How to Cite:
Özyurt, S., (2018). Performance of Orthogonal Frequency Division Multiplexing with
Signal Space Diversity Via Subcarrier Coordinate Interleaving over Nakagami-M and Rician
Fading Channels, Engineering Sciences (NWSAENS), 13(4):291-301,
DOI: 10.12739/NWSA.2018.13.4.1A0420.
Engineering Sciences
Status : Original Study
ISSN: 1308 7231 (NWSAENS)
Received: July 2018
ID: 2018.13.4.1A0420
Accepted: October 2018
Serdar Özyurt
Ankara Yıldırım Beyazıt University, sozyurt@ybu.edu.tr, Ankara-Turkey
DOI
http://dx.doi.org/10.12739/NWSA.2018.13.4.1A0420
ORCID ID
0000-0002-9612-6227
CORRESPONDING AUTHOR
Serdar Özyurt
PERFORMANCE OF ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING WITH SIGNAL
SPACE DIVERSITY VIA SUBCARRIER COORDINATE INTERLEAVING OVER NAKAGAMI-M
AND RICIAN FADING CHANNELS
ABSTRACT
In this work, orthogonal frequency division multiplexing
technique is combined with signal space diversity and error
probability performance of the system is inspected for binary phase
shift keying modulation over slow frequency-selective Nakagami-m and
Rician fading channels. First, appropriate subcarrier coordinate
interleaving techniques are obtained based on the number of
subcarriers, the number of resolvable multipaths, and the fading
parameters. It is shown that appropriate subcarrier coordinate
interleaving techniques under some of the investigated scenarios are
the same as the ones under the Rayleigh fading scenario. Subsequently,
the bit error rate of the studied system is compared with the bit
error rate of the original system. Further, it is demonstrated that
the considered system provides significant performance gain beyond the
original orthogonal frequency division multiplexing technique under
different scenarios.
Keywords: Orthogonal Frequency Division Multiplexing,
Signal Space Diversity, Subcarrier Coordinate
Interleaving, Nakagami-M Fading Channel,
Rician Fading Channel
1. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) has been
included in many contemporary communication standards such as WiFi
(IEEE 802.11), WiMax (IEEE 802.16), DAB (Digital Audio Broadcasting),
and DVB (Digital Video Broadcasting) [1]. Additionally, as a different
form of OFDM, discrete multi-tone transmission technique constitutes a
fundamental part of ADSL (Asymmetric Digital Subscriber Line) and VDSL
(Very-high-bit-rate Digital Subscriber Line) systems. In a wireless
communication system, when the channel between the transmitter and
receiver is frequency-selective, a complex filter must be applied on
the received signal in order to recover the original transmitted
signal by means of filtering. The frequency response of the filter
must be identical to the inverse of the frequency response of the
channel. Depending on how much frequency-selective the channel is, the
design and implementation of such a filter may be quite challenging if
possible. In OFDM, on the other hand, the frequency-selective channel
is divided into numerous subchannels in such a way that all the
subchannels can be considered to possess flat frequency responses. By
sending data in a low-rate fashion from each subchannel, the
communication between the transmitter and receiver is carried out over
many parallel subchannels. As all the subchannels exhibit flat
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Engineering Sciences (NWSAENS), 1A0420, 2018; 13(4): 291-301.
frequency responses, the channel equalization at the receiver can be
performed by means of a scalar multiplication instead of a complicated
filtering operation [2]. Additionally, the fact that Fourier transform
included in OFDM can be implemented effectively using the modern
baseband techniques has increased the popularity of OFDM. On the other
hand, when any of the subchannels experiences a low channel gain, this
can cause an incorrect detection on the relevant symbol at the
receiver. This poses a serious disadvantage for the original OFDM
method.
Signal space diversity (SSD) also known as modulation diversity
[3 and 9] is based on utilizing the orthogonal basis in the signal
space for the purpose of increasing the diversity order and hence
enhancing the error performance of the system. This benefit comes with
no increase in bandwidth or power consumption and no significant
growth in complexity. For the systems employing two dimensional signal
constellations, SSD can be achieved in two steps [3 and 9]. In the
first stage, it is ensured that no two symbols in the signal
constellation has identical in-phase (I) and quadrature (Q)
coordinates. In this way, any symbol can be uniquely identified by
either of its two coordinates. The I and Q coordinates of the
transmitted symbol are made sure to experience independent fading
states in the second step. The first condition mentioned above can be
achieved by properly rotating the signal constellation. There exist
multiple methods for performing the second step. For instance, this
requirement can be satisfied by employing time domain interleaves and
deinterleaves at the transmitter and receiver, respectively. However,
in such a case, the depth of the employed interleaves must be greater
than the coherence time of the channel. This condition can be quite
demanding for the transmission of latency-sensitive data. In OFDM
system, the second step involved in achieving SSD can also be managed
by transmitting the I and Q coordinates of the transmitted symbol over
distinct subchannels (subcarrier coordinate interleaving). As it
brings about no remarkable extra latency to the original system, this
technique is highly desirable as compared to the time domain
interleaving scheme.
