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Performance Analysis for Coded CDMA Systems
Jianqiu Zhang, Member, IEEE
Abstract—This correspondence analyzes the bit-error rate (BER) performance of coded synchronous code-division multiple-access (CDMA) systems assuming perfect channel state information (CSI) and optimal joint
multiuser detection/decoding (OJMUDD). Our analysis is conducted in the
same framework as that of uncoded systems. First, we derive the precise
probability of an error event, then we provide an upper bound on the BER
based on the sum of pairwise error probabilities, and, finally, we tighten
the upper bound by considering decomposable error events. Many new
concepts unique to coded systems are introduced. We propose to use quasi
parity checks for identifying permissible error events, introduce the concept of compatible probability of error matrices, extend the list of conditions for identifying decomposable error events, and introduce the concept
of conjugate sets to explore the symmetry among indecomposable error
events. Simulation results are given along with theoretical predictions.
Index Terms—Code-division multiple access (CDMA), error control
coding, joint multiuser detection and channel decoding, multiuser detection, performance analysis, turbo decoding.
I. INTRODUCTION
Most current code-division multiple-access (CDMA) systems
treat multiuser detection and channel decoding separately. However,
recently, by employing turbo decoding techniques, Wang et al. [2]
illustrated that near-single-user performance can be achieved in coded
CDMA systems even with large cross correlations when joint multiuser
detection (MUD) and channel decoding is performed. Obviously,
performance analysis of coded systems is needed if we want to explore
the potential capacity increase enabled by joint multiuser detection
and decoding. However, besides some information-theoretic results
[4], [5], and a brief mention of the asymptotic efficiency [3], there does
not exist detailed performance analysis for coded CDMA systems.
Manuscript received July 9, 2001; revised July 10, 2003. The material in this
correspondence was presented in part at the International Symposium on Information Theory, Washington, DC, June 2001.
The author is with the Department of Electrical and Computer Engineering, University of New Hampshire, Durham, NH 03824 USA (e-mail:
[email protected]).
Communicated by V. V. Veeravalli, Associate Editor for Detection and Estimation.
Digital Object Identifier 10.1109/TIT.2003.820030
0018-9448/03$17.00 © 2003 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
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Fig. 1. System model on the transmitting side.
The performance of uncoded systems is well studied [1]. While
many concepts for uncoded systems are useful, it is not possible to
analyze the performance of coded systems without introducing a
significant number of new concepts.
In this correspondence, we analyze the performance of synchronous
coded CDMA systems using the same framework as that of uncoded
systems [1]. First, we derive the precise probability of error events represented by two-dimensional error matrices. We find that the probability of an error event depends on its total error distance and its compatible probability. In uncoded systems, only the total error distance of
an error event significantly affects error performance while the compatible probability, first mentioned in [6], is a trivial constant that can be
easily calculatedy. However, in coded systems, the compatible probability is more difficult to calculate and it may notably affect the bit-error
rate (BER) performance at low signal-to-noise ratio (SNR). We establish that the compatible probability is determined by the employed error
control code. Next, we provide an upper bound on the BER based on
the sum of pairwise error probabilities. As demonstrated before [1],
some error events are decomposable, i.e., their maximum-likelihood
decision regions are completely encompassed by that of other pairwise
error events, and therefore such error events can be ignored when evaluating the upper bound. We extend the list of conditions for identifying
decomposable error events given in [7] and derive a tighter upper bound
by only considering indecomposable error events. Then, we explore the
symmetry among error events and define the new concept of a conjugate error event set. We show that error events in the same conjugate
set have the same probability and decomposability. We also discuss the
evaluation of conjugate sets size by introducing several propositions.
Finally, we derive a simple approximation of error performance if error
patterns with large number of errors are ignored. Such approximation
can be readily applied in practical situations.
In the last section, we simulate two cases. The first case is about a
simple coded system so that the exact BER bound analysis can be verified using the optimal joint multiuser detection/decoding (OJMUDD)
algorithm. In the second case, a possible coded CDMA system is simulated. Where exact analysis is not possible, the simulation results confirm our approximate analysis.
The correspondence is organized as follows. In Section II, we give
system descriptions and definitions, in Section III we evaluate the probability of error events, in Section IV we discuss the upper bound on
BER, and finally we show simulation results in Section V.
