Abstract
Multiple input multiple output orthogonal frequency division multiplexing (MIMO-OFDM)
systems have been proposed in the recent past for providing high-data rate services over wireless
channels. When combined with space time coding it provides the advantages of space-time
coding and OFDM, resulting in a spectrally efficient wideband communication system.
However, MIMO OFDM systems suffer with the problem of inherent high peak to average
power ratio (PAPR). In this paper, a generalized framework for PAPR reduction for space-time
coded OFDM systems based on modified PTS, with real and imaginary parts, separately
multiplied with phase factors is considered. To further reduce the PAPR, forward error-
correcting codes (FECs) such as Golay codes and Turbo codes are employed in the modified
PTS radix FFT, where, the PAPR is jointly optimized in both the real and imaginary part. The
simulation results show that the combined FEC with modified PTS technique significantly
provides better PAPR reduction with reduced computational complexity compared to ordinary
PTS with FEC for MIMO OFDM system.
1. Introduction
Recently space-time coded OFDM systems have been receiving wide spread attention. They
were mainly proposed in [1] to achieve data rates of 1.5-3 Mbps over a bandwidth of 1 MHz. In
[2], a space-time coded OFDM system capable of performing within 3 dB of the outage capacity
is proposed. In [3], space-time codes are designed for OFDM based local area networks (LAN)
and in [4] space-time codes have been designed for use with OFDM over frequency selective
channels, which can achieve a diversity order of the product of the number of transmit antennas,
number of receive antennas and the number of channel taps. A space-time-frequency coded
OFDM system which achieves maximum diversity is proposed in [5]. Despite its many
advantages, MIMO OFDM suffers with the problem of high PAPR and carrier frequency offset
sensitivity [6]. Hence, it is important to reduce the PAPR, otherwise, high power amplifiers
(HPA) in the transmitter need to have a linear region that is much larger than the average power,
which makes them expensive and inefficient. This is because if an HPA with a linear region
slightly greater than the average power is used, the saturation caused by the large peaks will
result in intermodulation distortion, which increases the bit error rate (BER) and causes spectral
widening, resulting in adjacent channel interference.
Although many techniques for reducing the PAPR of MIMO-OFDM systems have been
addressed, none of them have been developed yet. Earliest methods for PAPR reduction is the
use of clipping [7]. This is a very simple PAPR reduction technique but introduces in-band
distortion and out of band radiation, resulting in BER degradation. Other PAPR reduction
techniques include employing partial transmit sequences [8], and selective mapping [9]. Block
coding schemes for limiting the PAPR are also proposed. For example, by using complementary
Golay sequences one can limit the PAPR to only 3 dB [10]. Although, these codes are very
effective in reducing the PAPR, complexity remains the same as that of the conventional PAPR
reduction technique. A theoretical framework of PAPR reduction by channel coding is given in
[11]. Although there may be different frameworks for PAPR reduction for MIMO-OFDM,
here the case where the input to the space-time encoder is modified such that with real and
imaginary parts, separately multiplied with phase factors is considered. To further reduce the
PAPR, modified PTS is combined with forward error-correcting codes (FECs) such as Golay
codes and Turbo codes, where, the PAPR is jointly optimized in both the real and imaginary part.
In this paper, the main objective is to extend the modified PTS PAPR reduction techniques for
the single antenna transmission systems, to the case of MIMO systems.
2. PAPR IN MIMO-OFDM SYSTEM
Considering, N modulated data symbols from a particular signaling constellation, to create a
complex-valued symbol vector
( )
0, 1, 1
....
k N
X X X X
= ,
where,
k
X
is the complex value carried by the k
th
subcarrier of the m
th
transmit antenna.
The OFDM symbol can be written as
( )
0
1
2
0
1
N
j kf t
k
k
x t X e
N
t
=
=
, 0 t T s s (1)
where T is the symbol interval, and
0
f
=1/T is the frequency spacing between adjacent
subcarriers. Replacing t=n
b
T , where
b
T =T/N, gives the discrete time version denoted by
1
2 /
0
1
( )
N
j kn LN
k
k
x n X e
N
t
=
=
, n=0,1,NL-1 (2)
where, L is the oversampling factor. The symbol-spaced sampling sometimes misses some of
the signal peaks and results in optimistic results for the PAPR. The sampling can be
implemented by an IFFT.
