PERFORMANCE ANALYSIS OF CLIPPED STBC CODED MIMO OFDM SYSTEM

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PERFORMANCE ANALYSIS OF CLIPPED
STBC CODED MIMO OFDM SYSTEM
Koiloth S R S Jyothsna and Tummala Aravinda Babu
Department of Electronics and Comunication Engineering
Chaitanya Bharathi Institute of Technology
Gandipet, Hyderabad, India
ABSTRACT
A combination of Multiple-Input Multiple-Output Spatial Division
Multiplexing technology and Orthogonal Frequency Division Multiplexing
technique, namely MIMO-OFDM systems, been well-known as a potential
technology to provide high speed data transmission and spectrum efficiency to
attain throughput of 1 Gbit/sec and beyond improves link reliability for
modern wireless communications. The rising development of Internet related
contents and demand of multimedia services leads to increasing curiosity to
high speed communications. It has been shown that by using MIMO system, it
is possible to increase that capacity considerably. To use advantage of MIMO
diversity to overcome the fading then we need to send the same signals
through the different MIMO antennas and at the receiver end the different
antennas will receive the same signals travelled through diverse paths.
Efficient implementation of MIMO OFDM system is based on the FFT
algorithm and MIMO encoding like Alamouti Space Time Block coding
(STBC). In this paper, MIMO-OFDM based on Almouti Space Time Block
Codes is described and the BER performance of this system is observed for
various antenna configurations .The channel capacity per unit bandwidth is
evaluated as a function of SNR and the MIMO channel capacity for different
transmit and receive antennas is observed. Finally, Peak to Average Power
Ratio (PAPR) is reduced using clipping technique and the BER performance
of the ASTBC system with clipping and without clipping is observed.
Key words: MIMO, OFDM, STBC and PAPR
Cite this Article: Koiloth S R S Jyothsna and Tummala Aravinda Babu.
Performance Analysis of Clipped STBC Coded MIMO OFDM System.
International Journal of Electronics and Communication Engineering &
Technology, 7(1), 2016, pp. 28-44.
http://www.iaeme.com/IJECET/issues.asp?JType=IJECET&VType=7&IType=1

Performance Analysis of Clipped STBC Coded MIMO OFDM System
1. INTRODUCTION
MIMO (Multiple Input, Multiple Output) is an antenna technology for wireless
communications in which multiple antennas are used at both the source (transmitter)
and the destination (receiver). The antennas at each end of the communications circuit
are combined to minimize errors and optimize data speed. MIMO is one of several
forms of smart antenna technology, the others being MISO (Multiple Input, Single
Output) and SIMO (Single Input, Multiple Output). In conventional wireless
communications, a single antenna is used at the source, and another single antenna is
used at the destination. In some cases, this gives rise to problems with multipath
effects. When an electromagnetic field (EM field) is met with obstructions such as
hills, canyons, buildings, and utility wires, the wave fronts are scattered, and thus they
take many paths to reach the destination. The late arrival of scattered portions of the
signal causes problems such as fading, cut-out (cliff effect), and intermittent reception
(picket fencing). In digital communications systems such as wireless Internet, it can
cause a reduction in data speed and an increase in the number of errors. The use of
two or more antennas, along with the transmission of multiple signals (one for each
antenna) at the source and the destination, eliminates the trouble caused by multipath
wave propagation, and can even take advantage of this effect.
MIMO technology has aroused interest because of its possible applications in
digital television (DTV), wireless local area networks (WLANs), metropolitan area
networks (MANs), and mobile communications.
2. MIMO SYSTEM
MIMO systems are composed of three main elements, namely the transmitter (TX),
the channel (H), and the receiver (RX). In NT is denoted as the number of antenna
elements at the transmitter, and Nr is denoted as the number of elements at the
receiver. It is important to note that the system is described in terms of the channel.
For example, the Multiple-Inputs are located at the output of the TX (the input to the
channel), and similarly, the Multiple-Outputs are located at the input of the RX (the
output of the channel).
The channel with Nr outputs and Nt inputs is denoted as a Nr X Nt matrix
where each entry hi;j denotes the attenuation and phase shift (transfer function)
between the jth
transmitter and the ith
receiver. It is assumed that the MIMO channel
behaves in a “quasi-static” fashion, i.e. the channel varies randomly between burst to
burst, but fixed within a transmission. This is a reasonable and commonly used
assumption as it represents an indoor channel where the time of change is constant
and negligible compared to the time of a burst of data. The MIMO signal model is
described as
r = Hs + n (1)
where, r is the received vector of size NR×1, H is the channel matrix of size NR ×NT ,
s is the transmitted vector of size NT×1, and n is the noise vector of size NR×1. Each
noise element is typically modelled as independent identically distributed (i.i.d.) white
Gaussian noise, with variance NT=2. An explanation for this model is as follows. The

