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One-shot Linear Decorrelating Detector for Asynchronous
CDMA
M.
Peng,
Y. J.
Guo and
S.
K. Bar ton
Telecommunication Research Group
Depar tment
of
Electronic and Electrical Engineering
Bradford University, UK
Abstract
A one-shot linear decorrelating detector(LDD) for the detec-
tion of asynchronous CDMA is presented. By treating every
user as two independent users and using maximal ra tio com-
bining, a bit-by-bit detection is obtained. By applying a
phase estimation algorithm on the output of the one-shot
LDD, it is shown that the proposed detector can be used for
the detection of asynchronous CDMA without the knowledge
of carrier phase of any user. Simulation results show th at the
performance of the proposed detector approaches that of the
optimum detector for single user transmission.
1 Introduction
The fundamental limitation of the DS/CDMA system is
the so-called near-far problem, which leads to severe per-
formance degradation for the conventional receiver. In 1986,
Verdu investigated the optimuni maximum-likelihood mul-
tiuser receiver for multiuser communication
[l].
Verdus re-
ceiver is near-far resistant, but its complexity is exponen-
tial in the number
of
active users. Recently much attention
has been given to suboptimal architectures with less com-
plexity. Among many suboptimal detectors proposed, Lupas
and Verdus linear decorrelating detector (LDD)
[2]
[3] has
attracted wide attention. The LDD achieves the same near-
far resistance as the optimum detector while its complexity
is linear in the number of users.
Although the LDD is much simpler than Verdus optimum
detector, it is still too complicated and will lead to unac-
ceptable detection delay while dealing with asynchronous
CDMA. With sacrifice of data transmission rate, Zheng and
Barton proposed an isolation bit insertion (IBI) LDD
[4]
o
simplify the implementation and reduce detection delay. An-
other scheme, also proposed by Verdu and Lupas,
is
called
one-shot LDD [ 5 ] [ 6 ] ,where the detection of data is based
on the signal observed in one bit length period, called one-
shot window, so the detection of an asynchronous CDMA
is transformed into that of synchronous one. According to
Verdu and Lupass scheme, the one-shot window is syn-
chronised with one of the users,
so
for
a
K-user asynchronous
CDMA, a (2K-l)-user synchronous CDMA problem must be
solved for the detection of one user, and a total of K different
0-7803-3336-5196
5.00
0 1996 IEEE
(2K-l)-user synchronous CDMA problems must be solved for
the detection of the total
K
users.
In this paper, an improved one-shot LDD scheme is pre-
sented, in which the one-shot window is not synchronised
with any user. By treating every user as two independent
users and using maximal ratio combining, a bit-by-bit de-
tection is obtained. With this approach, the multiuser de-
tection of a K-user asynchronous CDMA is converted into
that of
a
2K-user synchronous CDMA.
For
the LDD and
many other detection schemes, a knowledge of the time de-
lay, the carriers frequency and phase, is normally assumed.
Noncoherent detection has been reported
[7]
for unknown
carrier phase. In-this paper, by applying a single-user phase
estimation algorithm at each output of the one-shot LDD,
it is shown that the proposed detector can be used
for
the
coherent detection of asynchronous CDMA without the pre-
knowledge of user phase. Simulation results show that the
performance
of
the proposed detector approaches that of the
optimum detector for single user transmission.
2
One-shot
LDD
Consider a BPSK CDMA system. The received signal can
be expressed as:
N K
r t )= b k i ) ~ S k t - - i T b - - 7 ; E ) c o s 2 . r r f , t + 8 k ) + n t )
i = O k = l
(1)
where K is the number of total users, N the number of trans-
mitted information bits, Tb the duration of each information
bit,
fc
the carrier frequency,
Pk
and 8 k are power and initial
phase of user k , respectively, b k i ) is the ith bit of user I C
n t )
the AWGN with variance
a2
nd
s k t )
user
ks
signa-
ture waveform which satisfies:
and
S i t ) d t
= 1 3)
The idea of the one-shot LDD is to transform the detection
of asynchronous CDMA into the detection of synchronous
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CDMA. Consider the one-shot window as shown in Fig-
ure
1.
Each user can be treated a s the superposition of two
independent users. For example, user k can be treated as the
superposition of two new users k and k ?whose signatures
skf ( t ) and sk( t )are defined as:
The fact that Sk(t) has unit energy gives:
pk pk
=
1
All new signatures have unit energy. In this paper, it is
also assumed that all new signatures, sk ( t ) and sk(t) (k =
1 , 2 ,
..,
K), are linearly independent. Figure 2 shows an ex-
ample of the decomposition of signature waveforms.