II. SYSTEM DESCRIPTION AND DEFINITIONS
Consider a coded synchronous CDMA system with K users in
Fig. 1. The information bits of each user are organized into blocks of N
bits and we denote the N 2 1 binary information bit vector of the k th
user as dk = [dk1 ; . . . ; dkN ] , where [1] stands for transpose. Then
d k passes through a user-specific channel encoder and interleaver
with rate R, and generates L = N=R output bits q k = G k dk , where
k represents the L 2 N combined code generating and interleaving
matrix. For convenience, G k is simply referred to as the k th code
generating matrix. We assume that only linear codes are used, and we
can find an L 2 (L 0 N ) parity-check matrix P k accompanying each
G k , such that q k P k = 0, where 0 stands for an all-zero vector.
After encoding and interleaving, the elements of q k are mapped to
symbols of duration T using binary phase-shift keying (BPSK). Let
bk = [bk1 ; . . . ; bkL ] represent the set of symbols of the k th user. The
user data matrix of all user symbols is represented by
G
K 2 f01; +1gK2L
b1 ; . . . ; b ]
B = [b
in which the columns are denoted as fbl gL
l=1 , and bl is the vector of
symbols to be transmitted at the lth symbol interval. The BPSK symbols are subsequently spread by their designated normalized signature
waveforms s(t) = [s1 (t); . . . ; sK (t)] , where sk (t) t 2 [0; T ] is the
signature waveform assigned to the k th user. With ideal time and frequency information, the received signal at the lth symbol interval is
l
ll
l
r (t) = s A b + n (t)
(1)
where A l = diag fA1l ; . . . ; Akl ; . . . ; AKl g is the receive amplitude
matrix known to the receiver, and nl (t) is the observation white
Gaussian noise with variance 2 .
^ , and
An error event happens when B is incorrectly detected as B
1
^
B 0 B ), with row error vecits error matrix is defined as E = 2 (B
tors f k gkK=1 and column error vectors fl gL
l=1 . Note that the error
pattern, or the absolute value of a row error vector, z k (k ) = j k j,
forms a codeword of the k th user’s linear block code. The 1’s and 0’s
in z k (k ) should not be interpreted the same as in q k , rather, they are
indicators of errors. Any element of E , say kl , either equals bkl if there
is no detection error or “0” otherwise. If for any k 2 f1; . . . ; K g and
l 2 f1; . . . ; Lg, such that kl = 0bkl , then E and B are not compatible. Conversely, it is possible to determine some BPSK symbols in B
corresponding to the nonzero elements of a given E that is compatible
with B , because the nonzero elements in E equal the corresponding
elements in B . Subsequently, the corresponding binary bits of q k 8k
can be determined. By converting “1” in E to “1,” “01” to “0,” and
“0” to the unknown element “x,” we derive a quasi codeword matrix
K=1 . Define modular two
^ (E
E ) with quasi row codewords fq
^k (k )gk
Q
addition between x and any other value in f1; 0; xg as x, then a quasi
parity check is formulated by multiplying a quasi row codeword with
P k = x k . If q
^k (k ) is a
its corresponding parity-check matrix q^k (k )P
legitimate quasi-codeword, then xk should only contain values of “x”
and “0.” Given an arbitrary E 2 f1; 0; 01gK 2L , it is permissible if
and only if
1) all its quasi row codewords can pass quasi parity check and
2) all the row error vectors have valid error patterns
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
^ ).
^ such that E = 1 (B 0 B
i.e., there exists at least one pair of B and B
2
The proof of this point is trivial and is not shown here.
III. THE PROBABILITY OF AN ERROR EVENT WITH OJMUDD
The probability of a permissible error event can be evaluated as
P (E ) =
B
2B
P (E ; B ) =
B
2B
P (E jB )P (B )
(2)
where B is the set of all possible user data matrices. If we assume independent and identically distributed (i.i.d.) information bits, the probability of a particular user data matrix is p(B ) = 2 1 , where KN
is the total number of information bits in the block. P (E jB ) equals
zero if E is not compatible with B . Otherwise, P (E jB ) is the probability that the i.i.d. Gaussian noise vector n = [n1 ; . . . ; nL ] falls in
^ = B 0 2E . The density function of
the pairwise decision region of B
n can be factored as
L
p(n) =
(3)
l=1
In [1], it was shown that p(nl ) is the Gaussian density function of the
observation noise and we have the probability of l as
d2 (l )
2
(4)
where ( l ) stands for the pairwise decision region of l
d2 (l ) = l Al RAl l
(5)
is its squared error distance, and R is the K 2 K cross-correlation
matrix whose k1 k2 th element is calculated according to
k
k
=
T
< sk ; sk > =
0
sk (t)sk (t)dt:
0
~ ( ))
rank (G
=2
D2 (E )
2
(6)
where D2 (E ) = ll=1 d2 (l ) is defined as the total error distance of
the error matrix E .
