I. J. Communications, Network and System Sciences. 2008; 1: 1-103
Published Online February 2008 in SciRes (http://www.SRPublishing.org/journal/ijcns/).
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
PAPR Reduction Scheme with SOCP for MIMO-
OFDM Systems
Basel RIHAWI, Yves LOUET
SUPELEC - IETR
Avenue de la Boulaie CS 47601 35576 Cesson Sévigné, France
Abstract
Combination of multiple-input multiple-output (MIMO) with orthogonal frequency division multiplexing (OFDM)
has become a promising candidate for high performance wireless communications. However one major disadvantage of
MIMO-OFDM systems lies in a prohibitively large peak-to-average power ratio (PAPR) of the transmitted signal on
each antenna. In this paper we extend from SISO to MIMO systems a method based on allocating dedicated subcarriers
for PAPR mitigation. These subcarriers are located on unused subcarriers of OFDM spectrum under the assumption
they all fall under the power mask. This is originally implemented with a SOCP optimization algorithm applied before
space time coding scheme. This jointly mitigates PAPR on each MIMO branch scheme. This approach does not
degrade the bit-error-rate (BER) and the data bit rate and no side information (SI) transmission is required. Simulation
results are presented in the IEEE 802.16 WiMAX standard contexts: an Alamouti space time code with two transmitted
antennas and 256 OFDM subcarriers are considered where 56 of which are unused and allocated for PAPR reduction.
PAPR gains up to 7dB are obtained depending on mean power increase limitation. Moreover, with a spectrum mask
constraint, this method is standard compliant.
Keywords: MIMO-OFDM, PAPR, SOCP
1. Introduction
MIMO radio systems have attracted considerable
interest and have been studied intensively since Telatar [3]
due to its potential of achieving high data rates in
multipath channel. It is shown that when multiple
transmit and receive antennas are used to form a MIMO
system, the system capacity can be improved by a factor
given by the minimum between transmitter and receiver
antennas number, compared to a single-input single-
output (SISO) system with flat Rayleigh fading or
narrowband channels [4] For high data rate wireless
wideband applications, MIMO combined with orthogonal
frequency division multiplexing (OFDM) is being
considered in a large number of current technology
applications as in IEEE 802.16 WiMAX standard for
example.
OFDM system has great advantages of having simple
equalization and capability of transmitting high data bit
rates over frequency selective channels. Nevertheless, its
associated signal exhibits high PAPR. To counteract this
well known problem, many PAPR reduction techniques
have been proposed in the literature [5].
In the same way, OFDM and MIMO association
(known as MIMO-OFDM) has faced the similar
difficulties. In that case, large peaks can occur in any
transmitted branch of the MIMO system. Some recent
works have investigated MIMO-OFDM PAPR reduction
via selective mapping [6] cross-antenna rotation and
inversion [7], unitary rotation [8], polyphase interleaving
and inversion [9] or optimal PAPR reduction [10]. All
these methods imply SI transmission so as to recover
useful data at the receiver which has to be inevitably
modified if one of these methods is implemented. To
avoid SI transmission, dedicated subcarriers have been
generated for PAPR mitigation. In [2], unused subcarriers
of OFDM based standards have been allocated and
optimized for PAPR reduction under the assumption that
they all fall under spectrum mask requirements. The
receiver does not take these subcarriers into account and
so needs no modification. It has been applied in a single
input single output (SISO) context and with a SOCP
optimization algorithm, what constitutes a natural
extension of the well know Tone Reservation (TR)
method [1].
In this paper, we apply SOCP algorithm (discussed in
[2] for SISO-OFDM) to MIMO-OFDM systems. The
objective is to mitigate the PAPR without any BER and
date bit rate degradation and without sending any SI.

30 B. RIHAVI ET AL.
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
These dedicated subcarriers reveal mean power increase
which should be controlled associated with spectrum
mask requirements.
This paper is organized as follows. Section 2 defines
OFDM signal, MIMO-OFDM systems and PAPR for
MIMO-OFDM used in this study. Section 3 shows how
the problem of PAPR reduction can be formulated as an
optimization problem for MIMO-OFDM systems. Section
4 is devoted to simulation results in the IEEE 802.16
WiMAX standard context and exposes the constraints
used for SOCP. Finally, Section 5 summarizes and
concludes the paper.
2. OFDM Signal and PAPR Definitions for
MIMO-OFDM
Before proceeding further, let us define the notations
used throughout the paper. Time and frequency domain
vectors are denoted by small and capital bold case letters
respectively. The matrices are denoted by capital italic
bold case letters.