(8)
- - - -
One shot
Window
- - - e- - -
Ooc-shot Window
- - - -
T h
...
Figure 1: One-shot window for
a
K-user asynchronous
CDMA.
Defining
b k - 1 ) =
0
(k = 1 , 2 ,
...,
K ) ,
equation
(1)
can be
rewritten as:
T t )
=
xzo
xf= i[bk( i ) d m s k f ( t
T b )
f
b k ( i ) d m s k t Tb)] o s ( 2 ~ f ~ t6,) + n ( t )
9)
Thus the original K-user asynchronous CDMA problem can
be treated as
a
2K-user synchronous CDMA problem, on
which a synchronous LDD can be applied.
Let y ( i )
= [y l t
i ) ,
l
i ) ,
.., K t i), y K i)]
represent the
matched filters output at time iTb, where
n
Figure 2: Decomposition of signature waveforms for the o
shot LDD.
with p
=
1, 1 ,..,
K,
K . Let R represent the correlat
matrix with its element given by:
(
n
where
p ,
q
=
1 , l ,2 ,
..,
K ,K
and b( i )
= [ b l (
1),
b1
i ) , 2 i ) , b2 i), ..
,
b~ i ) ,
b~
i)] represent the d
vector, then we have
(
( i )=
RWb(i) +
n( i )
where W e d i a g [ , / n , d f l ,
...,
/SI
and
n(i)
is a zero-mean Gaussian K-vector with covaria
matrix equal to
a2R.
With z ( i )
=
[zl/ i), zl i ) , ..,zK
i),
z ~ i)] represen
the LDD output, equation
(12)
gives
~ ( i )
R - l y ( i )
a
=
Wb(i)
+
n,(i)
(
where n,(i)
=
[ n 1 / ( i ) , n 1 ( i ) ,. . ,nK ( i ) ,nK , , ( i ) ]s a n
vector, whose elements nk and nk-3 are zero-mean Gaus
noise with variance
a 2 < k 1
and a 2 < k k , respectively, wh
< k l k : and < k k are the corresponding diagonal elements
matrix
R-l.
From equation (9) , it can be seen that users
k
and
in the synchronous CDMA share the power of user k in
original asynchronous system. Therefore, any detection a
rithm based on one single bit period will suffer from the
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of signal-noise-ratio (SNR). On the other hand, it can also
be seen that for any two consecutive one-shot windows, user
k in the first carries the same da ta information as user k' in
the following. Therefore, by combining the outputs of LDD
in two consecutive windows, the SNR loss can be recovered.
. ( i l
lI
A
Figure 3: One-shot LDD for asynchronous CDMA.
Define Z k ( 2 ) as:
(14)
A
z k ( i ) ZZ x ; z k i 1)+ x ; z k l i)
where A and
A
are weighting coefficients to be determined.
From equation (13), we have:
Zk i )
= [A;
s d z ] b k 2
1)
+ nkt
i )
+ & n k x , i 1)
(15)
As
nkl i )
and
n k
i
1)
are independent zero-mean Gaus-
sian noise with variances
c r c k l t
and
a 2 < k k ,
respectively,
the above two noise items can be represented as one noise
n k ( i ) ,
which is also zero-mean Gaussian but with variance
By solving an optimization problem, the maximum SNR
C2
[
ck' k' ( xk ) 2 ; e 1
in equation (15) is achieved with A and A given by:
Accordingly, equation (15) leads to:
where n k has variance 02 [ *
+
e]nd the maximal
SNR is:
k r k
The above SNR is the efficient SNR for the one-shot LDD to
detect the information bits of user k under a CDMA scenario.
The maximum SNR, which can be obtained in the absence
of multiuser interference, is 9 . he ratio
of
the two
S N Rs
characterises the near-far resistance of the one-shot LDD,
which is:
With the statistic z k ( i ) , the estimate of the dat a bit b k ( i
1)
can be obtained:
& k ( i 1)
=
s g n [ z k ( i ) ]
(20)
This means that the above detector has one bit detection
delay.
