International Journal of Scientific & Engineering Research, Volume 2, Issue 11, November-2011 1
ISSN 2229-5518
BER Analysis of MIMO-OFDM System Using OSTBC Code Structure for M-PSK under Different Fading Channels
Lavish Kansal, Ankush Kansal and Kulbir Singh
In this paper the analysis of high level of modulations ( i.e M-PSK for different values of M) on MIMO-OFDM system is presented. Here AWGN and Rayleigh channel have been used for analysis purpose and their effect on BER for high data rates has been presented. The proposed MIMO-OFDM system with OSTBC using 8X8 antenna configuration has better performance in terms of BER vs SNR than the other systems.
Keywords:- MIMO, OFDM, OSTBC, M-PSK
—————————— ——————————
s the demand for high-data rate multimedia grows, several approaches such as increasing modulation
order or employing multiple antennas at both
transmitter and receiver have been studied to enhance
the spectral efficiency [1],[2]. In today’s communication systems Orthogonal Frequency Division Multiplexing (OFDM) is a widespread modulation technique. Its benefits are high spectral efficiency, robustness against inter-symbol interference, ease of implementation using the fast Fourier transform (FFT) and simple equalization techniques.
Recently, there have been a lot of interests in combining the OFDM systems with the multiple-input multiple- output (MIMO) technique. These systems are known as MIMO OFDM systems.
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Lavish Kansal is currently Assistant Professor in Department of
Electronics & Electrical Engineering, Lovely Professional University, Jalandhar, Punjab, India, Email: - lavish.s690@gmail.com.
Ankush Kansal is currently Assitant Professor in Electronics & Communication Engineering Department, Thapar University, Patiala, India, Email: - akansal@thapar.edu.
Spatially multiplexed MIMO is known to boost
the throughput, on the other hand, when much higher throughputs are aimed at, the multipath character of the environment causes the MIMO channel to be frequency- selective. OFDM can transform such a frequency- selective MIMO channel into a set of parallel frequency- flat MIMO channels and also increase the frequency efficiency. Therefore, MIMO-OFM technology has been researched as the infrastructure for next generation wireless networks [3].
Therefore, MIMO-OFDM, produced by employing multiple transmit and receive antennas in an OFDM system has becoming a practical alternative to single carrier and Single Input Single Output (SISO) transmission [4]. However, channel estimation becomes computationally more complex compared to the SISO systems due to the increased number of channels to be estimated. This complexity problem is further compounded when the channel from the ith transmit antenna to the mth receive antenna is frequency-selective. Using OFDM, information symbols are transmitted over several parallel independent sub-carriers using the computationally efficient IFFT/FFT modulation/demodulation vectors [5]-[8].
These MIMO wireless systems, combined with OFDM, have allowed for the easy transmission of symbols in time, space and frequency. In order to extract diversity from the channel, different coding schemes have been developed. The seminal example is the
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Alamouti Space Time Block (STB) code [9] which could
extract spatial and temporal diversity. Many other codes
have also been proposed [10]–[12] which have been able to achieve some or all of the available diversity in the channel at various transmission rates.
In open-loop schemes, there are generally two approaches to implement MIMO systems. One is to increase the spatial transmit diversity (STD) by means of space-time coding and space-frequency coding. Another is to raise the channel capacity by employing spatial division multiplexing (SDM) that simultaneously transmits independent data symbols through multiple transmit antennas. STD mitigates impairments of channel fading and noise, whereas SDM increases the spectral efficiency [13], [14].
