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de Lamare, Rodrigo C. and Sampaio-Neto, Raimundo (2008) Minimum mean-squared
error iterative successive parallel arbitrated decision feedback detectors for DS-CDMA
systems. IEEE Transactions on Communications. pp. 778-789. ISSN 0090-6778
https://doi.org/10.1109/TCOMM.2008.060209
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Published paper
de Lamare, R.C. and Sampaio-Neto, R. (2008) Minimum mean-squared error
iterative successive parallel arbitrated decision feedback detectors for DS-
CDMA systems
, IEEE Transactions on Communications, Volume 56 (5), 778 -
779.
eprints@whiterose.ac.uk
778 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 5, MAY 2008
Minimum Mean-Squared Error Iterative Successive
Parallel Arbitrated Decision Feedback Detectors for
DS-CDMA Systems
Rodrigo C. de Lamare and Raimundo Sampaio-Neto
Abstract—In this paper we propose minimum mean squared
error (MMSE) iterative successive parallel arbitrated decision
feedback (DF) receivers for direct sequence code division multiple
access (DS-CDMA) systems. We describe the MMSE design
criterion for DF multiuser detectors along with successive,
parallel and iterative interference cancellation structures. A novel
efficient DF structure that employs successive cancellation with
parallel arbitrated branches and a near-optimal low complexity
user ordering algorithm are presented. The proposed DF receiver
structure and the ordering algorithm are then combined with it-
erative cascaded DF stages for mitigating the deleterious effects of
error propagation for convolutionally encoded systems with both
Viterbi and turbo decoding as well as for uncoded schemes. We
mathematically study the relations between the MMSE achieved
by the analyzed DF structures, including the novel scheme, with
imperfect and perfect feedback. Simulation results for an uplink
scenario assess the new iterative DF detectors against linear
receivers and evaluate the effects of error propagation of the
new cancellation methods against existing ones.
Index Terms—DS-CDMA systems, multiuser detection, deci-
sion feedback structures, iterative detection, iterative decoding.
I. INTRODUCTION
M
ULTIUSER detection has been proposed as a means to
suppress multi-access interference (MAI), increasing
the capacity and the performance of CDMA systems [1].
The optimal multiuser detector of Verdu [2] suffers from
exponential complexity and requires the knowledge of timing,
amplitude and signature sequences. This fact has motivated
the development of various sub-optimal strategies: the linear
[3] and decision feedback (DF) [4] receivers, the succes-
sive interference canceller [5] and the multistage detector
[6]. Recently, Verdu and Shamai [7] and Rapajic [8] et al.
have investigated the information theoretic trade-off between
the spectral and power efficiency of linear and non-linear
multiuser detectors in synchronous AWGN channels. These
works have shown that given a sufficient signal to noise
ratio and for high loads (the ratio of users to processing
gain close to one), DF detection has a substantially higher
Paper approved by X. Wang, the Editor for Multiuser Detection and
Equalization of the IEEE Communications Society. Manuscript received April
4, 2006; revised December 6, 2006.
R. C. de Lamare is with the Communications Research Group, Department
of Electronics, University of York, York Y010 5DD, United Kingdom (e-mail:
rcdl500@ohm.york.ac.uk).
R. Sampaio-Neto is with CETUC/PUC-RIO, 22453-900, Rio de Janeiro,
Brazil (e-mail: raimundo@cetuc.puc-rio.br).
Digital Object Identifier 10.1109/TCOMM.2008.060209.
spectral efficiency than linear detection. For uplink scenarios,
DF structures, which are relatively simple and perform linear
interference suppression followed by interference cancellation,
provide substantial gains over linear detection.
Minimum mean squared error (MMSE) multiuser detectors
usually show good performance and have simple adaptive
implementation. In particular, when used with short or re-
peated spreading sequences the MMSE design criterion leads
to adaptive versions which only require a training sequence
for estimating the receiver parameters. Previous work on DF
detectors examined successive interference cancellation [9],
[10], [11], parallel interference cancellation [13], [14], [15]
and multistage or iterative DF detectors [14], [15]. The DF
detector with successive interference cancellation (S-DF) is
optimal, in the sense that it achieves the sum capacity of the
the synchronous AWGN channel [10]. The S-DF scheme is
capable of alleviating the effects of error propagation despite
it generally leads to non uniform performance over the users.
In particular, the user ordering plays an important role in the
performance of S-DF detectors. Studies on decorrelator DF
detectors with optimal user ordering have been reported in
[11] for imperfect feedback and in [12] for perfect feedback.