In OFDM modulation, a block of Ndata
symbols )1,...,1,0( −= NkX k, of vector X, will be
transmitted in parallel such that each symbol (k X)
modulates a subcarrier )1,...,1,0( −
=
Nkfk. The N
subcarriers are orthogonal iffkfk∆= , where
Tfk/1=and
T
is the OFDM symbol period. The
resulting analog OFDM base band signal )(tx can be
expressed as
[] .,0,
1
)(
1
0
2
∑
−
=
∈= N
k
tfj
kTteX
N
txk
π
(1)
In practice, the frequency complex symbol vector X is
transmitted into a discrete-time signal x = [x0,x1,…,xN-1]
via an inverse discrete Fourier transform (IDFT) i.e.
).(Xx IDFT= (2)
Unfortunately, because of the statistical independence
of all subcarriers, time-domain samples are approximately
complex Gaussian distributed what results in high
amplitude values. This is characterized by PAPR of signal
x(t) defined by
{}
{}
,
)(
)( max
)( 2
2
],0[
txE
tx
txPAPRTt∈ = (3)
where E{.} denotes the expectation operation. To
compute easily PAPR, signal x(t) is sampled to get
∑
−
=−== 1 0
2,1,...,0,
1N
k
NL
kn
j
kn LNneX
N
x
π
(4)
where L is an oversampling factor. It has been proved that
L=4 is sufficient for capturing the continuous-time peaks
[11]. Then, the resulting temporal signal x is
,Xx L
Q
=
(5)
where QL is the IDFT matrix of size NL scaled by L (see
Appendix 6.2). Subsequently, PAPR of x is defined as
{}
{}
.
x
max
x2
2
10
E
x
PAPR k
NLk −≤≤
= (6)
Figure 1. Structure of the two antennas MIMO-OFDM
system
In the rest of this paper, all vectors are supposed to be
oversampled by factor L. Moreover, we assume a space-
time block code (STBC) for the studied MIMO-OFDM
system (see Figure 1) that employs Alamouti scheme [12].
In this case,
].......[ *
12
*
2
1
2
*
101 −−
−−−−= LNLNLNLN XXXXXXX
and
].......[ *
21
*
1
22
*
012 −−
−
=LNLNLNLN XXXXXXX
are the two outputs of Alamouti STBC (see Appendix
6.1). C is the corrective signal added to X in frequency
domain in order to reduce PAPR (see Figure 2).
],......[ 12
2
1
2
10 −−
−
=
LNLNLNLNCCCCCCCC1 and C2
are generated by the same STBC. To avoid in and out of
band distortions due to non linear power amplification,
PAPR of all transmit signals should be simultaneously as
small as possible. Since performance is governed by the
worst-case PAPR, we define MIMO-OFDM PAPR
PAPRMIMO as the maximum of all PAPR related to all NT
MIMO paths [14]. Subsequently,
{} .max
,....,1 i Ni
MIMO PAPRPAPR
T
x
=
=
(7)
PAPR being a random variable, an appropriate
description is its complementary cumulative distribution
function (CCDF). It gives the probability P0 of exceeding
a given threshold PAPR0, i.e.,
).( 00 PAPRPAPRP>
=
Pr (8)

PAPR REDUCTION SCHEME WITH SOCP FOR MIMO-OFDM SYSTEMS 31
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
The proposed strategy for PAPR reduction is to search
an additive corrective signal c in order to verify
PAPR(x+c) < PAPR(x). One way to find c is to use an
optimization approach, what is detailed in next section.
3. Minimizing PAPR with SOCP Technique
3.1. General Principle of the Propose Method
PAPR mitigation principle used through out this paper
is to add artificial subcarriers instead of unused ones. The
signal addition is achieved in frequency domain as
presented in [2] and illustrated in Figure 2.
Thus, PAPR of the resulting signal (x+c) is given by
Figure 2. Principle of adding corrective subcarriers instead
of unused ones for PAPR mitigation
{}
{}
.
cx
max
cx2
2
10
+
+
=+ −≤≤
E
cx
PAPR kk
NLk (9)
Ideal computing would be to mitigate PAPR by
minimizing the maximum peak of the combining signal
(x+c=x+QLC) while keeping constant its average power.
This objective can be formulated as
,maxmin ,C
row
Lkk
k
Cqx+ (10)
where row
Lk
q,is the k-th row of QL.
Such a minimization problem given by Eq. 10 can be
formulated as SOCP much easier than in semi definite
program (SDP) or quadratically constrained quadratic
program (QCQP) [13]. SOCP is a convex optimization
problem class that minimizes a linear function over the
intersection of an affine set and the product of second-
order (quadratic) cones [13].
3.2. SOCP Approach
Tone reservation (TR) is an efficient technique to
reduce the PAPR of a multicarrier signal [1]. This method
is based on adding a data-block-dependent time domain
signal to the original multicarrier signal to reduce its
peaks. This time domain signal can be easily computed at
the transmitter and stripped off at the receiver.