In section 2, general theory of OFDM and the necessary condition for orthogonality is discussed. In section 2.1, the signal model of OFDM system with SISO configuration is discussed in detail with the help of block diagram. In section 2.2, M-PSK (M- PHASE SHIFT KEYING) modulation technique is discussed in detail. In section 2.3, different channels used for analyses purpose are discussed namely AWGN and Rayleigh channel. In section 3, general theory about the MIMO system is presented. In section 4, MIMO-OFDM system with OSTBC is discussed. In this section general theory about OSTBC is presented. In section 4.1, idea about the linear detection technique i.e. Zero Forcing equalization for MIMO-OFDM system is presented. Finally in section 5, the simulated results based on the performance of MIMO-OFDM system in AWGN and Rayleigh channels have been shown in the form of plots of BER vs SNR for M-PSK modulation and for different antenna configurations.
OFDM is a multi-carrier modulation technique where data symbols modulate a sub-carrier which is taken from orthogonally separated sub-carriers with a separation of ‘fk’ within each sub-carrier. Here, the spectra of sub-carrier is overlapping; but the sub-carrier signals are mutually orthogonal, which is utilizing the bandwidth very efficiently. To maintain the orthogonality, the minimum separation between the sub-carriers should be ‘fk’ to avoid ICI (Inter Carrier Interference) .
By choosing the sub-carrier spacing properly in relation
to the channel coherence bandwidth, OFDM can be used
to convert a frequency selective channel into a parallel collection of frequency flat sub-channels. Techniques that are appropriate for flat fading channels can then be applied in a straight forward fashion .
Figure.1 shows the block diagram of a OFDM system with SISO configuration. Denote Xl ( l = 0,1,2,....,N −1) as the modulated symbols on the lth transmitting subcarrier of OFDM symbol at transmitter, which are assumed independent, zero-mean random variables, with average power .The complex baseband OFDM signal at output of the IFFT can be written as:
xn = l (1)
where N is the total number of subcarriers and the
OFDM symbol duration is T seconds.
At the receiver, the received OFDM signal is mixed with local oscillator signal, with the frequency offset deviated from Δf the carrier frequency of the received signal owing to frequency estimation error or Doppler velocity, the received signal is given by:
n = (xn 
hn) + zn (2)
where hn, , and zn represent the channel impulse response, the corresponding frequency offset of received signal at the sampling instants: Δf T is the frequency offset to subcarrier frequency spacing ratio, and the AWGN respectively, while denotes the circular convolution. Assuming that a cyclic prefix is employed; the receiver have a perfect time synchronization. Note that a discrete Fourier transform (DFT) of the convolution of two signals in time domain is equivalent to the multiplication of the corresponding signals in the frequency domain.
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Source
Data
Channel
Coding
Inter leaving
Modulation
I
S F P
/ F /
P T S
Cyclic
Prefix
CHANNEL
Remove Cyclic Prefix
F
S F P
/ T /
P S
Demodulation De-Inter leaving
Channel
Decoding
Receive
Data
Figure 1: Block Diagram of OFDM system
Then the output of the FFT in frequency domain signal
on the kth receiving subcarrier becomes:
k = lHlYl - k + Zk , k=0,….,N-1
= XkHkU0 + HlYl - k + Zk (3)
The first term of (3) is a desired transmitted data symbol Xk. The second term represents the ICI from the undesired data symbols on other subcarriers in OFDM symbol. Hk is the channel frequency response and Zk denotes the frequency domain of zn. The term Yl - k is the coefficient of FFT (IFFT), is given by:
Yl - k = (4)
when the channel is flat, Yl - k can be considered as a
complex weighting function of the transmitted data
symbols in frequency domain [15].
Modulation is the process of mapping the digital information to analog form so it can be transmitted over the channel. Consequently every digital communication system has a modulator that performs this task. Closely related to modulation is the inverse process, called demodulation, done by the receiver to recover the transmitted digital information [16].
Modulation of a signal changes binary bits into an analog waveform. Modulation can be done by changing
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the amplitude, phase, and frequency of a sinusoidal
carrier. There are several digital modulation techniques
used for data transmission. The nature of OFDM only allows the signal to modulate in amplitude and phase.