transmitted signals are mixed in the channel since they use the same carrier frequency.
At the receiver side, the received signal is composed of a linear combination of each
transmitted signal plus noise. The receiver can solve for the transmitted signals by
treating as a system of linear equations. If the channel H is correlated, the system of
linear equations will have more unknowns than equations. One reason correlation
between signals can occur is due to the spacing between antennas. To prevent
correlation due to the spacing, they are typically spaced at least c=2, where c is the
wavelength of the carrier frequency. The second reason correlation can occur is due to
lack of multipath components. It is for this reason that rich multipath is desirable in
MIMO systems. The multipath effect can be interpreted by each receive antenna
being in a different channel. For this reason, the rank of a MIMO channel is defined
as the number of independent equations offered. It is important to note that: rank (H)
< min (NR;NT) and therefore the maximum number of streams that a MIMO system
can support is upper-bounded by min(NR;NT). Since the performance of MIMO
systems depends highly on the channel matrix, it is important to model the channel
matrix realistically. The following section provides an overview of typical channel
models used for computer simulations.
3. SPACE–TIME BLOCK CODES (STBC)
Space–time block coding is a technique used in wireless communications to transmit
multiple copies of a data stream across a number of antennas and to exploit the
various received versions of the data to improve the reliability of data-transfer. The
fact that the transmitted signal must traverse a potentially difficult environment with
scattering, reflection, refraction and so on and may then be further corrupted by
thermal noise in the receiver means that some of the received copies of the data will
be 'better' than others.
Transmit antennas
Time slots
An STBC is usually represented by a matrix as shown above. Each row represents
a time slot and each column represents one antenna's transmissions over time. Here,
Sij is the modulated symbol to be transmitted in time slot i from antenna j. There are
to be T time slots and nT transmit antennas as well as nR receive antennas. This block
is usually considered to be of length T. The code rate of an STBC measures how
many symbols per time slot it transmits on average over the course of one block. If a
block encodes k symbols, the code-rate is r=k/T. Only one standard STBC can
achieve full-rate Alamouti's code. STBCs as originally introduced, and as usually
studied, are orthogonal. This means that the STBC is designed such that the vectors
representing any pair of columns taken from the coding matrix are orthogonal. The
result of this is simple, linear, optimal decoding at the receiver.
3.1 ALAMOUTI Space Time Block Code
Alamouti code is the first STBC that provides full diversity at full data rate for two
transmit antennas. Fig.1 shows the block diagram of the Alamouti space-time
encoder. S1= [ ,

Figure 1 A block diagram of the Alamouti space-time encoder
The information bits are first modulated using an M-ary modulation scheme. The
encoder takes the block of two modulated symbols s1and s2 in each encoding
operation and hands it to the transmit antennas according to the code matrix
(2)
The first row represents the first transmission period and the second row the
second transmission period. During the first transmission, the symbols s1 and s2 are
transmitted simultaneously from antenna one and antenna two respectively. In the
second transmission period, the symbol is transmitted from antenna one and the
symbol from transmit antenna two. It is clear that the encoding is performed in
both time (two transmission intervals) and space domain (across two transmit
antennas). The two rows and columns of S are orthogonal to each other and the code
matrix is orthogonal
(3)
Where I2 is a (2 × 2) identity matrix. This property enables the receiver to detect s1 and
s2 by a simple linear signal processing operation. Let us look at the receiver side now.
Only one receive antenna is assumed to be available. The channel at time t may be
modelled by a complex multiplicative distortion h1(t) for transmit antenna one and
h2(t) for transmit antenna two as shown in equation 3.3 and 3.4. Assuming that the
fading is constant across two consecutive transmit periods of duration T, we can write
h1(t)= h1(t+T)=h1= (4)
h2(t)= h2(t+T)= h2= (5)
Where, |hi | and θi , i = 1, 2 are the amplitude gain and phase shift for the path from
transmit antenna i to the receive antenna. The received signals at the time t and t + T
can then be expressed as
= (6)
= (7)
Where, r1 and r2 are the received signals at time t and t + T, n1 and n2 are complex
random variables representing receiver noise and interference.
Information
source
Alamouti
Code S
Modulator

