In this paper, the near-far resistance of the minimum mean square error (MMSE) detector is derived for the multicarrier direct sequence code division multiple access (MC-DS-CDMA) communication systems. It is shown that MC-DS-CDMA has better performance on near-far resistance than that of DS-CDMA.

1. ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
Near-Far Resistance of MC-DS-CDMA
Communication Systems
Xiaodong Yue1, Xuefu Zhou2, and Songlin Tian1
1
University of Central Missouri
Department of Mathematics and Computer Science
Warrensburg, MO USA
Email: {yue, tian}@ucmo.edu
2
University of Cincinnati
School of Electronics and Computing Systems
Cincinnati, OH USA
Email: {xuefu.zhou}@uc.edu
Abstract—In this paper, the near-far resistance of the minimum on near-far resistance or not. It is worth mentioning that near-
mean square error (MMSE) detector is derived for the far resistance also depends on the type of the receiver
multicarrier direct sequence code division multiple access detector. In this paper, the near-far resistance of the widely
(MC-DS-CDMA) communication systems. It is shown that
used MMSE detector is derived for the MC-DS-CDMA
MC-DS-CDMA has better performance on near-far resistance
than that of DS-CDMA.
systems. It is shown that MC-DS-CDMA has better
performance on near-far resistance than that of DS-CDMA
Index Terms—Code division multiaccess, Near-far resistance, systems.
wireless communication
II. SYSTEM MODEL
I. INTRODUCTION
Consider an asynchronous MC-DS-CDMA systems with
Multicarrier CDMA, which combines the Orthogonal Nc subcarriers and J active users in a multipath fading
Frequency Division Multiplexing (OFDM) based multicarrier channel. Spreading codes of length Lc are used to distinguish
transmissions and CDMA based multiuser access, is a different users.
promising technique for future 4G broadband multiuser It is shown in [4][5] that MC-DS-CDMA and DS-CDMA
communication systems. The application of OFDM greatly have the similar baseband MIMO system model. The total
resolves the difficulty raised by multipath fading that is received signal vector χ M ( n) observed in additive white
especially severe for broadband communication systems. On
Gaussian noise w(n) can be expressed as [4][5].
the other hand, the application of CDMA greatly simplifies
the multi-access and synchronization design. χ N ( n)  Ηs ( n )  w ( n ) (1)
There have been many different types of multicarrier Where N is the smoothing factor, H is the signature matrix
CDMA systems proposed [1][2]. One of them is MC-DS- and s(n) is the transmitted information symbol vector. A
CDMA [3], where each OFDM block (after IFFT and cyclic smoothing factor N is indispensable to capture a complete
prefix) is block-wise spreaded, i.e., the OFDM block is desired user symbol since the detector does not know the
spreaded into multiple OFDM blocks, each multiplied with staring time of each desired user symbol.
different chip of the spreading code. A major feature of MC- Since the channel effect on received energy can always
DS-CDMA communication system is that each OFDM be incorporated into a diagonal amplitude matrix A, without
subcarrier works like DS-CDMA. Specifically, if there is only loss of generality, we assume that the columns of the channel
one subcarrier, then the MC-DS-CDMA reduces to a matrix H are all normalized in the following sections. Then (1)
conventional DS-CDMA. One of the major advantages of can be rewritten as
MC-DS-CDMA is that each DS-CDMA signal (in each
subcarrier) of a user can be maintained orthogonal to that of χ N ( n)  ΗAs ( n)  w ( n) (2)
all the other users, when orthogonal spreading codes are
used. As a result, multi-access interference (MAI) is mostly III. NEAR-FAR RESISTANCE OF MMSE RECEIVER FOR MC-DS-CDMA
avoided, which may greatly enhance the performance over SYSTEMS
conventional DS-CDMA.
The following assumptions will be made throughout this
However, the well-known near-far problem in a multiuser
paper. AS1: The symbols in s(n) are independently and
setting still places fundamental limitations on the performance
identically distributed (i.i.d), with variance 1 (since symbol
of MC-DS-CDMA communication systems. Therefore, near-
energy can also be absorbed into the diagonal matrix A). AS2:
far resistance remains one of the most important performance
The noise is zero mean white Gaussian. AS3: The signature
measures for MC-DS-CDMA systems. Furthermore, it would
matrix H is of full column rank (known as the identifiability
be interesting to explore whether the multicarrier has benefits
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2. ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
condition in the blind multiuser detection literatures). Note H
H  H d [00100] , where 1 is in the dth position. Note the
AS3 is a reasonable assumption in practice considering the
first and last equalities are based on the assumption of full
randomness of the multipath channels [6].
column rank of the channel matrix H. From (3), it is seen that
Without loss of generality, we assume that the dth symbol
the AME does not depend on the interfering signal ampli-
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) tudes. Thus, it is equal to the near-far resistance  d [7]. It
only represents its position in s(n)). Therefore, the MMSE 1
detector weight vector is given by fmmse=R-1Hd [6] , where R  d  d  1
then follows that ( H H H )( d , d )
is the autocorrelation matrix of the received signal χ M ( n)
and Hd is the dth column in corresponding to the desired Proposition 1 can be carried one step further to reach an
transmitted symbol sd(n). It is also well known that when expression that facilitates comparison of near-far resistance
noise approaches zero, the zero forcing (ZF) detector is between multicarrier and single carrier DS-CDMA systems.
proportional to the MMSE detector [6], fzf = αfmmse, where α is Before proceeding further, however, we need to define some
a constant. Therefore, both detectors share the same near- useful matrices. Let I denote the subspace spanned by inter-
far resistance. We have the following Proposition on near-far ference signature vectors Hi, i  d where Hi denotes the ith
resistance of MC-DS-CDMA systems. 
column in the signature matrix H. H is the matrix obtained by
A. Proposition 1 deleting the dth column Hd from H. It is easy to show that