There can be coherent or non-coherent modulation techniques. Unlike non-coherent modulation, coherent modulation uses a reference phase between the transmitter and the receiver which brings accurate demodulation together with receiver complexity [17].
In M-ary PSK modulation, the amplitude of the transmitted signals was constrained to remain constant, thereby yielding a circular constellation. By allowing the amplitude to vary with the phase, a new modulation scheme called quadrature amplitude modulation (QAM) can also be obtained as shown in figure 2 [16].
Phase-shift keying (M-PSK) for which the signal set is: Si(t)= *(cos (2π*fc + 2 )) (5)
i=1,2,…..M & 0 < t < Ts
where Es the signal energy per symbol Ts is the symbol duration, and fc is the carrier frequency.
This phase of the carrier takes on one of the M possible values, namely
θi = 2(i-1)π/M where i=1,2,…,M
An example of signal-space diagram for 8-PSK is shown in figure 2
Figure 2 :- PSK Constelations
Wireless transmission uses air or space for its transmission medium. The radio propagation is not as smooth as in wire transmission since the received signal is not only coming directly from the transmitter, but the combination of reflected, diffracted, and scattered copies of the transmitted signal.
Reflection occurs when the signal hits a surface where partial energy is reflected and the remaining is transmitted into the surface. Reflection coefficient, the coefficient that determines the ratio of reflection and transmission, depends on the material properties.
Diffraction occurs when the signal is obstructed by a sharp object which derives secondary waves. Scattering occurs when the signal impinges upon rough surfaces, or small objects. Received signal is sometimes stronger than the reflected and diffracted signal since scattering spreads out the energy in all directions and consequently provides additional energy for the receiver which can receive more than one copies of the signal in multiple paths with different phases and powers. Reflection, diffraction and scattering in combination give birth to multipath fading. [18]
Additive white Gaussian noise (AWGN) channel is a universal channel model for analyzing modulation schemes. In this model, the channel does nothing but add a white Gaussian noise to the signal passing through it. This implies that the channel’s amplitude frequency response is flat (thus with unlimited or infinite bandwidth) and phase frequency response is linear for all frequencies so that modulated signals pass through it without any amplitude loss and phase distortion of frequency components. Fading does not exist. The only distortion is introduced by the AWGN. AWGN channel is a theoretical channel used for analysis purpose only.
The received signal is simplified to:
r(t) = s(t) + n(t) (6)
where n(t) is the additive white Gaussian noise [18].
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Constructive and destructive nature of multipath components in flat fading channels can be approximated by Rayleigh distribution if there is no line of sight which means when there is no direct path between transmitter and receiver. The received signal can be simplified to:
r(t) = s(t)*h(t) + n(t) (7)
where h(t) is the random channel matrix having Rayleigh distribution and n(t) is the additive white Gaussian noise. The Rayleigh distribution is basically the magnitude of the sum of two equal independent orthogonal Gaussian random variables and the probability density function (pdf) given by:
p(r) = 0 (8)
where σ2 is the time-average power of the received signal
[19],[20].
Multi-antenna systems can be classified into three main categories. Multiple antennas at the transmitter side are
where the general element, hnt,nr (τ, t) represents the
complex time-variant channel transfer function at the
path between the nt-th transmitter antenna and the nr-th receiver antenna. NT and NR represent the number of transmitter and receiver antennas respectively.
Tx antennas MIMO-channels Rx antennas
Tx 1 Rx 1
Tx 2 Rx 2
Tx M Rx N
Figure 3 Block Diagram of a generic MIMO system with
M transmitters and N receivers
Derived from Shannon’s law, for the capacity of MIMO channel the following expression was proven in [21] and [22]:
C = (det(I + HRssHH)) (10)
usually applicable for beam forming purposes.
where H denotes the channel matrix and HH
its
Transmitter or receiver side multiple antennas for realizing different (frequency, space) diversity schemes. The third class includes systems with multiple transmitter and receiver antennas realizing spatial multiplexing (often referred as MIMO by itself).
