JOURNAL OF TELECOMMUNICATIONS, VOLUME 6, ISSUE 1, DECEMBER 2010
1
Performance Analysis of MIMO Space-Time
Block Coded System in Nakagami Fading
Channel with Arbitrary Fading Parameters
Ritesh Pratap Singh, Dr. Amit Kumar Kohli
Abstract- As MIMO could increase the capacity of the wireless communication system considerably compared with Single Input Single Output (SISO)
and OFDM could obtain good performances in multipath frequency-selective fading channels, we combine them together with the Space-Time Block
Coding (STBC) scheme and the Nakagami-m fading model. The Nakagami-m (0.5≤ m<∞) fading model, covering a series of channel conditions in
different fading parameters, brings greater flexibility and accuracy than normal, logistic and other distributions. In this regard, we assume that STBC
(Space Time Block Codes) with and without CSIT (Channel State Information at Transmitter) for arbitrary Transmitter and Receiver antenna MIMO
(Multiple Input Multiple Output) systems. In the analysis, the transmit-receive link that maximizes the instantaneous received signal-to-noise ratio
(SNR) is selected for transmission and reception. Exact BER results for SIMO, MISO & MIMO system considering BPSK modulated signals for
arbitrary fading figure are validated by MATLAB simulations.
Index Terms - STBC (Space Time Block Codes), Nakagami-m fading model, CSIT (Channel State Information at Transmitter), MIMO (Multiple Input
Multiple Output) systems
-----------------------------------------------
♦
1 INTRODUCTION
Typically, the use of multiple antennas at both sides of a
communication link adheres to one of two distinct
approaches the aim of which is either to maximize the data
rate of the system or to improve its reliability, that is
minimize the system’s error probability. In this paper, we
focus on the latter type of MIMO systems; more specifically
we aim at analyzing the performance of MIMO diversity
systems employing orthogonal space-time block coding
(STBC) over Nakagami fading channels.
Space-time coding (STC) is a transmit diversity
mechanism that consists of spreading the information across
the transmit antennas in order to maximize the diversity gain
over fading channels [1]. An important family of STC is
space-time trellis coding (STTC) which provides both
diversity and coding gains. Nevertheless, STTC entails
significant implementation complexity which grows
exponentially as a function of the diversity order and the
transmission rate [2]. Due to this fact, space-time coding
remained somewhat of a theoretical concept until the
introduction of STBC, a second family of STC which was
originally discovered by Alamouti in his seminal work [3],
————————————————
• PhD student, (Signal Processing & Wireless Communication)
Thapar University, Patiala, India.
• Assistant Professor, ECE Department
Thapar University, Patiala, India.
----------------------------------------------where a simple and effective transmission paradigm for
systems equipped with two transmit antennas was presented.
The Alamouti scheme does not require channel
knowledge at the transmitter, supports maximum likelihood
detection, and provides a full diversity gain over fading
channels. As a result, it has been adopted as the open-loop
transmit diversity scheme by current 3GPP standards.
Furthermore, using the theory of orthogonal designs, Tarokh
et al. [4], [5] were able to generalize the Alamouti scheme to
an arbitrary number of antennas (MIMO systems), thereby
setting the basis for the so-called MIMO STBC concept.
While the approach to STBC in [4] focused on the theory
of orthogonal designs, a rather interesting approach from a
signal-to-noise ratio (SNR) maximization point of view [6]
also led to the STBC concept. The latter approach has been
exploited, first in [7], to assess the gap in performance
between the non-ergodic MIMO channel capacity using STBC