From (19), it can be seen th at the near-far resistance of
the one-shot LDD is determined by the diagonal elements of
the inverse of the cross-correlation matrix, k / and
< k k ,
as well as the energy p k f and
p k
defined by (6). Generally
speaking, both
c k l k ~
and
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Phase Delay '7-k
user 1
0 . 1 ~
0.12Tb
user
2
0 . 3 ~ 0.25Tb
user 3 - 0 . 1 5 ~
0.3Tb
user 4
0 . 2 3 ~
0.76Tb
user 5 -0 . 35 ~
0.8Th
Let c 'k~,k ,nd
c'k,.kj,
epresent the diagonal elements Of
matrix
R
- , and let represent the output of the LDD.
Similar to equation (17), we have:
z k i )
=a[ F ] b k i 1)eJek+ 6 k
(23)
e kt Ck
,
where Re{fik} and Im{6k} are independent zero-mean
Gaussian noise with variance
c2[$k -
+
1
From statistics
z k ,
the estimation
of
data
b k
and phase
0k can be carried out on
a
single user basis. It is seen that
by using the one-shot LDD, the problem of multiuser phase
estimation has been simplified into that of single user one,
on which many techniques can be employed [8].
In this paper,
a
decision-feedback algorithm, similar to
that given in
[8],
is employed for phase estimation. The
estimate of the phase is updated in every bit period and is
based on all the statistics 2, and the estimate of data i&
before:
k l k'
c
k , l
2 1
e k ( i ) =
arg xk(j )zk(j)]
(24)
,3=1
For the first bit period, the estimate of the phase is carried
out as:
m = arg [ (O)l (25)
(26)
The estimate of data bit is given as:
g k ( i 1)= Sgn[Rte(zk(i)e-3sk(2))]
Power (9/ 20 2)( dB)
5
50
10
20
Similar to other phase estimation techniques, the above
method results in
an
ambiguity of T, but this problem can
be solved by using differentially encoded PSK transmission
(DEPSK)
[8].
4
Computer simulation
511-chip Gold codes have been chosen as signatures in the
simulation. The first example is on the one-shot LDD with
known phases. A 5-user CDMA system is simulated. The
parameters of the 5 users are shown in Table 1. Fixing the
powers of noise and users
2,
3, 4 and 5, Figure
4
shows the
bit-error-ratio (BER) curve of user 1 with the change of its
SNR. For comparision, the theoretical BER curve of opti-
mum coherent detection for single user transmission is also
shown in Figure
4
(The two curves are very closely located.).
It
can be seen that the performance of the one-shot LDD
is very close to that of the optimum detection with single
user transmission, which is the upper bound of any detec-
tion scheme for CDMA.
The second example is on the near-far resistance of the
proposed detector. Fixing the SNRs of user 1, 2, 4 and 5
and changing the SNR of user 3, the BER of user 1 s tested.
Figure
5
clearly shows the near-far resistance of the proposed
detector.
The third example is on the one-shot LDD without the
knowledge of phases. The parameters of users are the same
Table
1:
Parameters of 5 users in the simulation system. T
power of user 1 will be changed during simulations.
0
1 2
3 4 5
6
7
8
9
SNRl = 1010g(Pl /2i~~)dB)
Figure
4:
User 1's BER with the change of its own pow
as that shown in Table 1. Differential coding is employ
Fixing the powers of noise and users 2, 3,
4
and 5, the B
curve of the user
1
with the change of its SNR is obtain
based on the phase estimation and one-shot detection te
niques described in section 3. The estimates of phases
updated in every bit period. The converging speed of
estimate of
a
phase to its real value depends on the SNR
the corresponding user. Simulation shows that the esti
tion error is less than 5O after about 200 bit period time
SNR=OdB, and about 10 bit period time for SNR=5dB.
BER curve of user 1 is also shown in Figure 4. From Fig
4, it is observed that the BER for DEPSK approxima
doubles that for BPSK with the same SNR. This is a re
similar to that of single user transmission [8].
5
Conclusion
A one-shot LDD for the detection of asynchronous CD
has been investigated. By treating every user as two indep
dent users and using maximal ratio combining,
a
bit-by
detection is obtained. Then by making use
of
the outpu
the one-shot LDD and employing a phase estimation te
nique,
a
one-shot LDD without the knowledge of user ph
has been obtained. Computer simulation showed the
formance of the proposed detector approaches that of
optimum detector for single user transmission, which is
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SNRl
=
8dB
0 10 20 30 40 50 60
S N R3
= 1010g(P~/20~)
dB)
Figure
5 :
User 1s BER with the change of user 3s power.
upper bound of any detection schemes for CDMA. As the
proposed detector has very good performance but simple ar-
chitecture, it is
a
promising scheme for practical application.
References
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