3.2 Equivalent Virtual (2 × 2) Channel Matrix (EVCM) of the Alamouti
Code
Conjugating the signal r2 in that is received in the second symbol period, the received
signal may be written equivalently as r
= (8)
= (9)
Thus the equations 3.8 and 3.9 can be written as
= + (10)
or in short notation as
y = Hvs + (11)
Where, the modified receive vector y = [ ] T has been introduced. Hv will be
termed the Equivalent Virtual MIMO Channel Matrix (EVCM) of the Alamouti
STBC scheme. It is given by
Hv= (12)
Thus, by considering of the elements of y as originating from two virtual receive
antennas (instead of received samples at one antenna at two time slots) one could
interpret the (2 × 1) Alamouti STBC as a (2× 2) spatial multiplexing transmission
using one time slot. The key difference between the Alamouti scheme and a true (2 ×
2) multiplexing system lies in the specific structure of Hv. Unlike to a general i.i.d.
MIMO channel matrix, the rows and columns of the virtual channel matrix are
orthogonall.
= )I2 = I2 (13)
where I2 is the (2 × 2) identity matrix and h 2 is the power gain of the equivalent
MIMO channel with =( ). Due to this orthogonality the receiver of the
Alamouti scheme (discussed in detail in the following subsection) decouples the
MISO channel into two virtually independent channels each with channel gain h2
and
diversity d = 2. It is obvious that the EVCM depends on the structure of the code and
the channel coefficients. The concept of the EVCM simplifies the analysis of the
STBC transmission scheme. The existence of an EVCM is one of the important
characteristics of STBCs.
3.3 ASTBC Based MIMO OFDM System Model
Consider a space time block coded MIMO-OFDM system equipped with transmit
antennas and receive antennas as illustrated in Figure 2. The message bit sequence is
mapped into a sequence of BPSK symbols which will be converted into N parallel
symbol streams after serial to parallel conversion. Each of the N parallel symbol
streams is then encoded by the space-time block code encoder i=1,2,3.....NT
into where is the antenna index and is the symbol time index.

Figure 2 Block Diagram of MIMO OFDM System using STBC coding.
The number of symbols in a space-time codeword is N=NT×NR. Then the symbol
streams are subjected to inverse fast Fourier transform operation followed by cyclic
prefix insertion between two consecutive OFDM symbols in order to reduce the effect
of the delay spread of the multipath channels. The length of the CP is adjustable and
must be set in order to keep a bandwidth efficient system without occurring inter
symbol interference or inter carrier interference. At the receiver, after removing the
CP and applying FFT, the transmitted symbol stream is estimated using the
received signal Assume the channel gain follows the Rayleigh distribution
from the ith
transmit antenna to jth
the receive antenna over the tth
symbol period. If the
channel gains do not change during T symbol periods, the symbol time index can be
omitted and as long as the transmit antennas and receive antennas are spaced
sufficiently apart, NT NR fading gain {hij} can be assumed to be statistically
independent. If is the transmitted signal from the ith
transmit antenna during tth
symbol period, the received signal at the jth
receive antenna during tth
symbol period is
given by
= (14)
where is the noise process at the jth
receive antenna during tth
symbol period, which
is modelled as the zero mean circular symmetric complex Gaussian (ZMCSCG) noise
of unit variance, and is the average energy of each transmitted signal.
In general we can write as
Y= (15)
3.3.1 BER Performance Evaluation
In order to make an investigation of performance analysis of the MIMO-OFDM
system with Alamouti Space Time Block Code as the transmit diversity and MRC
diversity technique as the receive diversity over a Rayleigh fading channel, we deal
with MATLAB simulation using the parameters based on IEEE802.a standard. BPSK
modulation was used to determine the BER versus SNR performance of the system.