Define the compatible number of E , C (E ), as the total number of
data matrices that are compatible with E . Then (2) can be further elaborated as
D2 (E )
2
=
Pc (E )Q
D2 (E )
2
(7)
where Pc (E ) is defined as the compatible probability of E .
Since the row error vectors of an error matrix are independent, the
compatible number of an error matrix is the product of the compatible
numbers of its row error vectors C (E ) = K
k=1 C ( k ). To find C ( k ),
we can start with the quasi-codeword q^k ( k ) of k . By removing the
qk (k ), we obtain a partial codeword ~qk (k ) and its
unknown bits in ^
~ ( k ) that are related to one another by
partial code-generating matrix G
G~ k (k )dk = q~k :
:
(9)
Proposition 1: If the number of nonzero elements or the weight of
k , w(k ), is less than or equal to N , then its partial code-generating
matrix has rank w( k ).
Proof: Because the full code-generating matrix has rank N , any
w(k ) N rows of it are linearly independent.
For example, suppose a user employs a (3; 2) linear block code with
Gk
=
1
0
1
0
1
1
(8)
:
Given a row error vector k = [1 0 0 1], we can determine its quasicodeword as q^k (k ) = [1 x 0], and subsequently its partial codeword
as q~k = [1 0]. Striking out the second row in Gk , (8) becomes
1
0
1
1
d1
d2
=
1
0
:
Because the rank of the partial code generating matrix is 2 and the total
number of information bits is 2, the compatible number of the error
vector is 2202 = 1.
IV. BER EVALUATION FOR CODED CDMA SYSTEMS
Suppose the maximum-likelihood decision region of E is (E ), then
the error probability for the nth bit of the k th user is derived by
2
^
E E
P (E jB ) = Q
2
2
Pkn () =
It is easy to verify that
(E )
P (E ) = CKN
Q
(E )
Pc (E ) = CKN
generating matrix
p(nl ):
P (nl 2 (l )) = P (l ) = Q
~ ( ) spans, i.e., the rank of
The dimension of the subspace that G
k k
~ ( ))
~ ( ), dictates that it can generate at most 2rank(G
G
distinct park k
tial codewords. Since dk spans a space of dimension N , and there exists 2N original codewords for the k th user, the number of original
codewords that are compatible with a given q~k becomes C ( k ) =
~ ( ))
N -rank(G
2
. With these findings, the compatible probability can
be written as
(E )
p(n)dnn
(10)
^ kn , it can be determined that a detection/dewhere for each E 2 E
coding error happens at the knth bit. However, it is a well known fact
that it is difficult to calculate the integration inside each summation
term. Instead, the terms of summation are usually replaced by pairwise
error probabilities and we obtain an upper bound
Pkn () P (E ):
(11)
2
For systematic codes, since the knth bit has direct correspondence
to an error element, say kl in E , it is straightforward to show that
E^ kn = fE : kl = 1; 01g. When dealing with nonsystematic codes,
^ , we need to test if the k th row
to check whether E belongs to E
kn
error vector results in an error at the knth bit. To begin with, we derive the error pattern of k and then convert it to the systematic form
by z~k ( k ) = T k z k (k ), where T k is the conversion matrix of the non^
E E
systematic code. Such conversion matrix exists for most nonsystematic
codes. Next, if we find that the knth bit in z~k ( k ) is one, we can con^ .
clude that E 2 E
kn
As in the uncoded case pointed out in [1], E is decomposable if
its maximum-likelihood decision region (E ) is totally encompassed
by the pairwise decision region of another error matrix (E 0 ). In
such cases, the term P (E ) in (11) becomes “superfluous,” and can be
deleted from the upper bound. In [1, Sec. 4.3.2], the conditions for
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
decomposable error vectors are listed with slight modification in the
context of this correspondence:
1)
2)
3)
l = 0l + l 00 ;
If (l )i = 0, then (0l )i
0l H l" l 00 0;
= (l )i = 0;
00
where H l = A l RAl . In coded systems, we can vectorize E and obtain
" = [ 1 1 1 1 L ] and then construct
H
=
H1
..
