func
ons wh
ch are of he form>

We noe ha such a ra
ona func
on doesn' ex
s s
nce
 means ha for cera
n for some  " ݇ " $ wh
ch
s
mposs
b e by Theorem 2. Amongs he func
ons sa
sfy
ng he cond

ons of sec
on 5, we have d
scarded a consan func
ons, bu as h
s sep
s unnecessary here, we can add hem back. For any " ݇ " $, we dec are wo ra
ona func
ons

o be

equ
va en. Then he d
fferen code mar
ces correspond o choos
ng prec
se y one po ynom
a

from each equ
va ence c ass. For each po ynom
a #ሺ ሻ he ሺܹ'ሻ code array

w
h s
e,

௖ሺ௞ሻᇱ ௤ା

and݇  (݇ᇱ


s g
ven by

. Th
s resu s
n aሺܹ ሺ$ ሻ  ܹ (݇ᇱ ሻ 2-D OO ܹ

$)



À
?  ࢗ ૚ ࢗÀ À e 'ȁሺ$  ሻ and *% ሺ$ሻhave mu 
p
ca
ve order'. Take he wave enghs as

ሺ ሺ$ሻሻ(he order of e emens doesn' maer). Here we cons
der ra
ona func
ons

mapp
ng 
me
no wave engh. e us assoc
ae o 
me s o+, he e emen*  . We def
ne wo ra
ona

func
ons


n for some

wh
ch ra
ona func
ons
s compued
n [9] as>

c

௞

o

be

equ
va en

. F
rs d
scard a ra
ona func
ons


f

Nሺ௫ሻ

௚ሺ௫ሻ

sa
sfy= for & . The number of rema
n
ng .

c
c c

hoos
ng one func
on from each of he rema
n
ng equ
va ence c asses and assoc
a
ng

o


,

he

ሺܹ'ሻ

code

by

array

e
ng

where

and

resu s
n a

2-D OO of s
e

… ! À  We ver
fy he auo-corre a
on J cross-corre a
on proper
es of 2-D OO codes by us
ng ATA5 .The wo

codes are chosen form he 8 codes shown ear
er
n h
s repor.

S
mu a
on
s done wo pars 1.corconv.m o f
nd convo u
on. J 2.ocdma.m o generae code, corre ae J p o hem. ¬c

Th
s
s represened as mar
x as 
  code1=   „

     

     

     

     

 . -,


  code5=   „

     

     

     

     

 . -,

"
 !    
 
!  
 #   
     
    
  
 
  

 

  ! !  

c

c

c c

! À  

c





 ÀÀ ÀÀ
 À
ÀÀ   c
c c

  "   


!#c?c
 5Cc

D?;c  E'9$CFcc% c;c
c EG c c
cCH?+c  ECcc&c cc7 EGc c
c
74I"c cc?EGc c
c
754I"c cc5?5EGc c
cCI  c cc?!cGc7 C(  "#Gc c
cCI  5c cc5?!5cGc7 C(  5"#Gc c c?c">CGc 5c?c5">CGc . -& c?cJ5EGc . -& c?c
 . -& Gc ?7 "CGc Cc?cc
 . -& +cGc

c%?">C("c cc%?cc
 . -& cC(%ccKcc
 . -& c(%cGc c c

    c c  c 
7?"0Gc LLc "c  ""?7 "3G """"?"Gc  "4?7 "3Gc  ";?7 "3G ";""?"Gc  "