COMPLEXITY ANALYSIS OF A HYBRID THRESHOLD-BASED
ZF-MMSE EQUALIZER FOR SIC-BASED
MIMO-OFDM RECEIVERS
Nikolaos I. Miridakis and Dimitrios D. Vergados
Department of Informatics, University of Piraeus, 80 Karaoli & Dimitriou St., Athens, Greece
Keywords: Hybrid Successive Interference Cancellation (Hybrid SIC), Computational Complexity.
Abstract: In this paper, we propose a new reception scheme for MIMO-OFDM systems, which can switch between
two different equalization methodologies, accordingly, on a subcarrier basis. In fact, the proposed receiver
is based on the ordered successive interference cancellation (SIC) technique and uses jointly zero forcing
SIC (ZF-SIC) and minimum mean squared error SIC (MMSE-SIC) according to a defined threshold value.
The modulation scheme considered in this study is QPSK. Our main objective, here, is the provision of the
computational complexity efficiency. Hence, a complexity analysis is provided and upper and lower
complexity bounds for the proposed scheme are also derived.
1 INTRODUCTION
Orthogonal frequency division multiplexing
(OFDM) is proposed as one of the key technologies
for modulation and signal propagation. On the other
hand, multiple-input multiple-output (MIMO)
technologies hold the premise of achieving
significant performance improvement and capacity
enhancement in such systems. Moreover, MIMO
fading channels can be explored to provide either
spatial diversity gain (SD) in order to enhance the
system robustness, or spatial multiplexing gain (SM)
to the scope of the system capacity increase and the
transmission gain. Due to the complementary
benefits of MIMO and OFDM, the realization of
MIMO-OFDM systems is, therefore, of a great
importance (Lee et al, 2006) to ensure both the
effectiveness and the reliability of future service
demands in modern wireless ad hoc networks.
The interference effect represents a major
efficiency inhibitor while induces a typical upper
bound to the performance of MIMO-OFDM
systems. Successive interference cancellation (SIC)
represents quite an effective methodology, which
tends to counteract the later limitation. The
equalization methodology used in conjunction with
the SIC-based reception, determines both the
reliability of SIC, in terms of the bit-error-rate
(BER) performance, and the overall computational
complexity of the process. The maximum likelihood
(ML) criterion is the most optimal yet most
demanding equalization technique, in terms of the
BER performance and the computational
complexity, respectively. Particularly, the
computational burden of an ML equalizer is found to
be overwhelmed for most practical applications
(Verdú, 1998). Hence, most of the research
community has rather focused on suboptimal
equalization techniques, based on either zero forcing
(ZF) or minimum mean squared error (MMSE)
detection. Generally, in a noisy environment,
MMSE-SIC is more error resilient than ZF-SIC at
the cost of a higher computational complexity and
vice versa (Kim et al, 2009).
We focus, in this paper, on the performance of
SIC for MIMO-OFDM systems and, more
specifically, we propose a joint detection
methodology based on ZF and MMSE equalization.
Upon a signal decoding, the decision statistic, which
determines whether a symbol will be carried out by a
ZF-SIC or by an MMSE-SIC, exclusively, depends
on a threshold. The value of the threshold is defined
on an OFDM frame (or block) basis by taking into
account the statistics extracted from the received
information on each subcarrier. In this study we
focus on multiuser MIMO-OFDM systems where all
the transmissions established using QPSK
modulation schemes. For the best of our knowledge,
109
I. Miridakis N. and D. Vergados D..
COMPLEXITY ANALYSIS OF A HYBRID THRESHOLD-BASED ZF-MMSE EQUALIZER FOR SIC-BASED MIMO-OFDM RECEIVERS.
DOI: 10.5220/0003614101090112
In Proceedings of the International Conference on Wireless Information Networks and Systems (WINSYS-2011), pages 109-112
ISBN: 978-989-8425-73-7
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
a hybrid ZF-MMSE-SIC reception methodology has
never been proposed for MIMO-OFDM
implementations.
2 SYSTEM MODEL
We consider a MIMO-OFDM system consisting of
N
C
subcarriers with N
T
and N
R
( N
T
) transmit and
receive antennas, respectively.
After the CP removal, the receiver
conventionally performs fast Fourier transform
(FFT) at each subcarrier, which yields to the actual
received signal expressed as
nnnn
+y=Hx w
(1)
where
=
+
+⋯+

and
=

+
+⋯+

represent the received and
the transmit signal vector, respectively, at the n-th
subcarrier. Also, the
=
+
+⋯+