In digital transmission, the number of bit errors is the number of received bits of a
data stream over a communication channel that has been altered due to noise,
interference, distortion or bit synchronization errors. The bit error rate or bit error
ratio (BER) is the number of bit errors divided by the total number of transferred bits
during time interval.
In a noisy channel, the BER is often expressed as a function of the normalized
carrier-to-noise ratio measure denoted Eb/N0, (energy per bit to noise power spectral
density ratio), or Es/N0 (energy per modulation symbol to noise spectral density).
While in wireless communication, BER (dB) vs. SNR (dB) is used. The BER may be
analyzed using stochastic computer simulations.
3.4 Channel Capacity of MIMO OFDM System
It is the maximum amount of information that can reliably be transmitted over any
communication channel at any given instant. It is denoted by ‘C’ and can be given as
(16)
Where, B is Bandwidth in Hertz , S/N is Signal to Noise Ratio in watts or volts2
. For
MIMO the capacity is given by
(17)
Where, M is the minimum of MT (number of transmitting antennas) or MR (number of
receiving antennas).
3.4.1 Performance Analysis of MIMO OFDM System
The water filling algorithm has been employed to measure the performance of MIMO
OFDM integrated system.
3.4.1.1. Water Filling Algorithm
Water filling refers to a technique whereby the power for the spatial channels are
adjusted based on the channels gain. The channel with high gain and signal to noise
ratio gives more power. More power maximizes the sum of data rates in all sub
channels. The data rate in each sub channel is related to the power allocation by
Shannon’s G formula C = B log(1 + SNR). However, because of t is a logarithmic
function of power, the data rate is usually insensitive to the exact power allocation.
This motivates the search for simpler power allocation schemes that can perform close
to the optimal. The water filling algorithm is based on an iterative procedure. The
process of water filling algorithm is similar to pouring the water in the vessel .The
total amount on water filled (power allocated) is proportional to the Signal to Noise
Ratio of channel.
Power allocated by individual channel is given by
(19)
Where Pt is the power budget of MIMO system which is allocated among the different
channels and H is the channel matrix of system. The capacity of a MIMO is the
algebraic sum of the capacities of all channels and given by
(20)

We have to maximize the total number of bits to be transported .As per the
scheme following steps are followed to carry out the water filling algorithm.
Algorithm Steps:-
1. Take the inverse of the channel gains.
2. Water filling has non uniform step structure due to the inverse of the Channel
gain.
3. Initially take the sum of the total power Pt and the inverse of the channel gain.
It gives the complete area in the water filling and inverse power gain
(21)
4. Decide the initial water level by the formula given below by taking the average
Power allocated
(22)
5. The power values of each subchannel are calculated by subtracting the inverse
Channel gain of each channel
(23)
In case the power allocated value become negative stop iteration.
3.5 Peak to Average Power Ratio (PAPR) in OFDM
The PAPR is the relation between the maximum power of a sample in a given OFDM
transmit symbol divided by the average power of that OFDM symbol. PAPR occurs
when in a multicarrier system the different sub-carriers are out of phase with each
other. At each instant they are different with respect to each other at different phase
values. When all the points achieve the maximum value simultaneously; this will
cause the output envelope to suddenly shoot up which causes a 'peak' in the output
envelope. Due to presence of large number of independently modulated subcarriers in
an OFDM system, the peak value of the system can be very high as compared to the
average of the whole system. This ratio of the peak to average power value is termed
as Peak-to- Average Power Ratio. For the discrete-time version x[n],PAPR is
expressed as
(24)
Where E[ ] is the expectation operator. PAPR is evaluated per OFDM symbol. An
OFDM signal consists of a number of independently modulated sub-carriers which
can give a large PAPR when added up coherently. When N signals are added with the
same phase they produce a peak power that is N times the average power of the
signal. So OFDM signal has a very large PAPR, which is very sensitive to
nonlinearity of the high power amplifier.
The performance of a PAPR reduction scheme is usually demonstrated by three
main factors: the Complementary Cumulative Distributive Function (CCDF), Bit
Error Rate (BER), and transmitted signal power. These factors are explained below