The problem with the optimal ordering algorithms in [11],
[12] is that they represent a very high computational burden
for practical receiver design. Conversely, the DF receiver
with parallel interference cancellation (P-DF) [13], [14], [15]
satisfies the uplink requirements, namely, cancellation of intra-
cell interference and suppression of the remaining other-cell
interference, and provides, in general, uniform performance
over the user population even though it is more sensitive to
error propagation. The multistage or iterative DF schemes
presented in [14], [15] are based on the combination of S-
DF and P-DF schemes in multiple stages in order to refine
the symbol estimates, resulting in improved performance over
conventional S-DF, P-DF and mitigation of error propagation.
In this work, we propose the design of MMSE DF detectors
that employ a novel successive parallel arbitrated DF (SPA-
DF) structure based on the generation of parallel arbitrated
branches. The motivation for the novel DF structures is to
mitigate the effects of error propagation often found in P-DF
structures [13], [14], [15]. The basic idea is to improve the
S-DF structure using different orders of cancellation and then
select the most likely estimate. A near-optimal user ordering
algorithm is described for the new SPA-DF detector structure
and is compared to the optimal user ordering algorithm, which
0090-6778/08$25.00
c
2008 IEEE
DE LAMARE and SAMPAIO-NETO: MINIMUM MEAN-SQUARED DECISION FEEDBACK DETECTORS 779
requires the evaluation of K! different cancellation orders.
The results in terms of performance show that the SPA-DF
structure with the suboptimal ordering algorithm can achieve
a performance very close to that of the S-DF with optimal
ordering. Furthermore, the new SPA-DF scheme is combined
with iterative cascaded DF stages, where the subsequent stage
uses S-DF, P-DF or the new SPA-DF system to refine the
symbol estimates of the users and combat the effects of
error propagation. The performance of the proposed SPA-
DF scheme and the sub-optimal ordering algorithm and their
combinations with other schemes in a multistage detection
structure is investigated for both uncoded and convolutionally
encoded systems with Viterbi and turbo decoding.
This paper is structured as follows. Section II brieflyde-
scribes the DS-CDMA system model. The MMSE decision
feedback receiver filters are described in Section III. Sections
IV is devoted to the novel SPA-DF scheme, the near-optimal
user ordering algorithm and the combination of the SPA-
DF detector with iterative cascaded DF stages and Section
V details the proposed SPA-DF receiver for convolutionally
coded systems with Viterbi and turbo decoding. Section VI
presents and discusses the simulation results and Section VII
draws the concluding remarks of this paper.
II. DS-CDMA SYSTEM MODEL
Let us consider the uplink of a symbol synchronous binary
phase-shift keying (BPSK) DS-CDMA system with K users,
N chips per symbol and L
p
propagation paths. It should be
remarked that a synchronous model is assumed for simplicity,
although it captures most of the features of more realistic
asynchronous models with small to moderate delay spreads.
The baseband signal transmitted by the k-th active user to the
base station is given by
x
k
(t)=A
k
∞
i=−∞
b
k
(i)s
k
(t − iT ) (1)
where b
k
(i) ∈{±1} denotes the i-th symbol for user k,the
real valued spreading waveform and the amplitude associated
with user k are s
k
(t) and A
k
, respectively. The spreading
waveforms are expressed by s
k
(t)=
N
i=1
a
k
(i)φ(t − iT
c
),
where a
k
(i) ∈{±1/
√
N}, φ(t) is the chip waveform, T
c
is the chip duration and N = T/T
c
is the processing gain.
Assuming that the receiver is synchronised with the main path,
the coherently demodulated composite received signal is
r(t)=
K
k=1
L
p
−1
l=0
h
k,l
(t)x
k
(t − τ
k,l
)+n(t) (2)
where h
k,l
(t) and τ
k,l
are, respectively, the channel coefficient
and the delay associated with the l-th path and the k-th user.
Assuming that τ
k,l
= lT
c
, the channel is constant during
each symbol interval, the spreading codes are repeated from
symbol to symbol and the receiver is synchronized with the
main path, the received signal r(t) after filtering by a chip-
pulse matched filter and sampled at chip rate yields the M-
dimensional received vector
r(i)=
K
k=1
A
k
b
k
(i)C
k
h
k
(i)+A
k
b
k
(i − 1)
¯
C
k
h
k
(i − 1)
+ A
k
b
k
(i +1)
˘
C
k
h
k
(i +1)+n(i)
=
K
k=1
A
k
b
k
(i)p
k
(i)+η
k
(i)
+ n(i)
(3)
where M = N + L
p
− 1, n(i)=[n
1
(i) ... n
M
(i)]
T
is the
complex gaussian noise vector with E[n(i)n
H
(i)] = σ
2
I, (.)
