ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012 condition in the blind multiuser detection literatures). Note AS3 is a reasonable assumption in practice considering the randomness of the multipath channels [6]. Without loss of generality, we assume that the dth symbol in s(n) is the desired transmitted symbol of the desired user and simply denote it by sd(n) (note the subscript d of sd(n) only represents its position in s(n)). Therefore, the MMSE detector weight vector is given by fmmse=R-1Hd [6] , where R

H H  H d [00100] , where 1 is in the dth position. Note the

first and last equalities are based on the assumption of full column rank of the channel matrix H. From (3), it is seen that the AME does not depend on the interfering signal amplitudes. Thus, it is equal to the near-far resistance  d [7]. It then follows that

is the autocorrelation matrix of the received signal χ M ( n) and Hd is the dth column in corresponding to the desired transmitted symbol sd(n). It is also well known that when noise approaches zero, the zero forcing (ZF) detector is proportional to the MMSE detector [6], fzf = αfmmse, where α is a constant. Therefore, both detectors share the same nearfar resistance. We have the following Proposition on near-far resistance of MC-DS-CDMA systems.

 )  , where C ( ) represents the column space. Denote C(H

1

H  and r  H  HH M H H H , R d H d  H d is a vector resulting from

1 ( H H H ) ( d ,d )

deleting the dth entry from the dth column of M. Note Rd is non-singular due to AS3. We have the following proposition.

, where the subscript (d, d) denotes choosing the element at the dth row and dth column.

B. Proposition 2 Proposition 2 The near-far resistance in Proposition 1 can be rewritten as

Proof: By applying the zero forcing detector to the received signal vector, the output contains only the useful signal and ambient Gaussian noise. The amplitude of the useful signal

 d

at the output is f H zf H d Ad s d ( n ) . Therefore, the energy of the useful

signal

at

the

output

* H 2 H H E s  E [f H zf H d A d s d ( n ) s d ( n ) Ad H d f zf ] Ad f zf H d H d f zf .

T he

Proof:  d

2  2 Es E  2 A2d f zfH Hd HdH f zf  2[Q1( Pd ( ))] n  lim  lim 2 2 2  0  0  0 Ad Ad  2f Hzf f zf A d

  H H H H f H H f H ( H 2 H  2I ) Hd Hd ( HA2 HH  2I ) Hd  lim mmseH d d mmse  lim d A H     0   0 f mmsef mmse 2 H 2 H 2 H 2 Hd ( H A H  I ) ( HA H  I ) Hd 

Ad4 1 2   2 H Hd HH A ( HHH) A H Hd

where P d ( ) Q (

(3)

(4)

Equation 1

1 ( H H H ) ( d ,d )

(3)

can

be

r ewritten

as

det( M )

 ( 1)

d d

det( R d ) , where det( ) represents the

exchange operations are executed, as a result we have det( M )det( M ) det( R d )r dH R dadj r d , where adj represents

1 1 ( HHH)( d ,d )

adjoint of a matrix and the second equality is resulted from a standard matrix equality [8] (pp. 50). Therefore, adj

Es ) , Q is the complementary Gaussian En

det( M ) det( R d )r dH R d r d  1r dH R d1r d . det( R d ) det( R d )

cumulative distribution function, “+” represents pseudoinverse. In (3), we have used the facts that  H 1 H ( H A 2 H H )  ( H  ) A 2 H  , ( H H H )  H  ( H  ) as well as

© 2012 ACEEE DOI: 01.IJCOM.3.1. 3

1r dH R d1r d

R d r d  M  H 1  . Since from M to M only even number of rd

H  A2 H H  A2 H H HH d H d H

1

determinant of a matrix. To compute det(M), let i = d and do the following row and column operations on M: 1) exchange the ith row column with the (i+1)th column; 2) exchange the ith row with the (i+1)th row and set i = i + 1. If i ‘“ col(H), where col(H) denotes the number of columns of H, go to step 1; else terminate the row and column operations. We finally obtain

 d  lim

H A2 H H HH H A2H H HH d d H d d H

1 ( H H H ) ( d ,d )

is

2 variance of the noise is E n  2f H zf f zf where  is the power spectral density of white Gaussian noise. Using the definition in [7], the asymptotic multiuser efficiency (AME) for the desired transmitted symbol is show below

1

column in the signature matrix H. H  is the matrix obtained by deleting the dth column Hd from H. It is easy to show that

Proposition 1 The near-far resistance of the MMSE  d

1 ( H H H )( d , d )

Proposition 1 can be carried one step further to reach an expression that facilitates comparison of near-far resistance between multicarrier and single carrier DS-CDMA systems. Before proceeding further, however, we need to define some useful matrices. Let I denote the subspace spanned by interference signature vectors Hi, i  d where Hi denotes the ith

A. Proposition 1

detector for the MC-DS-CDMA system (2) is

 d  d 

54

Near-Far Resistance of MC-DS-CDMACommunication Systems
Near-Far Resistance of MC-DS-CDMACommunication Systems

In this paper, the near-far resistance of the minimum mean square error (MMSE) detector is derived for the multicarrier direct sequence code...