is a zero-mean complex AWGN vector with
covariance matrix given as
Φ
w
= {
() ()
H
nn
ww
} =
2
R
wN
σ I
,
(2)
where
represents the noise variance introduced
by the communication channel and
R
N
I
is the N
R
×N
R
identity matrix. We also provide a compact input-
output relation between all the OFDM subcarriers at
all the transmit and receive antennas of all the
network transmitting nodes, which is expressed as
1
0
C
N
nnn n
n
=
+=
Y= H Px w HXP+W
(3)
where
(0) (1) ( 1)
[ , ,..., ]
C
TT TT
N
=Yy y y
,
(0) (1) ( 1)
[ , ,..., ]
C
TT TT
N
=Xx x x
,
(0) (1) ( 1)
[ , ,..., ]
C
TT TT
N
=PP P P
,
(0) (1) ( 1)
[ , ,..., ]
C
TT TT
N
=Ww w w
(0) (1) ( 1)
[ , ,..., ]
C
T
N
=HHH H
.
Note that
H
is a multi-block diagonal matrix,
assuming that both ISI and ICI which are caused
mainly due to time or frequency offsets, are
perfectly compensated by an appropriate CP length.
3 PROPOSED SCHEME
On MIMO-OFDM infrastructures, the ZF
equalization leads to the noise enhancement because
the pseudoinverse channel matrix is not always
added destructively and, hence, it could result to the
potentially colored additive noise at the receiver.
Moreover, the diversity gain provided by the
multiple receive antenna array for the interference
suppression is eliminated along with the channel
matrix coefficients by exploiting the ZF equalizer in
MIMO channels, which results in a lower overall
diversity order (Zhang et al, 2009). Particularly, the
selected stream for detection at the first SIC step has
a diversity gain of N
R
, while the stream at the last
SIC step has a diversity gain of N
R
N
T
+ 1, which
is a rather undesirable condition.
On the other hand, in case of MMSE-SIC, the
interference is not totally removed. However, the
imperfect interference cancellation is compensated
by providing a higher diversity performance in the
decoding process. Moreover, MMSE does not
enhance the noise coefficients in comparison to the
respective ZF equalization, whereas the higher
diversity order tenet is found to be beneficial,
especially in low SINR regions (Zijian et al, 2008).
It is straightforward that a conventional MMSE
equalizer results to a lower BER probability in
comparison to a respective ZF one. Nevertheless, the
higher computational cost is a fundamental
prerequisite for the enhanced BER performance.
Hence, the selection of the appropriate equalizer is
still debatable, depending mostly on the users’ QoS
requirements or the network manufacturer.
Taking into account the benefits and the
drawbacks of the abovementioned equalizers, we
propose a novel framework for MIMO-OFDM
systems, which is based on a hybrid detection-
switching technique, associated with the
conventional SIC method. Since we are focusing on
a QPSK modulation, the signal is demodulated as
two independent BPSK signals in quadrature,
thereby only the real or the imaginary symbol part is
captured in each branch. The detection criterion is
tightly predetermined on an OFDM frame basis
according to a threshold value, gained by the
statistics collected upon the reception of an OFDM
frame. In this content and while the transmitting
symbols are independent of each other, we introduce
a threshold S, representing the mean amplitude of
the overall received signal (Marabissi et al, 2006),
which is expressed as
WINSYS 2011 - International Conference on Wireless Information Networks and Systems
110
() ()
()
() ()
()
() ()
()
1
0
1
1
00
1
1
1
C
C
R
CR
N
nn
n
CR
N
N
jj
nn
nj
CR
S
NN
NN
yy
NN
β
β
β
=
==
⎛⎞
=ℜ+
⎜⎟
⎝⎠
⎛⎞
=ℜ+
⎜⎟
⎝⎠
⎛⎞
=ℜ+
⎜⎟
⎝⎠
∑∑
YY
yy
(4)
where β is a constant which plays the role of a
tuning parameter.
Therefore, upon a signal reception
, the
decision statistic is determined by using ZF
detection if both ℜ
≥ S and ℑ
≥ S or by
using MMSE detection if ℜ
< S and ℑ
< S.
Moreover, switching the more reliable signals to ZF-
SIC and the less reliable ones to MMSE-SIC, we
also balance the overall complexity of the process.
Under this regime, we set the accuracy of the
reception process upon a reliability classification
basis. Furthermore, an appropriate balance on the
tradeoff between the performance efficiency and the
complexity reduction, depending always on the S
value, is accomplished.
4 COMPLEXITY ANALYSIS
We provide computational complexity upper and
lower bounds of the proposed scheme and we also
present a cross-analysis demonstration between the
proposed scheme and the quite complex MMSE-
SIC, in a MIMO-OFDM environment. The
computational complexity has been evaluated with
respect to the number of the expected floating point
operations (flops). Since the signal coefficients as
well as the channel matrix coefficients are complex-
valued, all the appropriate operations including
multiplications, additions and divisions, are
conducted upon complex values. In the rest of the
paper, when we discuss computational complexity
we refer to the complex operations (COs) including
only complex multiplications (CMs) and complex
additions (CAs). We count each complex CA as two
flops and each complex CM as six flops (Luo et al,
2007), (Gan et al, 2009), in order to evaluate the
overall computational complexity of the proposed
scheme.
From the property of Hermitian matrices, only
the half of the complexity can be obtained in
comparison to the complexity of computing a
general matrix of the same size. Typically, the main
computational burden is obtained by the number of
CMs. Computing H
H
H requires N
N
1
2
N
+
1
2
N
CMs, while
3
2
N
+
(
N
+1
)
N
+
1
2
N
CMs are needed in order to calculate (H
H
H+σ
2
)
-
1
H
H
, assuming that the real-valued
is priori
estimated at the receiver. However, the exact
number of COs for a matrix inversion event may
vary, depending mostly on how the channel
coefficients are handled. Several elimination
approaches (e.g. Gauss-Jordan elimination) or
advanced signal processing techniques based on
matrix decomposition methods (e.g. QR or LDL
H
decomposition), result to potentially different
computational burden at the cost of either BER
performance or hardware gain. We, hence,
approximate the complexity of the channel
inversion, with respect to the CAs, CMs and COs, as
O
(
). In case of a SIC reception, the channel
inversion procedure is suppressed due to the channel
nulling operation at each SIC step. Thus, the
computational burden at the i-th SIC step is obtained
as
O
(
(
−
)
).
The lower complexity bound of the proposed
scheme can be obtained as β 0, where all
transmitting symbols at all the OFDM subcarriers
are carried out by a ZF-SIC reception. Therefore, the
total number of COs in the case where N
T
= N
R
is
obtained as
CO
Lower Bound
=
(