as compared to the true MIMO channel capacity, and then in
[8], to derive the ergodic capacity of STBC over independent
Rayleigh fading channels subject to different power and rate
adaptation policies, given imperfect channel estimation at the
receiver.
There are different models to characterize the fading
envelope of a received signal. The Nakagami-m fading
distribution fading model [9] is one of the most versatile, in
the sense that it has greater flexibility and accuracy in
matching some experimental data than Rayleigh, log-normal,
or Rician distributions. The Rayleigh distribution is a special
case when the fading parameter m=1. It can approximate Rice
distribution for m > 1.
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JOURNAL OF TELECOMMUNICATIONS, VOLUME 6, ISSUE 1, DECEMBER 2010
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This article is organized as follows. In Section II, the
system model is briefly described. The Nakagami-fading
channel model and its generation are provided in Section III.
The impact of fading among spatially separated MIMO
branches on BER performance is simulated and the analytical
results are presented in Section IV. Finally, concluding
remarks are presented in Section V.
A block diagram of a MIMO system with Nr ≥ 1 transmit and
Nr ≥ 1 receive antennas using STBC is illustrated in Figure 1.
xNt
IFFT
FFT
IFFT
FFT
Scattering
Fading
Channel
r1
(2)
r2
and decides in favour of the code word that minimizes the
sum.
Let H (k) be Nr , Nt channel matrix which contains complex
channel frequency responses at kth subcarrier. Time-domain
channel impulse responses can be modelled as a tapped
delay line and expressed as
L−1
hi, j (t ) = ∑ hi, j (l )δ (τ − τ )
l
t =0
rNr
Fig 1. Simple block diagram of MIMO-STBC system
We consider a discrete-time baseband channel model and
assume a narrow bandwidth system so that the channel can
be considered as frequency-nonselective. We also assume
quasi-static fading, which implies that the channel
characteristics remain constant at least for the period of
transmission of an entire frame (T symbol durations). The
channel may vary from frame to frame. Under these
assumptions, the input-output relationship within one frame
can be represented by a complex baseband matrix notation as
contrary to transmitters, receivers could be denoted in (1),
after passing through scattering fading channels and some
essential OFDM processes.
Nt
r j = ∑ hi , j xi + η j
i =1
x1x2 ....x n x1 x2 ....xn .........x1x2 ....x n
11
1 2 2
2
l l
l
3 NAKAGAMI-M CHANNEL MODEL
STBC Decoder
FFT
IFFT
Channel
Estimator
STBC Encoder
x2
2
l m j n
∑ ∑ rt − ∑ hi, j xi
t =1 j =1
i =1
over all code words
2 SYSTEM MODEL
x1
one, so that the average power of the received signal at each
receive antenna is η and the signal-to-noise ratio is SNR.
Assuming perfect channel state information is available;
the receiver computes the decision metric
(1)
where hi,j (i=1,2,…,Nt ; j=1,2,…,Nr) are independent and
identically distributed (i.i.d) complex random variables,
representing the channel coefficient from the ith transmitter
antenna to the jth receiver antenna. Here we suppose that
│hi,j│follows Nakagami-m distribution and E[│hi,j│] =1,
where│hi,j│ denotes the squared magnitude of complex
random variable hi,j .η (j=1, 2, …, Nr ) are noise samples and
i.i.d complex AWGN variables with zero mean and variance
N0/2 per complex dimension. The average energy of the
symbols transmitted from each antenna is normalized to be
(3)
Where δ (•) is a kronecker delta function and the taps
hi,j(l) = αi,j (l) ejθ uncorrelated with phase θl uniformly
distributed over [0,2π] for all receiver-transmitter antenna
pairs (i,j). Each tap amplitude, i.e., αi,j = │hi,j(l)│is modelled as
a Nakagami-m random variable, the probability density
function (PDF) is given as
2mα α 2m−1
 m