3.5.1. Complementary Cumulative Distributive Function (CCDF)
In practice, the empirical CCDF is the most informative metric used for evaluating the
PAPR. PAPR reduction capability is measured by the amount of CCDF reduction
achieved. CCDF provides an indication of the probability of the OFDM signal’s
envelope exceeding a specified PAPR threshold within the OFDM symbol and is
given by
CCDF[ PAPR(xn
(t))] =prob(PAPR(xn
(t)> )) (25)
Where PAPR (xn
(t)) is the PAPR of the nth
OFDM symbol and is some threshold.
Based on the CLT, the envelope of the OFDM signal follows the Rayleigh
distribution and consequently its energy distribution becomes an exponential, or
equivalently, a central chi-square distribution with two degrees of freedom and zero
mean with a CDF given by
CDF( )=(1- ) (26)
The probability that the PAPR of the OFDM signal with N subcarriers is below a
threshold is the probability that all the N samples are below the threshold. Assuming
that the OFDM samples are mutually independent, this probability can be given as
prob(PAPR< )= CDF[ PAPR(xn
(t))] = (27)
3.5.2. Bit Error Rate
The performance of a modulation technique can be quantified in terms of the required
signal to noise ratio(SNR) to achieve a specific bit error rate (BER). Although the
main focus of PAPR reduction techniques is to reduce the CCDF, this is usually
achieved at the expense of increasing the BER. Clipping the high peaks of the OFDM
signal by the PA causes a substantial in-band distortion that leads to higher BER.
Other techniques may require that side information be transmitted as well. If the side
information is received incorrectly at the receiver, the whole OFDM symbol is
recovered in error and the BER performance degrades.
3.5.3. Transmitted signal power
Some PAPR reduction techniques require that the average power of the transmitted
signal be increased. If the linear region of the PA is not stretched to accommodate the
new signal, the signal will traverse the nonlinear region leading to higher distortions
and degraded BER performance. However, this solution increases the hardware cost.
3.6 PAPR Reduction Techniques
PAPR reduction techniques vary according to the requirement of the system and are
dependent on various factors such as PAPR Spectral efficiency, reduction capacity,
increase in transmit signal power, loss in data rate, complexity of computation and
increase in the bit-error rate(BER) at the receiver end are various factors which are
taken into account before adopting a PAPR reduction technique of the system.
3.6.1 PAPR Reduction by Clipping
One of the simplest signal distortion methods is the method of clipping the high peaks
of the OFDM signal prior to passing it through the PA. This method employs a clipper
that limits the signal envelope to a predetermined clipping level (CL) if the signal
exceeds that level; otherwise, the clipper passes the signal without change as defined
by

(29)
where x[n] is the OFDM signal, CL is the clipping level and x[n] is the angle of x[n].
Clipping is a nonlinear process that leads to both in-band and out-of-band distortions.
While the latter one causes spectral spreading and can be eliminated by filtering the
signal after clipping, the former can degrade the BER performance and cannot be
reduced by filtering. However, oversampling by taking longer IFFT can reduce the in-
band distortion effect as portion of the noise is reshaped outside of the signal band
that can be removed later by filtering. Filtering the clipped OFDM signal can preserve
the spectral efficiency by eliminating the out-of band distortion and, hence, improving
the BER performance but it can lead to peak power re growth.
The simulations are conducted for the OFDM signal without clipping and when
clipping is used with a clipping ratio (CR) of 1dB and 5dB. The CR is related to the
clipping level by the expression
(30)
Where, E[x[n]] is the average of the OFDM signal x[n].
4 BER PERFORMANCE OF MIMO OFDM SYSTEM
At first the performance of Alamouti’s Space Time Block Coded MIMO-OFDM
system under Rayleigh fading channel is investigated with various antennas
configurations. The simulation model employs BPSK modulation scheme and
Alamouti’s coding scheme using two transmit antennas and more than one receive
antennas. Table 1 shows the OFDM parameters considered for simulation.
Table 1 OFDM parameters considered for simulation
Parameters Value
Modulation BPSK
FFT size 64
No of symbols 10^4
No of sub carriers 52
Figure 3 BER performance of ASTBC based MIMO OFDM system for various antenna
configurations
0 5 10 15 20 25
10
-5
10
-4
10
-3
10
-2
10
-1
SNR dB
BitErrorRate
BER Performance of Alamouti’s STBC for Various Antenna Configuration
Alamouti STBC (NT=2, NR=1)