C ( H )  , where C ( ) represents the column space. Denote
Proposition 1 The near-far resistance of the MMSE
1  
M H H H , R d H H H and r d  H H H d is a vector resulting from

 d 1
detector for the MC-DS-CDMA system (2) is ( 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
Proof: By applying the zero forcing detector to the received can be rewritten as
signal vector, the output contains only the useful signal and
ambient Gaussian noise. The amplitude of the useful signal
1 H 
 d 1r d R d 1r d
at the output is f H H d Ad s d ( n ) . Therefore, the energy of the
zf 1
( H H H ) ( d ,d ) (4)
useful signal at the output is
E s  E [f H H d A d s d ( n ) s* ( n ) Ad H d f zf ] Ad f H H d H H f zf .
zf d
H 2
zf d T he Proof: Equation (3) can be r ewritten as
variance of the noise is E n  2f H f zf where  2 is the power 1 det( M )
zf  d 1
 d d
( H H H ) ( d ,d ) ( 1) det( R d ) , where det( ) represents the
spectral density of white Gaussian noise. Using the definition
in [7], the asymptotic multiuser efficiency (AME) for the determinant of a matrix. To compute det(M), let i = d and do
desired transmitted symbol is show below the following row and column operations on M: 1) exchange
the ith row column with the (i+1)th column; 2) exchange the
 2[Q1( Pd ( ))]
2  2 Es E  2 A2 f zf Hd Hd f zf
d
H H ith row with the (i+1)th row and set i = i + 1. If i ‘“ col(H), where
n
 d  lim 2  lim 2  lim 2 col(H) denotes the number of columns of H, go to step 1; else
 0 Ad  0 Ad  0  2f H f zf A
zf d
terminate the row and column operations. We finally obtain
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 f mmsef mmse  0 H 2 H 2  2 H 2
 R d r d 
Hd ( H A H  I ) ( HA H  I ) Hd M  H
rd 1  . Since from M to M only even number of

 
HH HH A2 H Hd Hd HH A2H Hd
H
 d
 
(3) exchange operations are executed, as a result we have
HH HH A2 H HH A2 H Hd
d
A4 1 H adj
det( M )det( M ) det( R d )r d R d r d , where adj represents
d
  2 1 2 
 1
H
Hd HH A ( HHH) A H Hd ( HHH)( d ,d )
adjoint of a matrix and the second equality is resulted from a
standard matrix equality [8] (pp. 50). Therefore,
Es ) H adj
where P d ( ) Q ( , Q is the complementary Gaussian det( M ) det( R d )r d R d r d H 
En  1r d 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
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3. ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
IV. COMPARISON OF NEAR-FAR RESISTANCE OF MMSE DETECTOR A. Proposition 3
BETWEEN MULTICARRIER AND SINGLE CARRIER DS-CDMA
COMMUNICATION SYSTEMS Proposition 3 Denote  1 as the expectation of the near-
d
It is interesting to investigate how does the multicarrier far resistance of the MMSE detector in DS-CDMA and  2 as
d
scheme affect the near-far resistance in DS-CDMA systems.
the expectation of the near-far resistance of the MMSE
To this end, we analyze the near-far resistance of the MMSE
detector in MC-DS-CDMA. Then under AS3, AS4, and AS5,
detector for 1). DS-CDMA and 2). MC-DS-CDMA.
In order to facilitate fair comparison between different
1
 d  d
2
. Proof: under AS3, starting from (4), the conditional
scenarios, we make the following assumption. AS4: the expectation of  id ,conditioning on the interference subspace
processing gain Lc, the system load J, the smoothing factor
I, is given by
N, the distributions of multipath delay spread and
asynchronous user delay are the same under different
1 1
scenarios. Since the dimension of the signature matrix will E [ id |I ]1 E[r iH ( R i ) r id | I ]1 E[tr{r id r iH ( Rid ) }|I ]
d d d
prove to be useful for our following derivations, we now 1
1 E[tr{HiH Hid H iH Hi ( R id ) }| I ]
 d

