Minimum total-squared-correlation design of DS ... - Semantic Scholar

Minimum Total-Squared-Correlation Design of DS-CDMA Binary Signature Sets George N. Karystinos and Dimitris A. Pados Department of Electrical Engineering State University of New York at Buffalo Buffalo, NY 14260 USA

Abstract— The Welch lower bound on the total-squaredcorrelation (TSC) of signature sets is known to be tight for realvalued signatures and loose for binary signatures whose number is not a multiple of 4. In this paper, we derive new bounds
on the TSC of binary signature sets for any number
  inof signatures        ,and any signature length  For almost all we develop simple algorithms for the design of optimum binary signature sets that achieve the new bound.

I. I NTRODUCTION In direct-sequence code-division-multiple-access (DSCDMA) systems, multiple users are assigned individual binary antipodal signatures (spreading codes) to access a common, in time and frequency, communication channel. In conjunction with channel and receiver design specifics, the overall system performance is determined by the selection of the user signature set. Since each user signal acts as interference for the signals of the other users, an appropriately selected/designed user signature set contains signatures with low pairwise cross-correlation. A fundamental measure of the cross-correlation properties 1 of a signature is " the       set  !#  $&total-squared-correlation %('*)+-, $,.0/12&0/34(TSC).   56 If , is 5 a set of normalized (complex-valued in general) user signatures of length (processing gain) 7 , then the TSC of set is the sum of the squared magnitudes of all inner products between signatures [2]: TSC 8

:9+;
=? @ =?&AA AA
       ?   we      5 consists of nor- of %ž WA2…Yc  ¡   ,   ¤!
(

K  ¡   M ¥.  Recall and ¢ define ¦
3theP for3any Then, the signature set 1(
2
matrix¡" that ? ,§
( =?  and ¤C
$ §) malized signatures and after straightforward calculations we ¤ !
3  obtain 1)) ”
 : Remove one arbitrary from to 7F G H,@       4   row

       . Then, (24) TSC 7B1C@D
 E M obtain the ¨ ` matrix
  ¢  ¦  §) We design the ` ª signature Hence, TSC 7B1C@ is equal to the bound in (10) for 
(E_/J
G  for any ¤©
$    (mod 2) and the optimality of the binary signature set design matrix as follows:* + + + in (23) is established. < 5 O d e 4a) 
`a  E and bJ`P (mod 2): Let V`W`2Yc be , 4- /.
0    (29) + + + an arbitrary  % binary vector. We design the _ L signa+ ture matrix as follows: *+ + + 1U
“¡ 2 «<      ?   -  5 has normalized signaThe signature set  
 # 3
 ) and , 4- /.
0  V  V   (25) tures ( 8 K8 • gfh V  :V  ij E ¬  TSC 7R1C@\
(30) k This case includes the underloaded Gold signature sets with nl )m 9o rp tq s as well as the signature sets obtained by u cyclic shifts of an v -sequence of TSC 7R1C@ is equal to the bound in (18) for œ
`•tJ length p xq ?s y the familiar Rademacher-Walsh orthogonal codes [6] for Hence, z ThisLl casewm o includes  (mod 2). We conclude that the set that we designed in (29) is {l wm ?o /| 9} U v ~m ,o €} , } ‚y ‚y ‚y } and  u ~ƒ „l } used in current CDMA technology. TSC-optimum. is equal to the bound in (8) for Hence, TSC (mod 4) and (mod 2). The set design in (25) is TSCoptimum. 4b) and (mod 2): Let be an arbitrary binary vector. We design the signature matrix as follows:

6

804

C IV. C  signature matrix as folIn Section II we derived new bounds on the TSC of binary      
    antipodal signature sets for both underloaded and overloaded (31)  CDMA systems (given by (8), (10), (16), and (18)). In Sec     ! #" has normalized signa- tion III we identified sufficient conditions on the values of The signature set 
 (number of users) and  (processing gain) which guarantee that tures. We calculate  the corresponding new bounds on the TSC are tight. In addition, we were able to design optimum (minimum-TSC) (32) TSC $%'&(


 for whichbinary  the antipodal signature sets for all values of which is equal to the bound in (16) for )
+*-, (mod 4). sufficient conditions hold true. design of the optimum signature sets (and the tightness The set-design in (31) is TSC-optimum.  "4 be an arbitrary 5  of The the TSC bounds as it is described in Proposition 3)) 
/. : Let 021/3  1) depends binary vector. We design the 6( signature matrix as follows: on the existence of a Hadamard matrix of size  A necessary condition for a Hadamard matrix to exist is that its size is a  !   !  
  7   098  (33) multiple of 4 (except for the trivial cases of size 1 or 2). In deed,  is a multiple of 4 by definition (see Proposition 1). Therefore, necessary condition for the design algorithm to

 ! ;  !  " consists of nor- work is thetheexistence Then, the signature set -
: of a Hadamard matrix for the specific  malized signatures and we can calculate  Many Hadamard matrices are known for specific value of   multiples of 4. Assuming that in CDMA applications values >=  . TSC $. 

 



 

     



  " :.
  o * p A 4), and (vi) (mod 4). Then, the signature set 
O has normalized It is interesting to note that these (i)-(vi) combinations consignatures and it can be shown that  stitute a small percentage (q A  q  %) all possible com  A among "  Therefore,   R  > = A binations of and  in  A r , , the new TSC $%P&(
(36) bounds together with the sufficient conditions and the design  .QA  We conclude that TSC $