.
0 111
..
.
0 111
0 111
Hl
..
.
0
111 0
:
HL
Now we can apply the three conditions of decomposability listed above,
i.e., " is decomposable if
1) " = " 0 + " 00 ;
2) If (")i = 0, then ("0 )i = ("00 )i
3) " 0 H "00 0;
and specific to coded systems
4) both "0 and " 00 are permissible.
= 0;
Proposition 2: If " can pass quasi parity check, then its decomposition "0 and " 00 that satisfy Conditions 1 and 2 can pass quasi parity
check as well.
Proof: The quasi parity check of " is completed by first converting it back to the matrix form and then conducting quasi parity
checks on the rows. The known bits in q^("0 ) are a subset of the known
bits in q^("), and the rest of known bits in q^(") are replaced with “x”s
in q^("0 ). Parity checks involving these replaced “x”s will result in “x”s
and are treated as valid results according to the definition of quasi parity
check. Therefore, q^("0 ) can pass the quasi parity check, and the same
argument applies to q^("00 ).
Proposition 3: Given that " 0 and " 00 satisfy Conditions 1 and 2, if " 0
is permissible, then "00 is permissible.
Proof: To prove that " 00 is permissible, we need to show that it
can pass quasi parity check which is already shown in Proposition 2,
and it has a valid error pattern. Due to the assumption of Conditions 1
and 2 of decomposability, the error patterns are related to each other
according to z (") = z ("0 ) + z ("00 ). Since we assume that both " 0 and
" are permissible, i.e., z ("), and z ("0 ) must be valid codewords, z ("00 )
must also be a valid codeword in the context of linear codes.
Proposition 4: Suppose that for all l 2 f1; . . . ; Lg, l can be decomposed to l 0 and l 00 . By stacking these decomposed column vectors, we obtain " 0 and " 00 . If either one of the stack vectors are permissible, then " is decomposable.
Proof: We assume that either " 0 or " 00 is permissible and therefore using Proposition 3, we can verify that Condition 4 of decomposability is satisfied. Conditions 1 and 2 are satisfied automatically by the
0
00
stacked error vectors. Finally, since " 0 H "00 = L
l=1 l H l l and
each term in the sum is greater or equal to 0, it becomes evident that
Condition 3 is satisfied.
This proposition can simplify the process of searching for a decomposition of an error matrix because if a decomposition is found for
every column error vector and one of the stacked vectors is verified
to be permissible, a decomposition of the error vector is found. However, the reverse of this proposition is not necessarily true, i.e., if " is
decomposed to " 0 or " 00 , the portions corresponding to the lth column,
l 0 and l 00 , do not necessarily form a decomposition of the lth column
error vector. Given an error matrix, we should first try using this proposition to find a decomposition and if not successful, all other forms of
3285
decomposition should be tested before we can pronounce the decomposability of the error matrix.
^ kn , the
Suppose the set of indecomposable matrices is E kn 2 E
BER bound in (11) can be tightened
Pkn ( ) <
P (E ):
(12)
E
E 2E
Note that a set of several error matrices may have the same probability with E , which can be denoted as
(knE ) = fE 0 2 E kn ; P (E 0 ) = P (E )g:
Suppose that E kn can be partitioned to mutually exclusive groups
of matrices with identical error probability. Define a new set Ekn =
(E )
(E )
[ 1 1 1 [ kn
= E kn ; now (12) can
fE 1 ; . . . ; E g, such that kn
be simplified to
Pkn ( ) <
(E )
kn
M (
)P (E )
E 2E
(13)
kn ), denotes the size of kn , or the multiplicity number
where M (
of E .
(E )
Now our problem becomes how to find kn given E . By setting one
or any combination of columns in E , say l ; l ; . . . to their opposites
0l ; 0l ; . . ., we can obtain a conjugate of E , E 3 .
(E )
(E )
Proposition 5: If E 3 is permissible, then P (E 3 ) = P (E ).
Proof: First, it is straightforward to verify that D2 (E ) =
2
D (E 3 ) by using (5). Next, we verify that Pc (E 3 ) = Pc (E ). Since
any corresponding row error vectors in E and E 3 , say k and 3k , have
the same set of indeterminable elements, they share an identical partial
~ (k ) = G~ (3k ), 8k 2 f1; . . . K g. In
code generating matrix, i.e., G
reference to (9), it is evident that Pc (E 3 ) = Pc (E ).