In this study, OFDM is combined with SSD and the error
performance of the system is investigated over slow Nakagami-m and
Rician fading channels under binary phase shift keying (BPSK)
modulation. In [9], an OFDM/SSD combination based on subcarrier
coordinate interleaving has been proposed and analyzed over slow
Rayleigh fading channels with BPSK modulation. The scheme in [9]
functions based on minimizing the correlation coefficient between the
subchannels affecting I and Q coordinates of the transmitted symbol.
To this end, several distinct interleaving techniques have been
proposed for different cases in [9]. This work assumes Nakagami-m and
Rician fading channel scenarios which are more general than the
Rayleigh fading scenario (Rayleigh fading is a special case of
Nakagami-m and Rician fading scenarios). A comparison with the
original OFDM system is provided in terms of the bit error rate (BER).
A literature survey about the inclusion of SSD into OFDM systems can
be found in [9].
Throughout the paper, the following notation is utilized: E{.},
exp(.), |.|, (.)
∗
, j, ℜ{.}, and ℑ{.} respectively denote the expected
value, exponential function, norm for complex numbers (absolute value
for real numbers), complex conjugate,
1−
, real and imaginary parts of
a complex number. The probability density function (PDF) of X is
represented by f
X
(x) and f
XY
(x,y) stands for the joint PDF of X and Y.
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The correlation coefficient between two complex random variables X and
Y is denoted by
,XY
and given by
( )
( ) ( )
,
cov ,
var var
XY
XY
XY
=
(1)
where
( )
**
cov ,X Y E XY E X E Y=−
is the covariance between the
random variables X and Y. Additionally,
( )
2
2
var X E X E X=−
stands for
the variance of the complex random variable X. Two complex random
variables X and Y are said to be uncorrelated only if
( )
cov , 0XY =
and
0E XY E X E Y−=
, i.e., the pseudo-covariance between X and Y is zero.
2. RESEARCH SIGNIFICANCE
This study generalizes the work in [9] by adopting more general
fading channel models. Unlike the Rayleigh fading channel assumption
in [9], two more comprehensive channel models (Nakagami-m fading and
Rician fading channel models) are adopted in this study. It is
important to note that Rayleigh fading scenario can be recovered by
substituting m=1 in Nakagami-m fading case and K=0 in Rician fading
case. Due to the mentioned reasons, the current work contains original
contribution.
3. SYSTEM MODEL
It is assumed that a single-antenna transmitter communicates
with a single-antenna receiver over a frequency-selective fading
channel. Using OFDM, the frequency-selective channel is converted into
multiple parallel subchannels each of which experiences flat fading.
The communication channel stays constant throughout one OFDM frame and
alters from one frame to another in a statistically independent
fashion. Full channel state information is available only at the
receive side. Assume that the number of subcarriers and the number of
resolvable multipaths are respectively represented by N (N is a power
of two) and L (L≪N). Each transmitted OFDM frame is composed of N data
symbols and L cyclic prefix symbols. In the original OFDM method, the
value of N is assumed to be known at both ends of the transmission
link. The scheme inspected in this work assumes that the values of N
and L are both known at the transmitter and receiver. In practice, L
takes small integer values and the value of L can be sent back to the
transmitter from the receiver using a low-rate feedback channel. With
the successful transmission of each frame, N modulated symbols are
conveyed to the receiver. Modulated symbols are produced from a BPSK
signal constellation which is rotated by an angle of counter-
clockwise. Let the modulated symbol sequence and the energy allocated
for each bit be denoted by {s
1
, s
2
, …, s
N
} and E
b
, respectively. Because
of the energy loss while transmitting the cyclic prefix, one can write
|s
i
|
2
=E
b
′
=E
b
(N/(N+L)). For any modulated symbol s
k
, s
k
=s
kI
+j s
kQ
can be
written where k ∈ {1, 2, …, N}. Here, s
kI
and s
kQ
stand for the I and Q
coordinates of the symbol s
k
. At the transmitter, coordinate
interleaving (CI) is applied on the sequence {s
1
, s
2
, …, s
N
} just
before the inverse fast Fourier transform block. Likewise, coordinate
deinterleaving (CD) is performed just after the fast Fourier transform
(FFT) operation at the receive side. Let {X
1
, X
2
, …, X
N
} denote the
sequence obtained after the CI operation at the transmitter. Then, we
can write X
k
=s
kI
+j s
iQ
for i, k∈{1, 2, …, N} and i≠k. In other words,
only Q coordinates are interleaved amongst themselves. Assume that the
impulse response of the frequency-selective channel is represented by
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the sequence {h
0
, h
1
, …, h
L−1
}. Here, h
k
for k ∈ {0, 1, …, L−1} are
independent complex random variables. Further, when written in the
polar coordinate system as h
k
=|h
k
| e
j
k
, |h
k
| and
k
are independent
random variables. If the output of the FFT operation at the receiver
is denoted by {Y
1
, Y
2
, …, Y
N
}, then one can write Y
k
=H
k
X
k
+Z
k
for
1,2, ,kN
. We have
1
0
exp 2 .