4.1. BER Performance with and without clipping
The simulations are conducted for the OFDM signal without clipping and when
clipping is used with a clipping ratio (CR) of 1dB and 5dB. The CR is related to the
clipping level by the expression. Figure below shows the BER performance of the
system with and without clipping and Empirical CCDF with and without clipping for
different values of CR.
Figure 4 BER performance with and without clipping for different values of CR
Figure 5 Empirical CCDF with and without clipping for different CR
Table 2 shows the parameters used for simulation of clipping technique.
Table 2 Parameters used for simulation of clipping technique
Parameters Value
Modulation BPSK
FFT size 64
No of symbols 10^4
No of sub carriers 52
Clipping Ratio 1dB,5Db
4.2 Deterministic Channel Capacity of MIMO-OFDM System
For a MIMO system with NT transmits and NR receive antennas, a narrowband time-
invariant wireless channel can be represented by NR NT deterministic matrix
H . Consider a transmitted symbol vector x which is composed of NT
0 2 4 6 8 10 12 14 16 18 20
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR(dB)
BER
BER without clipping and with clipping for different values of CR
no clipping
CR = 5dB
CR = 1dB
0 2 4 6 8 10 12
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
PAPR threshold(dB)
CCDF
Empirical CCDF without clipping and with clipping for different values of CR.
no clipping
CR = 1dB
CR = 5dB

independent input symbols x1,x2,x3......... . Then, the received signal y can be
written in a matrix form as follows.
Y= (31)
where (z=z1,z2,z3.......... T
is a noise vector which is assumed to be zero mean
circular symmetric complex Gaussian (ZMCSCG). The autocorrelation of transmitted
signal vector is given by
(32)
The capacity of a deterministic channel is defined as
bits/channel use in which f(x) is the probability density function (PDF) of the transmit
signal vector x, and is the mutual information of random vectors x and y.
From the fundamental principle of the information theory, the mutual information of
the two continuous random vectors x and y is given as
(33)
in which H(y) is the differential entropy of y and is the conditional
differential entropy of y when x is given. Using the statistical independence of the two
random vectors z and x in Equation 31, we can write equation 33 as follows
(34)
From the equation 33 we observe that H(z) is a constant, we can see that the
mutual information is maximized when H(y) is maximized. Now, the auto-correlation
matrix of y is given as
(35)
Putting the value of equation 31 in equation 35 we find
(36)
where, Ex the energy of the transmitted signals and N0 is the power spectral density of
the additive noise The differential entropy H(y) is maximized when y is
ZMCSCG which consequently requires x to be ZMCSCG. The mutual information
can be found from equation 34 as follows
= + ) bps/Hz (37)
Then, the channel capacity of deterministic MIMO channel in the case of CSI
known to both receiver and transmitter side is expressed as
bps/Hz (38)
When H is not known at the transmitter side, one can spread the energy equally
among all the transmit antennas so that the autocorrelation function of the transmit
signal vector x is given as
(39)
Finally the channel capacity is given as

+ )
(40)
where r=min(NT,NR) denotes the rank of H and denotes the ith
eigen value.
4.2.1 Performance of Deterministic MIMO Channel Capacity
The performance of deterministic channel capacity per unit bandwidth is evaluated as
a function of SNR. In this simulation, a highly scattered environment is considered.
The capacity of a MIMO channel is analyzed with the antenna configuration as shown
in Table 3 below. Each channel is considered as a parallel flat fading channel. The
power in a parallel channel (after decomposition) is distributed as water filling
algorithm. Channel matrix H is measured using Rayleigh distribution function.
Table3 Antenna Configuration for MIMO channel capacity
Combination No of Transmitting
antennas
No of Receiving
antennas
1 2 2
2 3 3
3 4 4
4 5 5
This simulation computes channel capacity and PDF of elements in SVD of
matrix H, by varying the SNR from -10 dB to 20 dB, where 104 iterations are
performed.
Figure 6 Deterministic MIMO Channel Capacity in Terms of SNR
4.3 Ergodic Channel Capacity of MIMO-OFDM System
In general case, MIMO channels change randomly and hence is a random matrix
which means that its channel capacity is also randomly time varying and follows an
ergodic process in practice Then, we consider the following statistical notion of the
MIMO channel capacity.
bps/Hz (41)
-10 -5 0 5 10 15 20
0
5
10
15
20
25
30
SNR in dB
ChannelCapacity(bps/Hz)
Deterministic MIMO Channel Capacity in Terms of SNR
nt=2, nr=2
nt=3, nr=3
nt=4, nr=4
nt=5, nr=5