2
+2
(
1−
2
)
+2
(
−+1)+
O
(
(
−
)
)))
(5)
Similarly, the upper complexity bound of the
proposed scheme can be obtained as β + , where
all transmitting symbols at all the OFDM subcarriers
are carried out by an MMSE-SIC reception,
exclusively. Hence, the total number of COs
considering the case of N
T
= N
R
, is obtained as
CO
Upper Bound
=
(3
+
(
2−6
)
+


(3
+2−)+
O
(
(
−
)
)))
(6)
In order to show the fraction of saved complexity
of the proposed scheme, we introduce the quotient
proposed
M
MSE
Flops
Flops
ξ
=
(7)
where Flops
proposed
denotes the total number of flops
for the proposed hybrid scheme and Flops
MMSE
denotes the respective number of flops for the
conventional MMSE-SIC. In fig. 1, the performance
of ξ is depicted with respect to β for varying number
of N
T
= N
R
antennas, considering a MIMO-OFDM
COMPLEXITY ANALYSIS OF A HYBRID THRESHOLD-BASED ZF-MMSE EQUALIZER FOR SIC-BASED
MIMO-OFDM RECEIVERS
111
network with 128 subcarriers. As an illustrative
example, in case of N
T
= N
R
= 6 and when β = 3.5,
the proposed scheme requires 0.78 × Flops
MMSE
number of flops in order to complete an OFDM
frame reception. Consequently, this reflects to a
strong reduction in the overall computational
complexity, since the proposed methodology does
not produce any additional overhead to the hardware
gear or to the system latency. Furthermore, it is
worth mentioning that the complexity efficiency of
the proposed scheme is a result of the diversity of
the detection approach, while the computational cost
for the decision of the most appropriate equalizer
depends on the calculation of S, which is negligibly
small.
Figure 1: Fraction of saved complexity of the proposed
scheme in comparison to the conventional MMSE-SIC for
different β values, in a MIMO-OFDM system with N
C
=
128. A small number of transmit and receive antennas is
considered in order to approach realistic scenarios,
suitable for practical implementations.
5 CONCLUSIONS
In this paper, we presented a new detection-
switching approach for SIC-based receivers in
MIMO-OFDM systems. The proposed scheme is
implemented by two well-known equalizers jointly.
More specifically, it switches between ZF and
MMSE equalization according to a certain threshold,
which is determined by the mean received amplitude
of the overall signal within an OFDM frame. All the
included transmissions have been implemented
under QPSK modulation alphabets. Upper and lower
asymptotic complexity bounds have derived through
a computational complexity analysis. We showed
that by applying the proposed detection switching
approach, up to 22% complexity savings can be
obtained. Some of our most important future aspects
are the study of the proposed hybrid SIC under
MIMO-OFCDM systems and a definition of an
appropriate threshold under QAM modulation
schemes.
ACKNOWLEDGEMENTS
This work is partly supported by UPRC.
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0.5 1 1.5 2 2.5 3 3.5 4
0.8
0.85
0.9
0.95
1
β
ξ
N
T
= N
R
= 2
N
T
= N
R
= 4
N
T
= N
R
= 6
N
T
= N
R
= 8
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