exp  − α 2  , α ≥ 0.
m
Γ(m)Ω
Ω


E α 2 
 
2


Ω = E α  , m =
.
 
2

E (α − E (α 2 ))2 


p(α ) =
(4)
Where Γ (•) is the Gamma function. For each tap and all (i, j)
pairs, the fading parameter m ≥ ½ is considered. In the
OFDM based system, the channel frequency response
between a pair of antennas (i, j) can be expressed as
H i, j ( k ) =
1 L−1
∑ h (l ) exp(− j 2π kl / N )
N l =0 i, j
(5)
Where Hi, j (k) is the frequency response of the channel for the
kth subcarrier, which is exactly the (i, j)th element of the
channel matrix H (k).
3.1. Nakagami Fading Generation
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For simulation purpose we have generated the nakagami
fading by using sum of sinusoidal using Rayleigh and Ricean
fading as given in [10] Equation 6. The received signal for
Rayleigh can be expressed as
(1−m )
R
= Rray e
nakagami
(1−m)
+ Rrice (1 − e
)
(6)
Nt X Nr
SNR (DB)
BER (SNR=6)
BER (SNR=8)
BER (SNR=10)
Where Rray and Rrice are envelops of Rayleigh and Ricean
channels respectively. Their method of generation is given in
[10]. By adopting sum of sinusoidal approach for received
signal generation and expressing it in inphase and
quadrature form for both rayleigh and ricean fading.
R (t ) = [ I (t )]2 + [Q (t )]2
Table I
BER of STBC for fading m=0.5
(7)
Where I(t) and Q(t) are the inphase and quadrature
components [10].
BER (SNR=12)
2X2
1X4
0.0767
0.0511
0.0338
0.0217
0.0204
0.0135
0.0087
0.0055
0.0085
0.0038
0.0017
0.0007
0.0012
0.0006
0.0002
0.0001
Nt X Nr
SNR (DB)
BER (SNR=6)
2X1
1X2
2X2
1X4
0.0459
0.0078
0.0038
0.0002
BER (SNR=8)
0.0226
0.0034
0.0010
0.0001
BER (SNR=10)
0.0104
0.0014
0.0002
0.0000
BER (SNR=12)
0.0047
0.0006
0.0000
0.0000
Table III
BER of STBC for fading m=2
In this section, we present a series of simulation results to
illustrate the effects of employing arbitrary number of
transmitter and receiver antenna on the BER performance of
space-time block codes in different Nakagami-m fading
channels. In all the calculations, it has been assumed that the
fading channels are normalized in the sense that
Ω nt, nr = Ω= 1. Uncoded single transmission antenna and
Linear STBC-BPSK transmission scheme with Alamouti code
C2, and four receive antennas STBC have been used. The
analytical results are verified using simulation; the
simulation was carried out using Matlab with the following
parameters
•
1X2
Table II
BER of STBC for fading m=1
4 SIMULATION RESULTS
•
•
•
2X1
Nt X Nr
SNR (DB)
BER (SNR=6)
BER (SNR=8)
BER (SNR=10)
BER (SNR=12)
2X1
1X2
2X2
1X4
0.0223
0.0070
0.0018
0.0004
0.0018
0.0004
0.0001
0.0000
0.0011
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
We see from these Tables I, II, III that simulated results for
different fading figure ‘m’ the value of BER decreases as the
number of receivers are increases.
A BPSK transmission scheme.
Frame length 10000.
Fading-independent MIMO flat fading Nakagami
statistics with fading parameter m = 0.5, 1 and 2.
Doppler spread fd = 100 Hz.
The information source is encoded using a space–time
block code, and the constellation symbols are transmitted
from different antennas. The receiver estimates the
transmitted bits by using the signals of the received antennas.
The results are reported for a BPSK and our space–time block
codes using one and two transmit antennas in Nakagami
fading environment for fading figure m=0.5, 1, 2. Simulation
results in Figure 2, 3 and 4 are given for one, two and four
receive antenna using BPSK modulation.
Fig. 2 BER Performance of MIMO-STBC for BPSK modulation in
Nakagami fading channel [m=0.5]
The transmission using two transmit antennas employs
the BPSK constellation and the code C2. It is seen in figure 2
that for Nakagami fading figure m=0.5 at the bit error rate of
10- 3, an improvement greater than 20 db is obtained for 1 X 4
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system relative to 2 X 1 system, 12 db relative to 1 X 2 system,
4 db relative to 2 X 2 system, for independent fading
channel.
depth increases. As well as increasing the diversity order at
transmitter and receiver BER performance deteriorates. It can
also be seen from the table I, II, and III that more the number
of receiver (exploiting receive diversity) is used more the BER
performance deteriorates as compared to exploiting transmit
diversity.
5 CONCLUSIONS
This article has presented the analytical evaluation of the
average BER performance of space-time block codes, based
on BPSK modulation scheme, over independent Nakagami-m
MIMO fading channels. Analytical expressions have been
shown for arbitrary fading parameters along the antenna