For this technique, the transmitter optimizes a given
subset of subcarriers for PAPR reduction. The objective
is to find the time domain signal to be added to the
original time domain signal x. If we add a frequency
domain vector ],...,,[ 1
10 −
=LN CCCC to X, the new
time domain signal can be represented as x+c =
IDFT(X+C), where c is the time domain signal due to C.
The TR technique restricts the data block X and peak
reduction vector C to lie in disjoint frequency subspaces,
i.e.
{}
{}
⎩
⎨
⎧
∉=
∈=
.,...,,,0
,...,,,0
21
21
vn
vn
iiinC
iiinX
The v nonzero positions in C are called peak reduction
carriers. Due to orthogonality, these additional subcarriers
cause no distortion on the useful data subcarriers. To find
the value of
{
}
vn iiinC ,...,,, 21
∈
, we must solve a convex
optimization problem that can easily be modeled as a
linear programming (LP) problem. To reduce the
computational complexity of LP, a simple gradient
algorithm is proposed in [1]. Nevertheless, TR proposed
in [1] allocates subcarriers among those which carry
useful information. This results in a data rate loss. In [2]
TR has been extended to unused subcarriers of standards
with SOCP algorithm. Significant PAPR reduction gains
have been obtained, under spectrum mask constraints (all
corrective subcarriers must fall under the spectrum mask)
and with mean power increase control.
3.3. Application of MIMO-OFDM
From Figure 1, signals 1 x and 2 xat the outputs of the
OFDM modulators are expressed as
),( 2
*
1
1
ACACx
cxx
1
11
•−•+=
+=
L
Q
(11)
),( 21
*' ACACx
cxx
2
22
2
•+•+=
+=
L
Q
(12)
where (.)* denotes complex conjugate, •the element wise
(dot) product, '
L
Qthe matrix deduced from L Qby pair
and odd rows permutations of L Q(see Appendix
6.2), T ]10...1010[
1=Aand T
]01...1010[
2=ANL×1
vectors (T denotes the transpose of the vectors).
Now PAPR of signals to be transmitted are
{}
{}
,
cx
max
x2
11
2
11
1+
+
=
E
cx
PAPR kk
k (13)

32 B. RIHAVI ET AL.
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
{}
{}
.
cx
max
x2
22
2
22
2+
+
=
E
cx
PAPR kk
k (14)
According to Eq.(7), PAPR of MIMO system is
{
}
{
}
{
}
. x,x max21 PAPRPAPRPAPRMIMO= (15)
The objective of this method is to reduce MIMO PAPR
by jointly reducing
{
}
1
xPAPR and
{
}
2
xPAPR . At first,
SOCP algorithm does not take into account the mean
power increase. So, the problem of PAPR minimization
can be formulated as
,)(min
minmaxmin
2
2
*
1
2
1
2
11
11
∞
∞
•−•+=
+=+
ACACx
cx
1
C
1
Cc
L
kk
k
Q
cx (16)
.)(min
minmaxmin
2
21
*'
2
2
2
22
22
∞
∞
•+•+=
+=+
ACACx
cx
2
C
2
c
L
C
kk
k
Q
cx
(17)
Eqs.(16) and (17) are convex optimization problems
formulated as one with SOCP :
)18(.)(
)(
min
21
*'
2
*
1
β
β
β
≤•+•+
≤•−•+
∞
∞
ACACx
ACACx
2
1
L
L
Q
Qtosubject
SOCP algorithm provides optimized solution Copt
according to Eq.(18). Nevertheless, the two resulting
signals )2,1( =+ iQ optLCxi have larger mean powers
compared to x what has to be taken into account for
practical applications. Moreover, Copt components must
fall under the spectrum mask requirements to make the
signal standard compliant. These two points are now
discussed.
4. Constraints Statement and Simulation
Results
4.1. Mean Power Increase Constraint
Figure 3 shows PAPR CCDF of original and
optimized signals for randomly generated QPSK symbols.
MIMO-OFDM system uses two antennas with N=256
subcarriers per antenna. 56 unused subcarriers are used as
the corrective signal. This system is inspired from IEEE
802.16 WiMAX standard. We set the oversampling factor
L to a value of 4. As we can see, PAPR is reduced by 7dB
for a 10-3 exceed probability level without any mean
power constraint.
Figure 3. PAPR reduction using SOCP without any
constraint in IEEE 802.16 context
However, adding a signal c to an original signal x to
reduce its PAPR increases the mean transmit power. The
relative mean increase power ∆E is defined as [1]
.