specify those parameters. Let H1 denote the signature matrix 1
1 E[tr{Hid HiH Hi ( R id ) HiH }| I ]
 
d
of the DS-CDMA system and H2 denote the signature matrix i iH | I ]  ( i ) 1  H }
1tr{E[ H d H d Hi R d Hi
of MC-DS-CDMA systems. Under AS4, the dimensions of
1 1
1  iH Hi ( R id ) }
tr{H 
the signature matrix H1 and H2 are N L c J ( L h CDMA  N 1) and
DS
row( H i )
1 1 (7)
N L c N c J N c ( L Mc  DS  CDMA N 1)
h respectively [5][9], where Lh 1 tr{R id ( R i ) }
d
row( H i )
(non-negative integer) is related to the maximum multipath col ( Hi )1
1
delay spread and the maximum asynchronous user delay, and row( H i )
is defined in [5][9] as follows.
 L c  L g 1 d j  
where tr ( ) represents the trace of a matrix, Hi , r id and R id are
L DS CDMA  max 
h  (5)
j  Lc  defined similar as in section 3 for the ith scenario. The sixth
equality is based on the property of conditional expectation
 N cL  L g 1 d j 
L MC  DS CDMA  max 
h
c  (6) and seventh equality is based on the fact that
j  N cL c 
 1
E[ H id H iH |I ] E[ Hid HiH ] I
where Lg denotes the maximum multipath delay spread and dj d d
row( H i ) due to AS5. Based on (5)
denotes the jth user’s asynchronous user delay.
(6) and AS4, it is straightforward to show that
Next we will compare the near-far resistance of MMSE
detector under different scenarios. The value of the near-far J ( L h CDMA N 1) 1
DS
E[ 1 |I ]1
d (8)
resistance derived in Proposition 1 clearly depends on the N Lc
multipath channels and asynchronous transmission delays.
Since these parameters are random in nature, it is more 2 J N c ( L h  DS CDMA N 1) 1
MC
E[ d | I ]  1  (9)
meaningful to compare the statistical average of the near-far N Lc N c
resistance rather than a particular random realization. To this
end, we need an additional assumption. AS5: Under the ith Subtracting (8) from (9), we have
scenario, assume H id (the vector in Hi which is corresponding
to the desired transmitted symbol of the desired user) is a
2 1
random vector with a probability density function E[ d | I ]  E[ d | I ]
1 J ( LDS CDMA N 1)1 J N c( Lh DS CDMA N 1)1
h
MC
 c (0 row ( H i ), I row( H i ) ) and is statistically independent  
row ( Hi ) N Lc N Lc N c
(10)
of the interference subspace I (since it won’t affect the J N c( Lh CDMALh DS CDMA)1 N c
DS MC
derivation, here I is a general expression which includes all 
N Lc N c
scenarios), where  c represents the complex normal
Under AS4, J, N and Lc are the same under different scenarios
distribution, row( H i ) denotes the number of rows in Hi,
x  x 
0 row ( H i ) represents the row ( H i )1 zero vector, and for each realization of I. It is also easy to show that     
 a   ab 
I row( H i ) represents the row ( H i )row( Hi ) identity matrix. The
when x > a and b > 1. In other word, L DS CDMA  L h  DS CDMA .
h
MC
fact that the variance is 1 row( Hi ) is because H id is normalized.
We then have the following proposition. Based on (10), we have E[ 1 |I ] E[ 2 |I ] for each realization
d d
of I.
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4. ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012
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