Proposition 6: If E is decomposable and E 3 is permissible, then
E is decomposable.
Proof: Suppose the vector form of E is decomposed into " =
" 0 + " 00 and we have
E3
"0 H "00
=
L
l=1
l 0 H l l 00 :
Since l H l l = 0 l H l 0l 8l, and if we decompose E 3 by setting elements of "0 and " 00 corresponding to the conjugated columns in
E to their opposite sign, we derive a decomposition "3 = "3 +"3 . We
can verify that Conditions 1–3 of decomposability are satisfied. Next
we prove that "3 is permissible. It is a conjugate of " 0 , therefore, it
must have a valid error pattern and since we assume that E 3 is permissible, "3 can pass quasi parity check as observed in Proposition 2. We
conclude that "3 is permissible and so is " 3 based on Proposition 3.
Now all four conditions of decomposability are satisfied.
0
00
0
00
Suppose E has (E ) number of nonzero columns, then there could
be at most 2(E ) number of conjugates. We need to determine how
many of these conjugates are permissible. First, we consider the
number of permissible conjugates of a given k . From the nonzero
~ (k )
elements of k , we can find its partial code-generating matrix G
~
which can generate at most 2rank (G( )) distinct partial codewords.
From these partial codewords, we can derive the corresponding permissible row error vectors that are conjugates. Therefore, the number
~
of permissible conjugates of k is 2rank (G( )) . Note that w(k ), the
number of nonzero elements of k , equals the number of rows in
~ (k ), and, therefore, it is greater or equal to rank(G~ (k )). Since the
G
total number of possible conjugates of k is 2w( ) , every conjugate of
~ (k )). Otherwise, there are
k is permissible when w( k ) = rank(G
~
at most 2rank(G( )) permissible conjugates.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
~ (k )).
Proposition 7: When w(k ) N , w(k ) = rank (G
Proof: For properly designed linear codes, G ( k ) has rank N .
~ (k ) consists of w(k ) rows from G(k ), these w(k ) rows
Since G
~ (k )) = w(k ).
must be linearly independent, i.e., rank (G
kn ), we can employ the following steps.
To find M (
1) Find all row error vectors with weight greater than the rank of
their partial generating matrix. Taking out the remaining rows in
E , we obtain a reduced form of E with a total of ^(E ) nonzero
columns. Find the number of permissible conjugates of the reduced error matrix by checking each variation and suppose the
^ (
Ekn ).
number is M
(E )
kn ) = M^ (
Ekn )2((E )0^(E )) because every variation is
2) M (
permissible for the rest of rows.
Note that in Step 1, due to Proposition 7, we can quickly exclude row
error vectors with less than N nonzero elements.
Since error events with large total error distances and small compatible probabilities have little impact on the overall performance, in
approximate analysis, it is quite safe in most cases to ignore events
whose error matrices have one or more row vectors with weight larger
than N . For the remaining error events, according to Proposition 7 and
(E )
kn
the algorithm outlined above, M (
) = 2(E) , and (9) becomes
(E )
( ) = 20
w( ))
Pc E
. Let
E~kn = fE 2 Ekn ; w(k ) N
kn )P (E ) Th g
and M (
(E )
where k 2 f1; . . . ; K g and Th is a threshold that one selects depending
on the type of application. For example, if the tolerable BER in an
application is in the range of 1003 0 1002 , then one may set Th to
a number less than 1004 . An approximation of the error performance
can be written as
( )
Pkn where ( ) =
ranges of SNR
w E
2(E) Q
2w(E)
~
E 2E
( )
D2 E
2
( )) is the sum of weights. While at higher
( )
D2 E
2
determines the performance, the term 22
may have a significant impact at lower ranges of SNR. We can see that the larger the minimum
Hamming weight, the better the performance because it increases both
2
w (E ) and D (E ). Suppose that identical error control coding is used
for all users, then the introduction of interleaver may increase D2 (E )
which lowers error probability, and (E ) which increases error probability. However, the overall effect of (E ) is much lower than that of
2
D (E ) generally. In summary, for a coded CDMA system, it is preferable to use error control codes with larger minimum Hamming weight
and interleavers under limited bandwidth.