The PAPR of the transmitted OFDM signal, x(t), is then given as the ratio of the peak
instantaneous power to the average power, written as
( )
2
0
2
max ( )
t T
x t
PAPR
E x t
s s
=
(
(3)
where E[] is the expectation operator.
From the central limit theorem, for large values of N, the real and imaginary values of ( ) x t
becomes Gaussian distributed. The amplitude of the OFDM signal, therefore, has a Rayleigh
distribution with zero mean and a variance of N times the variance of one complex sinusoid. The
complementary cumulative distribution function (CCDF) is the probability that the PAPR
exceeds a certain threshold
0
PAPR
.
0
( ( ( ))) P ( ( ( )))
r
CCDF PAPR x n PAPR x n PAPR = > (4)
Due to the independence of the N samples, the CCDF of the PAPR of SISO-OFDM as a data
block with Nyquist rate sampling is given by
0
0
P ( ( ( )) ) 1 (1 )
PAPR N
r
P PAPR x n PAPR e
= > = (5)
This expression assumes that the N time domain signal samples are mutually independent and
uncorrelated and it is not accurate for a small number of subcarriers. Therefore, there have been
many attempts to derive more accurate distribution of PAPR.
For a MIMO-OFDM system, analysis of the PAPR performance is the same as the SISO case on
each single antenna. For the entire system, the PAPR is defined as the maximum of PAPRs
among all transmit antennas [8], i.e.,
1
max
t
MIMO OFDM i
i M
PAPR PAPR
s s
= (6)
where
i
PAPR denotes the PAPR at the i
th
transmit antenna. Specifically, since in
MIMO-OFDM, M
t
N time domain samples are considered compared to N in SISO-OFDM, the
CCDF of the PAPR in MIMO-OFDM can be written as
0
0
( ) 1 (1 )
t
PAPR M N
r MIMO OFDM
P PAPR PAPR e
> = (7)
Comparing (7) with (5), it is evident that MIMO-OFDM results in even worse PAPR
performance than SISO-OFDM.
3. GOLAY SEQUENCES AND REED- MULLER CODE
3.1 Coding Theory
The binary complimentary sequences were proposed by M.J.E. Golay in 1961 [12]. The
complimentary sequences are sequences pairs for which the sum of aperiodic autocorrelation
functions is zero for all delay shifts. It was mentioned in [13] that the autocorrelation properties
of complemnentaty sequences can be used to construct the OFDM signal with low PAPR.
3.2 Complementary Sequence Theory
The pair of sequence x and y of length N, ie.,
| |
0 1 2 1
, , ,....
N
x x x x x
= and
| |
0 1 2 1
, , ,....
N
y y y y y
= ,
are said to be complementary if the following condition hold on the sum of both autocorrelation
functions:
1
0
( ) 2 ;
N
k k i k k i
k
x x y y N
+ +
=
+ =
0 i =
0 = ; 0 i = (8)
After taking the Fourier transform on both sides of Eq. (8) the above condition is translated into
the following equation.
2 2
( ) ( ) 2 X f Y f N + = (9)
where
( ) X f and
( ) Y f are the power spectrum of x and y respectively. From the spectral
condition of (9), it is observed that the maximum value of the power spectrum is bounded by 2N.
2
( ) 2 X f N s (10)
Because the average power of X(f) is equal to N, assuming that the power of the sequence x is
equal to 1, the PAPR of X(f) is bounded as
2
2
N
PAPR
N
s =
(=3 dB) (11)
Hence. using complimentary sequences as input to generate an OFDM symbol, it is guaranteed
that the maximum PAPR of 3dB can be achieved.