branches. Simulated results indicate that increasing the
number of antennas properly will achieve both great capacity
and BER performances. The increase of the parameter ‘m’ will
improve the performance of the system for a certain extent,
but not obviously.
Fig. 3 BER Performance of MIMO-STBC for BPSK modulation in
Nakagami fading channel [m=1]
Similarly, in figure 3 for fading m=1 (Rayleigh fading) at
the bit error rate of 10- 4, an improvement greater than 12 db
is obtained for 1 X 4 system relative to 2 X 1 system, 10 db
relative to 1 X 2 system, 4 db relative to 2 X 2 system.
Similarly, in figure 4 for fading m=2 at the bit error rate of
10- 5, an improvement greater than 10 db is obtained for 1 X 4
system relative to 2 X 1 system, 6 db relative to 1 X 2 system,
3 db relative to 2 X 2 system.
Fig. 4 BER Performance of MIMO-STBC for BPSK modulation in
Nakagami fading channel [m=2]
6 REFERENCES
[1] R. W. Heath Jr. and A. J. Paulraj, “Switching between multiplexing
and diversity based on constellation distance,” in Proc. of Allerton
Conf. on Comm. Control and Comp., Oct. 2000.
[2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for
high data rate wireless communication: Performance criteria and
code construction,” IEEE Trans. Inform. Theory, vol. 44, no. 2, pp.
1744–1765, Mar. 1998.
[3] S. M. Alamouti, “A simple transmitter diversity scheme for wireless
communications,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp.
1451–1458, Oct. 1998.
[4] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block
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45, no. 5, pp. 1456–1467, July 1999.
[5] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block
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[6] G. Ganesan and P. Stoica, “Space-time block codes: A maximum SNR
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1656, May 2001.
[7] S. Sandhu and A. Paulraj, “Space time block codes: A capacity
perspective,” IEEE Commun. Lett., vol. 4, no. 12, pp. 384–386, Dec.
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[8] A. Maaref and S. A¨ ıssa, “Capacity of space-time block codes in
MIMO Rayleigh fading channels with adaptive transmission and
estimation errors,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp.
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[9] M. Nakagami, “The m-distribution, a general formula of
intensity distribution of rapid fading,” in Statistical Methods in
Radio Wave Propagation, W. G. Hoffman, Ed, Oxford, England:
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[10] Li Tang, Zhu Hongbo,”Analysis and Simulation of nakagami fading
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As expected, the average BER performance deteriorates as
the fading parameter ‘m’ decreases; and the Nakagami-fading
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JOURNAL OF TELECOMMUNICATIONS, VOLUME 6, ISSUE 1, DECEMBER 2010
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Ritesh Pratap Singh is pursuing his Ph.D. from Thapar University,
Patiala in Signal Processing and Wireless Communication. He has
completed his B. Tech in Electronics and Instrumentation Engineering
from U.P.T.U, Lucknow and M. Tech in Communication Engineering from
SU, Meerut, India in 2006 and 2009 respectively. Currently working as an
Assistant Professor in the ECE Department at IIMT Engineering College,
Meerut. Areas of research interest include signal processing for highspeed digital communications, signal detection, MIMO, multiuser
communications, Image Processing, Image Sequence Restoration and
Enhancement. He is a Life Time member of IETE.
Dr. Amit Kumar Kohli received the B.Tech. (Honor) degree from Guru
Nanak Dev Engineering College, Ludhiana, the M.E. (Highest Honor)
degree from Thapar University Patiala (previously Thapar Institute of
Engineering and Technology) and the Ph.D. degree from Indian Institute
of Technology, Roorkee, India, in 2000, 2002 and 2006 respectively, all
in electronics and communication engineering. He is presently Assistant
Professor in Electronics and Communication Engineering Department of
Thapar University Patiala, India. His research interests include signal
processing and its applications, modeling of fading dispersive channels,
wireless and high data rate advanced communication systems, neural
networks and adaptive system design. He is reviewer of distinguished
international journals like IEEE, Springer and Elsevier etc. He received
National Scholarship, State Scholarship, MHRD Fellowship, Institution
Medal, and Gold Medal consecutively for academic performance.
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