We would also like to point out that (13) can be reduced to the upper
bound of uncoded CDMA system performance as given in [1]. First of
all, in an uncoded system, L = N and the subscript l and n can be
interchanged. Suppose l can be decomposed into l 0 and l 00 ; then it is
trivial to prove that E = [. . . ; l01 ; l ; l+1 ; . . .] can be decomposed
into E 0 = [. . . ; 0; l 0 ; 0; . . .] and E 00 = [. . . ; l01 ; l 00 ; l+1 ; . . .].
Using the same notation as in [1], define F k as the subset of indecomposable vectors from the set of all possible l ’s, it is obvious that
Ekn = fE ; s.t. l 2 F k ; and l0 ;l
0:
1 +1:L
= 0g:
(E )
(14)
K
k=1 w k
Q
kn ) = 2(E) = 2,
Now we can see that for E 2 Ekn , (E ) = 1, M (
w
(
))
=
w
(
)
.
Plugging
these
values
in (13), we obtain
and K
k
l
k=1
the same result as in [1, eq. (4.74)] after considering the symmetry that
exists in the indecomposable subset by replacing w(l ) with w(l ) 0 1.
To perform the exact upper-bound performance analysis, we first utilize the fact that permissible row error vectors must have valid error
patterns, and error patterns are codewords of the linear block codes, to
conclude that there are 2KN possible error patterns. Excluding error
patterns that have a zero element at the position of the bit to be analyzed, there remains 2KN 01 error patterns. The nonzero elements in
each error pattern can take values from f01; +1g, and by combining
all possible values of the nonzero elements, we obtain a set of possible
matrices. If we number all the error patterns, and let w(E j ) be the total
number of nonzero elements of the j th error pattern, then there exists a
total of 2j =1
2w(E ) possible matrices. Then these matrices are
organized to conjugate sets, and finally, one matrix from each conjugate
set is analyzed for its decomposability, probability, and the number of
permissible conjugates. The complexity of the algorithm is determined
by the error-control codes used. In approximate analysis, only a limited
number of error patterns with small total error distance and nontrivial
compatible probability is analyzed, and (14) can be used to derive the
approximation of BER efficiently. The complexity in such cases depends on the number of conjugate sets in E~kn , which is determined by
the threshold Th one selects. Note that the calculations of the quantities (E ), w(E ), and D2 (E ) in (14) are all straightforward, and the
biggest challenge is to identify the conjugate sets that belongs to E~kn .
The identification process is conceptually equivalent to finding error
patterns with small Hamming distance in the performance analysis of
error-control codes. For the analysis of nonsystematic codes, an additional procedure is needed for identifying the 2KN 01 error patterns
that would indicate an error at the bit to be analyzed. The extra step is
as described in Section IV.
V. SIMULATION
Case 1: First we consider the simple case of two equal unit power
users using the (3; 2) linear block code described earlier. Suppose the
cross correlation between the signature waveforms of the two users
equals 0:75. We consider the error probability of the first bit of the first
user P11 ( ). Since each permissible row error vector must have a valid
error pattern, user 2 can have four valid error patterns and for user 1, excluding the error pattern that (z 1 )1 6= 1, there are only two valid error
patterns. Except the all-zero error pattern, each error pattern has two
nonzero elements for this code. There could be 2 2 4 = 8 possible row
error vectors for user 1, 3 2 4 + 1 = 13 possible row error vectors for
user 2, and the total number of possible error matrices is 104. We can
identify 18 conjugate sets among these matrices. Since we have proven
that matrices within the same conjugate set have identical decomposability and probability, we only need to examine 18 error matrices each
from a conjugate set. The examination resulted in the following eight
matrices that belongs to E11 :
1 1 0 ;
0 0 0
1 0 1 ;
01 0 01
1 0
0 1
1 0 1 ; 1 1 0
0 0 0
01 01 0
1 1 0 ; 1 1 0
0 01 1
01 0 1
1 ; 1 0 1 :
01
01 1 0
Since in all these error matrices, none of the row error vectors have
more than two nonzero elements, according to Proposition 5, all
conjugates of these error matrices are permissible and the multiplicity
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
Fig. 2.
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Analysis and simulation results of Case 1.
number of each error matrix is simply 2(E ) . Based on these calculations, we can upper-bound the error probability of the first bit of the
first user using (13)
P11 () <
2
2 4Q
24
1
2
+
2
2 4 2 4Q
2
24
+
4
2
2 8Q
24
2:5
2
(15)
where the noise variance is related to the energy per bit and noise variE
ance ratio Eb =N0 as 1 = N
R, when unit receiving amplitude is assumed. In our example, the code rate is R = 2=3. We may also derive a
lower bound based on the minimum error distance error events, and in
this case the first term in (15) can serve for this purpose. In Fig. 2, we
plot the bounds and the simulation results. We note that after 10 dB, the
simulated BER is slightly “higher” than the upper bound. However, at
this point the upper bound and lower bound have converged and the imperfect simulation result is caused by limited number of Monte Carlo
trials. We accumulated 1000 bit errors for each simulated point, and the
simulation follows the bounds very closely.