3.3 Error Correction using Complementry Code
In this work, complementary sequence to suppress the PAPR in the MIMO-OFDM systems is
considered. Complementary sequences are encoded by the generator matrix G
N
and b
N
[14]. Let
A
N
denotes the corresponding codeword sequences of length N and u is the integer sequences
between [0, M-1] of length k. Then A
N
can be written as
A
N
= u . G
N
+ b
N
(mod M), (12)
where G
N
is a k X N matrix and b
N
is a phase shift sequence of length N while k is related to
N=2
k-1
for k=3,4,5,If the M-ary PSK (Phase shift keying) for modulation. i
th
phase sequence of
A
N
can be given by
2
,
i i
a
M
t
| | = +A (13)
where | A is the arbitrary phase shift, and
i
a is the i
th
sequence of A
N
. Even if the integer
i
a to
the phase sequence
i
| using (13), the complementary property is not change.
For example, the generator matrices for QPSK modulation. N=8 are represented as
G
8
=
10030332
01010101
00110011
00001111
(
(
(
(
(
(14)
b
8
=
| | 00020020 (15)
Decoding method taking the advantage of the complementary code redundancy is also proposed
in [15]. It can be applied conventional syndrome decoding using parity-check matrix H
N
. The
syndrome is calculated from A
N
, which are received codeword sequences of length N, that is
( ) .
T
N N
S A b H = (16)
A most likely error pattern can be estimated from the calculated syndrome. For example, the
parity-check matrix of the generator matrix G
8
in (14) is derived as
H
s
=
13310000
13003100
10303010
23303001
(
(
(
(
(
(17)
The large sets of binary complementary pairs of length 2
m
can be obtained from the 2
nd
order
cosets of the well-known 1
st
order Reed-Muller code R(1, m). This Reed-Muller code results in
low PAPR in addition to its error-correcting capability.
The r
th
order Reed-Muller code is designated as R(r, m), where m is the parameter related to the
length of the code, n=2
m
, and 0 s r s m. Half of the codes of R(r, m) are complements of the
other half. R(1, m) is also known as a bi-orthogonal code since it can be obtained from the
generator matrix of an orthogonal code by adding all-ones codeword to it [16].
In this work, PTS is combined with Golay complementary sequence, PTS is based on combining
signal sub blocks which are phased-shifted by different phase factors to generate multiple
candidate signals, so as to select the low PAPR signal. In general, the procedure for PTS is
obtained as following.
First, consider the data block,
0, 1, 1
.....
N
X X X X
=
is encoded with space-time encoder and following
two vectors
1
X and
2
X is given by
* *
1 0 1 2 1
[ , ,..... , ],
N N
X X X X X
=
* *
2 1 0 1 2
[ , ,..... , ].
N N
X X X X X
=
Encode the data blocks by using Reed Muller code. Define the codeword as a vector,
* *
1 0 1 2 1
[ , ,..... , ] ,
T
N N
S C C C C
=
* *
2 1 0 1 2
[ , ,..... , ] .
T
N N
S C C C C
=
where,C is an encoded data such as Reed Muller code.
Secondly, S to be transmitted is divided into several sub-blocks, V, by using subblock partition
scheme. In general, subblock partition scheme can be classified into 3 categories. The three
partition methods are adjacent, interleaved and random.
