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

 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

subcarriers with N

and N

(≥ N

) 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

= {

() ()

} =

σ I

(2)

where







represents the noise variance introduced

by the communication channel and

is the N

×N

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

nnn n

−

∑

Y= H Px w HXP+W

(3)

where

(0) (1) ( 1)

[ , ,..., ]

TT TT

N −

=Yy y y

(0) (1) ( 1)

[ , ,..., ]

TT TT

N −

=Xx x x

(0) (1) ( 1)

[ , ,..., ]

TT TT

N −

=PP P P

(0) (1) ( 1)

[ , ,..., ]

TT TT

N −

=Ww w w

(0) (1) ( 1)

[ , ,..., ]

N −

=HHH H

Note that

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

, while the stream at the last

SIC step has a diversity gain of N

– N

+ 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

() ()

()

() ()

()

() ()

()

−

⎛⎞

=ℜ+ℑ

⎜⎟

⎝⎠

⎛⎞

=ℜ+ℑ

⎜⎟

⎝⎠

⎛⎞

=ℜ+ℑ

⎜⎟

⎝⎠

∑

∑∑

(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 requires N







−











CMs, while

N





(



)









CMs are needed in order to calculate (H

H+σ







)

, 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

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

(





). 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

(





−

)



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

= N

obtained as

Lower Bound

∑

(









2





+2





(

1−

2

)

+2



(



−+1)+

(





−

)



)))

(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

= N

, is obtained as

Upper Bound

∑

(3





+





(

2−6

)













(3



+2−)+

(





−

)



)))

(6)

In order to show the fraction of saved complexity

of the proposed scheme, we introduce the quotient

proposed

MSE

Flops

(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

= N

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

= N

= 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

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

= N

= 2

= N

= 4

= N

= 6

= N

= 8

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