)(
)(
log10 2
2
10 x
cx
E
E
E+ =∆ (19)
This parameter must be as small as possible in order to
avoid power amplification saturation. Indeed, it is easy to
understand that if average power increases indefinitely
x+c would obviously have a resulting PAPR of 0dB but
the signal cannot be transmitted. Thus, the relative mean
power must be upper bounded and can be written as
,
γ
≤
∆
dB
E (20)
what implies
)()( 22 xcx EE
λ ≤+ , (21)
where 10
10
γ
λ =.γ is a constant related to the power
amplifier characteristics. The above condition can be
added as a constraint in the optimization problem. By
taking these conditions into account, SOCP algorithm
becomes
,)(
)22()(
)(
)(
min
221
*'
12
*
1
21
*'
2
*
1
KQ
KQ
Q
Qtosubject
L
L
L
L
λ
λ
β
β
β
≤•+•+
≤•−•+
≤•+•+
≤•−•+
∞
∞
∞
∞
ACACx
ACACx
ACACx
ACACx
2
1
2
1
where ).2,1(),(. 2== iENLKii x

PAPR REDUCTION SCHEME WITH SOCP FOR MIMO-OFDM SYSTEMS 33
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
Then, it has been shown that if γ=0.2 dB, the
additional allocated power for PAPR reduction represents
4.7% of the total amount. Figure 4 plots CCDFs of PAPR
for several γ values. As we can see, PAPR decreases as γ
increases.
Figure 4. PAPR reduction with SOCP with mean power
constraint
Nevertheless, even if mean power control is mandatory
as explained above, optimized subcarriers for PAPR
reduction has to fit spectrum requirements referring to
standard masks. It is clear that in some cases, x+c values
exceed spectrum mask as shown in Figure 5 for γ=0.2dB.
Subsequently, an additional constraint has to be included
in SOCP formulation related to spectrum mask, which is
considered in next subsection.
Figure 5. OFDM power spectral density (PSD) and IEEE
802.16 mask under mean power constraint (γ=0.2dB)
4.2. Spectrum Mask Constraint
A second constraint is added to SOCP. It concerns
spectrum mask and is formulated as
,,
,Ω∈≤ nC nni
δ
(23)
where i = {1,2}, δ refers to spectrum mask values and Ω
is the set of all allocated subcarriers indexes.
Figure 6. PAPR reduction with mean power and spectrum
mask constraints compared to that with only mean power
constraints γ=0.2dB in both cases
Figure 6 compares performance of two SOCP schemes:
the first one considers only mean power constraint and
the second one considers both mean power and spectrum
mask constraints. In both cases the mean power limitation
is the same (γ=0.2dB). As a result, it is shown that this
additional mask constraint slightly degrades PAPR
compared to the case where only mean power constraint
is considered but with the gain that spectrum
requirements are fulfilled. Figure 7 shows the power
density function of the OFDM signal with IEEE 802.16
spectrum mask when the two constraints are considered.
As expected, the resulting spectrum respects the mask.
Figure 7. OFDM power spectral density (PSD) and IEEE
802.16 mask under mean power (γ=0.2dB) and spectrum
mask constraints
5. Conclusion
In this paper, we have proposed a novel PAPR

34 B. RIHAVI ET AL.
Copyright © 2008 SciRes. I. J. Communications, Network and System Sciences. 2008; 1:1-103
reduction approach for MIMO-OFDM. It is based on
SOCP and provides significant PAPR reduction gains
without transmitting any side information and without
degrading BER and data rate at the same time. To do so,
we have used the unused subcarriers of standards to
generate the corrective signal that jointly mitigates the
PAPR of the MIMO-OFDM scheme. We have shown that
PAPR can be reduced by 7dB at the 10-3 probability
exceed level. To avoid an uncontrolled increase of the
relative mean power, we have included a mean power
constraint in SOCP algorithm. Finally, to be standard
compliant, the allocated subcarriers for PAPR reduction
have to fall under the spectrum mask, which has been
done in IEEE 802.16 WiMAX standard context.
6. Appendices
6.1. Alamouti STBC
Figure 8. Alamouti STBC for 2 antennas
6.2. Fourier matrices QL and Q’L
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
−−−
−−−
−
)1)(1(1).1(
2
)1)(2(1).2(
2
)1.(1
2
1.1
2
1
1
1
111
1
LNLN
LN
jLN
LN
j
LNLN
LN
jLN
LN
j
LN
LN
j
LN
j
L
ee
ee
ee
LN
Q
ππ
ππ
ππ
L
L
MOMM
L
L
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
−−−
−−−
−
)1)(2(1).2(
2
)1)(1(1).1(
2
)1.(1
2
1.1
2
'
1
1
111
1
1
LNLN
LN
jLN
LN
j
LNLN
LN
jLN
LN
j
LN
LN
j
LN
j
L
ee
ee
ee
LN
Q
ππ
ππ
ππ
L
L
MOMM
L
L
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PAPR REDUCTION SCHEME WITH SOCP FOR MIMO-OFDM SYSTEMS 35
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