S is partitioned into M disjoint sets, which is represented by the vector,
m
S , m=1,2,.., M (12)
In this work, the codeword vector S is partitioned by using adjacent method. Assume that the
subblocks or clusters consist of a contiguous set of subcarriers and are of equal size. The
objective is to optimally combine the M clusters, which in frequency domain is given by
'
1
M
m m
m
S b S
=
=
(13)
where, {b
m
, m=1, 2,, M} are weighting factors and are assumed to be perfect rotations. In other
words, the time domain is given by
1
M
m m
m
s b s
=
=
(14)
where,
m
s which is called the partial transmit sequence, is the IFFT of
m
S .The phase factors
m
b
are chosen to minimize the PAPR of s. By using 256 sub-carriers, M=4, Peak to Average Power
Ratio (PAPR) is reduced from 1% to more than 3 dB. PTS generates a signal with a low PAPR
through the addition of appropriately phase rotated signal parts. The codeword to be transmitted
are divided into several subblocks, V, of length N/V. Mathematically, expressed by
( )
1
V
v
k k
v
A A
=
=
(15)
All subcarriers positions in
( ) v
k
A
which are occupied in another subblock are set to zero. Each of
the blocks, v, has an IFFT performed on it,
{ }
( ) ( ) v v
n k
a IFFT A =
(16)
The output of each block (except for first block which is kept constant) is phase rotated by the
rotation factor as given by
( )
[0, 2 ]
j v
e
u
t e (17)
The blocks are then added together to produce alternate transmit signals containing the same
information as given by
( ) ( )
1
.
V
v j v
n n
v
a a e
u
=
=
(18)
Each alternate transmit signal is stored in memory and the process is repeated again with a
different phase rotation value. After a set number of phase rotation values, W, the OFDM symbol
with the lowest PAPR is transmitted as given by
2 3
, ,.... argmin(max )
v
n
a | | | = (19)
The weighting rotation parameter set is chosen to minimize the PAPR. The computational
complexity of PTS method depends on the number of phase rotation factors allowed. The phase
rotation factors can be selected from an infinite number of phases
( )
(0, 2 )
v
| t e
. But finding the best
weighting factors is indeed a complex problem.
By using multicarrier modulation technique, the crosstalk between the subcarriers should be
minimized. So it is required to maintain the orthogonality between the different modulated
carriers. The basic idea of PTS is to combine signal sub-blocks which are phased-shifted by
different phase factors to generate multiple candidate signals, so that the phase factor that results
in low PAPR can be selected. These phase factors combination correctly maintain the
orthogonality between the different modulated carriers. The coding method adds pattern of
redundancy to the input data in order to reduce the PAPR. In MIMO communication, data rate or
diversity order can be improved by exploiting the spatial dimension. In the same spirit, treating
the parallel transmit signals jointly, PAPR reduction may be improved. A modified PTS
technique with forward error correcting codes such as Golay complementary sequences with
Reed-Muller code is proposed to provide better PAPR reduction in the MIMO-OFDM systems
with lower computational complexity is shown in Figure 2.
4. TURBO CODING
In this work, a turbo encoder is employed which offer two advantages, significant PAPR
reduction and good bit error rate performance. Figure 1 shows the block diagram of turbo
encoder. Turbo codes [17] are parallel concatenated convolutional codes in which the
information bits are first encoded by a recursive systematic convolutional (RSC) code and then,
after passing the information bits through an interleaver, are encoded by a second RSC code. The
purpose of interleaving the coded data transforms burst error into independent errors. The result
of interleaving makes error burst to spread out in time, so that errors within a codeword appear to
be independent. The role of puncture is to periodically delete the selected bits to reduce the
coding overhead. Turbo decoder is used to recover the transmitted signal at the receiver side.
Figure 1 Turbo encoder
Also the turbo encoder can be used to generate different sequences and sequence with lowest
PAPR is selected for transmission. Figure 2 shows the transmitter side of MIMO-OFDM
systems, where the turbo coding and PTS are used for PAPR reduction.
The procedure for turbo PTS is same as Golay PTS except Reed-Muller complement sequence is
replaced with turbo encoder sequence. So by combining these two methods, significant
performance improvement can be achieved.
Figure 2 Block diagram of system model
5. RESULTS AND DISCUSSION
The analysis of the modified PTS with forward error correcting codes such as Golay
complementary sequences with Reed-Muller code and turbo code techniques has been carried
out using MATLAB 7.0. The simulation parameters considered for this analysis is summarized
in Table 1.