which is frequently known as an ergodic channel capacity. The ergodic channel
capacity for the open-loop system without using CSI at the transmitter side from
equation 41 is given as
)} (42)
Similarly, the ergodic channel capacity for the closed loop (CL) system using CSI
at the transmitter side is given as
) (43)
Sometimes the ergodic channel capacity is expressed as a function of the outage
channel capacity. The outage probability can be defined as
(44)
4.3.1 Performance of Ergodic MIMO Channel Capacity
The performance of ergodic MIMO channel capacity per unit bandwidth is evaluated
as a function of SNR. Cumulative density function is also evaluated for ergodic
channel capacity.
Figure 7 Ergodic MIMO Channel Capacity in Terms of SNR
4.4 Capacity of MIMO Correlated Fading Channel
In general, the MIMO channel gains are not independent and identically distributed
(i.i.d.) and the capacity of the MIMO channel are closely related to the channel
correlation. For this reason, consider the capacity of the MIMO channel when the
channel gains between transmit and received antennas are correlated. We model the
correlated channel as follows:
(45)
Where Hw denotes the independent and identically distributed (i.i.d) Rayleigh fading
channel gain matrix and Rt is the correlation matrix taking correlations between the
transmit antennas, Rr is the correlation matrix taking correlations between the receive
antennas. Then the correlated channel capacity can be represented as
) (46)
-10 -5 0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
F(X)
Empirical CDF
NT=NR=1
NT=NR=2
NT=NR=3
NT=NR=4

From the above equations, let us consider two cases for simulation.
Case1: Correlation exists between transmit and receive antennas, transmit
antennas and receive antennas but the correlation matrix Rt and Rr are identical
Figure 8 Capacity of i.i.d and Correlated Channel in Terms of SNR with Correlation exist
between the Transmit Antennas and Receive Antennas but Same Correlation Matrix
Observation: Figure8 shows the capacity of i.i.d and correlated channel in terms of
SNR with correlation exists between the transmit antennas and receive antennas but
same correlation matrix. From figure 8 I observe that at 15 dB of SNR value 4×4 i.i.d
channel provide 16.22 bps/Hz whereas 3×3 i.i.d channel provides 11.8 bps/Hz and
4×4 correlated channel provides12.34 bps/Hz. So i.i.d channel outperforms the
correlated channel.
Case 2: Correlation exists between transmit and receive antennas, transmit
antennas and receive antennas but the correlation matrix Rt and Rr are not identical.
Figure 9 Capacity of i.i.d and Correlated Channel in Terms of SNR with Correlation Exists
between the Transmit Antennas and Receive Antennas but different Correlation Matrix
Observation:
Figure 9 shows the capacity of i.i.d and correlated channel in terms of SNR with
correlation exists between the transmit antennas and receive antennas but different
-10 -5 0 5 10 15 20
0
5
10
15
20
25
SNR in dB
Capacity of i.i.d and Correlated Channel in Terms of SNR with Correlation Exists between the Transmit Antennas and Receive Antennas but Same Correlation Matrix
3×3 correlated channel
3×3 i.i.d channel
4×4 i.i.d channel
-10 -5 0 5 10 15 20
0
5
10
15
20
25
SNR in dB
Capacity of i.i.d and Correlated Channel in terms of SNR with Correlation Exists between the Transmit Antennas and Receive Antennas but Different Correlation Matrix
3×3 i.i.d channel
4×4 i.i.d channel