Case 2: In this case, we examine the impact of compatible probability on BER. The CDMA system considered uses the rate 1=2, constraint length 5, (23; 35) recursive systematic convolutional code for
all users. Suppose the cross correlation among four equal power users
is k k = 0 13 ; 8k1 6= k2 and k1 ; k2 2 f1; . . . ; 4g, and only one interleaver is used. Due to the identical channel encoder used for all users,
there certainly exist error matrices in which all four nonzero row error
vectors are identical, and without diversified interleavers, the nonzero
row error vectors can be aligned vertically in the error matrix. Consequently, the nonzero columns are all in the form of l = [1 1 1 1] ,
or l = [01 0 1 0 1 0 1] . Plugging these values in (5), we find
that d2 (l ) = 0, and in turn the total error distance is zero. According
to [3, eq. (10)], the asymptotic efficiency in this case would be zero
and very bad BER performance is expected due to such error matrices.
However, because of their low compatible probability, the impact of
these types of error matrices on BER is very low. Suppose the number
of nonzero columns of is , then the compatible probability becomes
Pc = (2 1 ) according to (9). With zero total error distance, the total
contribution to the BER bound in (14) becomes (2 1 ) if is less than
the block size. Since the convolutional code we used has a minimum
Hamming distance of 7, the contribution of these types of error matrices is at most (2 1 ) = 4:77 2 1007 . Therefore, in the usual BER
ranges around 1005 0 1004 , the impact of such matrices are trivial.
In fact, except error matrices with single nonzero row error vector, it is
not hard to examine and find that all other types of error matrices have
negligible impact on BER because of the combined effect of their total
error distance and compatible probability. Therefore, near-single-user
performance is expected for this CDMA system.
In Fig. 3, we simulated the performance of this four-user CDMA
system using the (23; 35) code. The block size of the information bits
for each user is 128. Since OJMUDD is not possible here, we use the
suboptimum exact soft-input soft-output (SISO) iterative (turbo) decoding/detection algorithm as proposed in [2] for the joint detection
and decoding of the system. The exact SISO turbo detector/decoder
consists of an exact multiuser detector and a bank of channel decoders.
Extrinsic information are exchanged between the multiuser detector
and channel decoders until the algorithm converges. The exact multiuser detector extracts soft information from received signals and the
extrinsic information fed from the channel decoders at each symbol interval. After the multiuser detector processes the whole block, the extrinsic information from the channel decoders are subtracted from the
soft information and the results (extrinsic information from the multiuser detector) are fed to the channel decoders. This process is repeated until there are no significant changes to the soft information
and finally a decision of each information bit is made based on the soft
information.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
Fig. 3. Simulation of Case 2 compared with single-user performance.
As predicted from our analysis, near-single-user performance is still
observed in the BER range of 1004 as shown in Fig. 3. Three iterations are used in the simulation. This example clearly shows that although some error matrices have very small error distance, because of
their small compatible probability, they will not affect the BER performance in lower ranges of SNR. That is, even though some system have
very bad asymptotic efficiency, these systems can still perform well
in lower ranges of SNR. In Fig. 3, we have also compared the performance of the (23; 35) code with that of the (15; 17) code while other
aspects of the system remain unchanged. We can see that both codes
follow their single-user bounds closely. The single-user bounds are obtained through experiments using the bit maximum a posteriori (MAP)
(Bahl-Cocke–Jelinek–Raviv(BCJR)) channel decoder. It is quite safe to
claim that the performance of coded CDMA systems with interleavers
is largely determined by the error distance properties of the error-control codes.
Finally, we want to point out that by using different interleavers
for different users, the scenario that all nonzero row error vectors are
aligned vertically rarely occurs. Therefore, the total error distance will
be larger as pointed out in [3], and in many cases, single-user performance can still be achieved even when the cross correlations are high
as documented in [2].
ACKNOWLEDGMENT
The author would like to thank the anonymous reviewers for their
extremely useful comments on this correspondence.
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