Table 1.Simulation parameters
Simulation parameters Type/Value
Number of subcarriers
64, 128, 256, 512, 1024
Number of subblock
4
Oversampling factor
4
Number of antennas
2X2
Modulation Scheme
QPSK
Phase factor 1, -1, j, -j
For comparison, PAPR reduction techniques with phase rotation factor and combined FEC-PTS
with phase rotation are deemed. In the MIMO-OFDM systems under consideration, modified
PTS technique is applied to the subblocks of uncoded information, which is modulated by
QPSK, and the phase rotation factors are transmitted directly to receiver through subblock. The
performance evaluation is done in terms of complementary cumulative distribution function.
Figure 3 Modified PTS performance for different
number of subcarriers and M
t
=2 with V=4
Figure 3 shows the performance of modified PTS technique in MIMO-OFDM system. Here
subblock size V=4 is considered for different number of subcarriers. Multi-carrier transmission
usually results in high peak to average power ratio. This PAPR value increases significantly as
number of carriers used in the MIMO OFDM transmission increase is shown in Figure 3.
Significant performance gain is seen for N=64 compared to N=128, 256, 512, and 1024. An
improvement of 0.9, 1.5, 2.3, and 2.8 dB respectively in PAPR can be attained for CCDF of 0.6
for M
t
=2.
Figure 4 PAPR reduction using PTS combined
with Golay sequence for different number
of subcarriers and M
t
=2 with V=4
Figure 4 demonstrates the performance of combined Golay with PTS for subblock size V=4 for
different number of subcarriers N=64, 128, 256, 512, and 1024. It is observed from the figure
that even with increase in the number of subcarriers PAPR remains constant as Golay sequences
is employed. Referring to Figure3 and Figure 4 it can be inferred that, combined Golay with PTS
results in significant performance gain in terms of PAPR reduction compared to a scheme
without FEC. Furthermore, for CCDF of 0.6 with N=1024, Golay complementary sequence
results in 4 dB reduction in PAPR as compared to uncoded scheme.
Figure 5 shows the performance results of modified PTS with Turbo coding for a subblock size
V=4 for different number of subcarriers. It can be observed that as the number of subcarriers
increases PAPR also increases. Here, N=64 results in better PAPR reduction compared to
N=1024. However, as N is increased from 64 to 128, 256, 512 and 1024 the PAPR is increased
by 1, 1.7, 2.3, and 3 dB for CCDF of 0.6 from 4.2 dB.
Figure 5 PAPR reduction using PTS combined
with turbo codes for different number
of subcarriers and M
t
=2 with V=4
Figure 6 Comparison of the CCDF of the PAPR
for original, modified PTS, Turbo PTS
and Golay with PTS for Mt=2,V=4 and
256 subcarriers.
The PAPR reduction performance for the STBC MIMO-OFDM is also shown for the purpose of
comparison. The total number of sub-carrier (N) is 256, number of cluster (V) is 4, and the
number of weighting factor (W) is 4. In figure 6, the PAPR reduction performance is evaluated
by using the Complementary Cumulative Distribution Function (CCDF). It can be seen that the
PAPR of modified PTS is 6.4 dB, Turbo-PTS is 5.9 dB and Golay-PTS is 3 dB at CCDF of 0.6,
respectively.
6. Conclusions:
In this paper, PAPR reduction technique based on modified PTS with FEC in MIMO-OFDM
systems using STBC has been considered. This approach, which combines the PTS technique
with Golay complementary sequences and Reed-Muller code, divides the subcarriers of OFDM
into several disjoint subblocks resulting in significant performance gain in terms of PAPR
reduction with low complexity. As a result, the CCDF of PAPR exhibits a steeper decay,
increased by a factor equal to the number of transmit antennas. The employment of MIMO
configuration along with PAPR reduction technique improved the capacity and the performance
of the OFDM system. The simulation results indicate that the proposed technique has a PAPR
reduction capability more than that of the modified PTS technique. Turbo PTS provides good
PAPR performance but as the number of subcarrier increases PAPR increases. Golay PTS results
in a performance improvement of 3.4 dB in terms of PAPR reduction compared to Turbo PTS
for N=256 and a CCDF value of 0.6 with low complexity as the PAPR is jointly optimized in
both the real and imaginary part.
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