L
ki
i
i
H h j k
N
−
=
= −
(2)
Additionally, Z
1
, Z
2
, …, Z
N
are independent and identically
distributed (IID) circularly symmetric complex Gaussian random
variables each having a variance of N
0
/2 per dimension. These variables
represent the additive white Gaussian noise (AWGN) components at the
receiver and N
0
is the one-sided power spectral density of the AWGN.
Moreover,
12
, , ,
N
H H H
are identically distributed correlated complex
random variables. By performing single-tap equalization on Y
k
, we can
write (H
k
∗
/|H
k
|)Y
k
=|H
k
|X
k
+Ẑ
k
where Ẑ
k
=(H
k
∗
/|H
k
|)Z
k
for k∈{1, 2, …, N}.
Assume that the Q coordinate of the kth modulated symbol is carried on
the tth subcarrier. Additionally, let d
k
be the decision variable used
by the maximum likelihood detector for the kth modulated symbol. In
this case, after the application of CD, d
k
=ℜ{Y
k
}+jℑ{Y
t
}=|H
k
|s
kI
+j
|H
t
|s
kQ
+Ž
k
can be written. Here, Ž
k
denotes the noise component after
the application of CD.
3.1. Nakagami-m Fading Case
In the first scenario, it is assumed that the magnitude |h
k
| has
a Nakagami-m distribution with the following PDF
( )
( )
( )
2 1 2
2 exp
k
m
m
h
m
f r r m r
m
−
=−
(3)
for
0r
where m represents the shape parameter of the PDF. Also,
the average power in the kth multipath (tap) is E{|h
k
|
2
}=1, i.e., all
the taps have identical average power (uniform power delay profile).
When m increases, the severity of the fading decreases. Additionally,
k
has a uniform distribution between 0 and 2 for any k. We assume
that all the taps have the same m value. Under the Nakagami-m fading
scenario, the expression in (2) is equal to the sum of L IID Nakagami-
m random phase vectors. Using statistical methods, it can be shown
that
0
k
EH =
and
1
22
0
L
ki
i
E H E h L
−
=
==
.
The absolute value of the correlation coefficient (ACC) between
the kth and tth subchannels
(
k
H
and
)
t
H
can be shown to be equal to
1
* * *
0
2
exp 2
L
k t k t k t
a
kt
k
kt
ja
E H H E H E H E H H
N
LL
EH
−
=
−
−
−
= = =
(4)
for k∈{1, 2, …, N} and t∈{1, 2, …, N}, with 0≤
kt
≤1. By
evaluating the sum, we can write
sin
.
sin
kt
kt
L
N
kt
L
N
−
=
−
(5)
The preceding expression indicates that ACC between any pair of
subchannels is independent of m. Also, the equation in (5) is in the
same form as the one obtained for the Rayleigh fading scenario in [9].
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Özyurt, S.
Engineering Sciences (NWSAENS), 1A0420, 2018; 13(4): 291-301.
Hence, the CI strategies proposed in [9] for the case of Rayleigh
fading can be used for the considered system model in a direct manner.
Assuming N=512 and L=5, the ACC is illustrated in Figure 1 where the
ACC is minimized for (k-t)=205 with a corresponding ACC of 0.00128.
Figure 1. The ACC for N=512 and L=5 under Nakagami-m fading scenario
3.2. Rician Fading Case
We consider Rician fading scenario in this part. To this end, we
adopt the model presented by International Telecommunication Union
[10]. In this respect, the first tap h
0
with the shortest path delay
includes multiple scattered paths and a specular or line-of-sight
component [11]. Therefore, h
0
is a nonzero-mean complex Gaussian random
variable, which can also be written as
00
1
1
1
K
hh
K
K
=+
+
+
where
0
h
is a
zero-mean complex Gaussian random variable and K is the Rician K-
factor [2]. All the remaining taps,i.e., h
1
,...,h
L-1
, are IID zero-mean
complex Gaussian random variables, which are also independent from
0
h
.
The PDFs of
0
h
and
1
h
(or of
21
,,
L
hh
−
) are respectively given by
( ) ( )
( )
( )
(
)
0
2
0
2 1 exp ( 1) 2 1
h
f r K r r K K I r K K= + − + − +
(6)
for
0r
where
( )
0
.I
is the modified Bessel function of the first
kind zero order [12] and
( )
( )
1
2
2 exp
h
f r r r=−
(7)
for
0r
. All the multipaths have the same average power as
E{|h
k
|
2
}=1 for any k. Under the Rician fading scenario, the expression
in (2) is equal to the sum of L IID complex Gaussian random variables.
Using statistical methods, it can be shown that
1
k
K
EH
K
=
+
and
2
1
1
11
k
K
E H L L
KK
= + + − =
++
.