T
and (.)
H
denote transpose and Hermitian transpose, respec-
tively, E[.] stands for ensemble average, b
k
(i) ∈{±1+j0} is
the symbol for user k, the amplitude of user k is A
k
, the user
k channel vector is h
k
(i)=[h
k,0
(i) ...h
k,L
p
−1
(i)]
T
with
h
k,l
(i)=h
k,l
(iT
c
) for l =0,...,L
p
− 1, the ISI is given by
η
k
(i)=A
k
b
k
(i −1)
¯
C
k
h
k
(i −1) + A
k
b
k
(i +1)
˘
C
k
h
k
(i +1)
and assumes that the channel order is not greater than N ,
i.e. L
p
− 1 ≤ N, s
k
=[a
k
(1) ...a
k
(N)]
T
is the signature
sequence for user k and p
k
(i)=C
k
h
k
(i) is the effective
signature sequence for user k,theM ×L
p
convolution matrix
C
k
contains one-chip shifted versions of s
k
and the M × L
p
matrices
¯
C
k
and
˘
C
k
with segments of s
k
have the following
structure
C
k
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
k
(1) 0 ... 0
.
.
. a
k
(1)
.
.
.
.
.
.
a
k
(N)
.
.
.
.
.
.
0
0 a
k
(N)
.
.
.
a
k
(1)
.
.
.
.
.
.
.
.
.
.
.
.
00
.
.
.
a
k
(N)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
¯
C
k
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 a
k
(N) ... a
k
(N − L
p
+1)
.
.
. 0
.
.
.
.
.
.
0
.
.
.
.
.
.
a
k
(N)
.
.
. 0
.
.
.
0
0
.
.
.
.
.
.
0
00... 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
˘
C
k
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 ... 00
.
.
. ...
.
.
.
.
.
.
0 ... 00
a
k
(1)
.
.
.
00
.
.
.
.
.
.
.
.
.
.
.
.
a
k
(L
p
− 1) ... a
k
(1) 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
The MAI comes from the non-orthogonality between the
received signature sequences, whereas the ISI span L
s
depends
on the length of the channel response, which is related to the
length of the chip sequence. For L
p
=1,L
s
=1(no ISI),
for 1 <L
p
≤ N, L
s
=2,for N<L
p
≤ 2N,L
s
=3.
780 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 5, MAY 2008
III. MMSE DECISION FEEDBACK RECEIVERS
Let us describe in this section the design of syn-
chronous MMSE decision feedback detectors. The input to
the hard decision device corresponding to the ith symbol is
z(i)=W
H
(i)r(i) − F
H
(i)
ˆ
b(i), where the input z(i)=
[z
1
(i) ... z
K
(i)]
T
, W(i)=[w
1
... w
K
] is M × K the
feedforward matrix,
ˆ
b(i)=[b
1
(i) ... b
K
(i)]
T
is the K × 1
vector of estimated symbols, which are fed back through the
K × K feedback matrix F(i)=[f
1
(i) ... f
K
(i)]. Generally,
the DF receiver design is equivalent to determining for user k
a feedforward filter w
k
(i) with M elements and a feedback
one f
k
(i) with K elements that provide an estimate of the
desired symbol:
z
k
(i)=w
H
k
(i)r(i) − f
H
k
(i)
ˆ
b(i) ,k=1, 2,...,K (4)
where
ˆ
b(i)=sgn[ℜ(W
H
r(i))] is the vector with initial deci-
sions provided by the linear section, w
k
and f
k
are optimized
by the MMSE criterion. In particular, the feedback filter f
k
(i)
of user k has a number of non-zero coefficients corresponding
to the available number of feedback connections for each type
of cancellation structure. The final detected symbol is:
ˆ
b
f
k
(i)=sgn
ℜ
z
k
(i)
=sgn
ℜ
w
H
k
(i)r(i) − f
H
k
(i)
ˆ
b(i)
(5)
where the operator (.)
H
denotes Hermitian transpose, ℜ(.)
selects the real part and sgn(.) is the signum function.