correlation matrix. In this case I noticed that 4×4 i.i.d channel provide 22bps/Hz
whereas 4×4 correlated channel provides 14 bps /Hz. So i.i.d channel outperforms the
correlated channel.
5. CONCLUSION
The performance of the ASTBC based MIMO OFDM system under Rayleigh fading
channel is evaluated and it is observed that the performance of two transmit antennas
with more receive antennas is much better than that of the system with two transmit
antenna and less receive antennas in term of BER due to the more diversity gain of
Alamouti’s code. The performance of deterministic, ergodic and correlated MIMO
channel capacity is evaluated. It is observed that the channel capacity increases with
the number of antennas added to the system, independent and identically distributed
channel outperforms the correlated channel. Finally, to reduce PAPR clipping
technique is applied and I observed that as the CR is reduced, the CL is lowered down
and more parts of the OFDM signal are clipped and hence, the BER is increasing and
the empirical CCDF is decreasing.
REFERENCES
[1] Md.Mejbaul Haque, Mohammad Shaifur Rahman and Ki-Doo Kim” Performance
Analysis of MIMO-OFDM for 4G Wireless Systems under Rayleigh Fading
Channel”, International Journal of Multimedia and Ubiquitous Engineering Vol.
8, No. 1, January, 2013.
[2] Yasir Rahmatallah, Seshadri Mohan” Peak-To-Average Power Ratio Reduction
in OFDM Systems:A Survey And Taxonomy”,IEEE Communications Surveys &
Tutorials, VOL. 15, NO. 4, Fourth Quarter 2013.
[3] Hemangi Deshmukh , Harsh Goud,” Capacity Analysis of MIMO OFDM System
using Water filling Algorithm”, International Journal of Advanced Research in
Computer Engineering & Technology (IJARCET) Volume 1, Issue 8, October
2012.
[4] K.Hariprasad reddy, M.Anusha”MIMO-OFDM using Power allocation in
WATERFILLING algorithm based on SVD process”, International Journal of
Engineering Science & Advanced Technology Volume-3, Issue-5, 197-204.
[5] Shilpa Bavi, Sudhirkumar Dhotre” PAPR Reduction in OFDM System Using
Clipping and Filtering Method” , International Journal of Advanced Research in
Computer Science and Software Engineering, Volume 5, Issue 2, February 2015.
[6] Nimay Chandra Giri, SK Mohammed Ali, Rupanita Das” BER Analysis and
Performance of MIMO-OFDM System using BPSK Modulation Scheme for Next
Generation Communication Systems”,International Journal of Engineering
Sciences & Research Technology, March, 2014.
[7] Arun Gangwar, Manushree Bhardwaj” An Overview: Peak to Average Power
Ratio in OFDM system & its Effect”,International Journal of Communication and
Computer Technologies Volume 01 – No.2, Issue: 02 September 2012.
[8] Parneet Kaur , Ravinder Singh” Complementary Cumulative Distribution
Function for Performance Analysis of OFDM Signals,“IOSR Journal of
Electronics and Communication Engineering, ISSN : 2278-2834 Volume 2, Issue
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[9] Prof. A.K Jaiswal, Er.Anil Kumar, Anand Prakash Singh”Performance Analysis
of MIMO OFDM system in Rayleigh fading channel” ,International Journal of
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[10] Mir Muhammad Lodro, Muhammad Hanif Abro” Ergodic Capacity of MIMO
Correlated Channels in Multipath Fading Environment with known Channel State
Information”, International Journal of Electrical and Computer Engineering,
Vol.2, No.5, October 2012.
[11] Yong Soo Cho,Jaekwon Kim,Won Young Yang,Chung Gu-Kang”MIMO OFDM
Wireless communications with MATLAB”,IEEE press.
[12] Ke-Lin DU and M.N.S.Swamy, “Wireless Communication Systems”, Cambridge
University Press,2010.
[13] HELMUT BOLCSKEI, ETH ZURICH”MIMO OFDM Wireless Systems:
Basics, Perspectives and Challenges”, IEEE Wireless Communications, August
2006.
[14] Asha Ravi, J.Nalini, Kanchana S.R.”Antenna Management of Space-Time Shift
Keying Systems” International Journal of Computer Science And Technology,
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