To describe the optimal MMSE filters we will initially
assume perfect feedback, that is
ˆ
b = b, and then will
consider a more general framework. Consider the following
cost function:
J
MSE
= E
|b
k
(i) − w
H
k
r(i)+f
H
k
b(i)|
2
(6)
Let us divide the users into two sets, similarly to [14]
D = {j :
ˆ
b
j
is fed back } (7)
U = {j : j/∈ D} (8)
where the two sets D and U correspond to detected and
undetected users, respectively. Let us also define the matrices
of effective spreading sequences P =[p
1
... p
K
], P
D
=
[p
1
... p
D
] and P
U
=[p
1
... p
U
]. The minimization of
the cost function in (6) with respect to the filters w
k
and f
k
yields:
w
k
= R
−1
U
p
k
(9)
f
k
= P
H
D
w
k
(10)
where the associated covariance matrices are R =
E[r(i)r
H
(i)] = PP
H
+ σ
2
I, R
U
= P
U
P
H
U
+ σ
2
I =
R −P
D
P
H
D
. Thus, assuming perfect feedback and that user k
is the desired one, the associated MMSE for the DF receiver
is given by:
J
MMSE
= σ
2
b
− p
H
k
R
−1
U
p
k
(11)
where σ
2
b
= E[|b
2
k
(i)|]. The result in (11) means that in the
absence of error propagation, the MAI in set D is eliminated
and user k is only affected by interferers in set U .
For the successive interference cancellation DF (S-DF)
detector , we have for user k
D = {1, ... ,k− 1},U= {k, ... ,K} (12)
where the filter matrix F(i) is strictly upper triangular. The
S-DF structure is optimal in the sense of that it achieves
the sum capacity of the synchronous CDMA channel with
AWGN [10]. In addition, the S-DF scheme is less affected
by error propagation although it generally does not provide
uniform performance over the user population. In order to
design the S-DF receivers and satisfy the constraints of the S-
DF structure, the designer must obtain the vector with initial
decisions
ˆ
b(i)=sgn[ℜ(W
H
(i)r(i))] and then resort to the
following cancellation approach. The non-zero part of the filter
f
k
corresponds to the number of used feedback connections
and to the users to be cancelled. For the S-DF, the number
of feedback elements and their associated number of non-zero
filter coefficients in f
k
(where k goes from the second detected
user to the last one) range from 1 to K − 1.
The parallel interference cancellation DF (P-DF) [14] re-
ceiver can offer uniform performance over the users but it
suffers from error propagation. For the P-DF in a single cell,
we have [14]
D = {1, ... ,k− 1 k +1, ...,K},U= {k} (13)
w
k
= R
−1
U
p
k
=
p
k
A
2
k
+ σ
2
(14)
The MMSE associated with the P-DF system is obtained by
substituting R
U
= R − P
D
P
H
D
into (9), which yields:
J
MMSE
= σ
2
b
− p
H
k
(p
k
p
H
k
+ σ
2
I)
−1
p
k
=
σ
2
A
2
k
+ σ
2
(15)
where for P-DF F(i) is full and constrained to have zeros
along the diagonal to avoid cancelling the desired symbols. In
order to design P-DF receivers and satisfy their constraints,
the designer must obtain the vector with initial decisions
ˆ
b(i)=sgn[ℜ(W
H
(i)r(i))] and then resort to the following
cancellation approach. The non-zero part of the filter f
k
corresponds to the number of used feedback connections and
to the users to be cancelled. For the P-DF, the feedback
connections used and their associated number of non-zero
filter coefficients in f
k
are equal to K −1 for all users and the
matrix F(i) has zeros on the main diagonal to avoid cancelling
the desired symbols.
Now let us consider a more general framework, where the
feedback is not perfect. The minimization of the cost function
in (4) with respect to w
k
and f
k
leads to the following filter
expressions:
w
k
= R
−1
(p
k
+ Bf
k
) (16)
f
k
=(E[
ˆ
b
ˆ
b
H
])
−1
B
H
w
k
≈ B
H
w
k
(17)
where E[
ˆ
b
ˆ
b
H
] ≈ I for small error rates and B =
E[r(i)
ˆ
b
H
(i)]. The associated MMSE for DF receivers subject
to E[
ˆ
b
ˆ
b
H
] ≈ I and imperfect feedback is approximately given
by
J
MMSE
≈ σ
2
b
− p
H
k
R
−1
p
k
− p
H
k
R
−1
Bf
k
(18)
In Appendix I we show that the expression in (18) equals (11)
under perfect feedback, and provide several other relationships
between DF structure with and without perfect feedback. Note
that the MMSE associated with DF receivers that are subject
to imperfect feedback depends on the matrix B = E[r
ˆ
b
H
],
that under perfect feedback equals P
D
, and the feedback