In radio communications MIMO means multiple antennas both on transmitter and receiver side of a specific radio link. In case of spatial multiplexing different data symbols are transmitted on the radio link by different antennas on the same frequency within the same time interval. Multipath propagation is assumed in order to ensure the correct operation of spatial multiplexing, since MIMO is performing better in terms of channel capacity in a rich scatter multipath environment than in case of environment with LOS (line of sight). This fact was spectacularly shown in [21]. MIMO transmission can be characterized by the time variant channel matrix:
H( ) = (9)
transpose conjugate, I represents the identity matrix and
Rss the covariance matrix of the transmitted signal s.
The transmit diversity scheme designed by Alamouti can be used only in a system with two transmit antennas. It turns out that this technique belongs to a general class of codes named Space–Time Block Codes or, more precisely, Orthogonal STBCs, since they are based on the theory of orthogonal designs. The authors of [5] introduced the theory of generalized orthogonal designs in order to create codes for an arbitrary number of transmit antennas.
The general idea behind STBCs construction is based on finding coding matrices X that can satisfy the following condition:
X. XH = p. . InT (11)
In this equation, XH is the Hermitian of X, p is a constant,
InT is the identity matrix of size nT × nT, nT represents
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the number of transmit antennas, and n is the number of symbols xi transmitted per transmission block in X. The generalized theory of orthogonal design is exploited to provide codes that satisfy Equation 11.
The orthgonality property of STBCs is reflected in the fact that all rows of X are orthogonal to each other. In other words, the sequences transmitted from two different antenna elements are orthogonal to each other for each transmission block. For real signal, it is possible to reach full rate. However, it has been proven in [5] that this statement is false for two-dimensional constellations, i.e., complex signals. The encoding and decoding approaches follow the pattern described in Alamouti’s scheme [24].
For complex signals, the theory of orthogonal designs can be used to generate coding matrices that achieve a transmission rate of 1/2 for the cases of 3 and 4 transmission antennas:
X1/2 =
X1/2 = (12)
Using the theory of orthogonal design to construct STBCs is not necessarily the optimal approach. There exist some sporadic STBCs mentioned in the literature, [27], that can provide a transmission rate of 3/4 for schemes of either 3 or 4 transmit antennas.
X3/4 =
X3/4 = (13)
It is important to notice that the channel coefficients must remain constant during the transmission of a block of coded symbols X.
The decoding of the STBCs described above can be easily deduced from the encoding matrix. Let us assume that we wish to estimate symbols xp and that we have defined by the received signal from antenna j at time instance k. The values to be added at the linear combiner are:
+ (hj,i). if we have xp at column k and line
(transmit antenna) i of X.
- (hj,i). if we have -xp at column k and line
(transmit antenna) i of X.
+ (hj,i). ( )* if we have (xp)* at column k and line
(transmit antenna) i of X.
- (hj,i). ( )* if we have -(xp)* at column k and line
(transmit antenna) i of X.
The linear combiner sum is realized for all receive antennas j [25].
It is important to remember that STBCs based on orthogonal design do not achieve a rate of 1 for complex signal constellations. In [8], it has been shown for 3 and 4 transmit antennas the maximum possible rate is 3/4 with
4 delays. For 5 to 8 transmit antennas, the achievable
rate is 1/2 with 8 delays, and for the 9 to 16 case, the rate becomes 5/16 in 16 time instances. In order to achieve the rate of a SISO system, the orthogonal property of STBCs must be broken as described in [26].
Signal detection of MIMO-OFDM system can be carried out by various sub-carrier channel signal detection. Although the whole channel is a frequency-selective fading, but various sub-carriers channel divided can be regarded as flat fading, so the flat fading MIMO signal detection algorithm for MIMO-OFDM system can be directly into the detection of all sub-channels, and signal detection algorithm of the corresponding MIMO-OFDM system can be obtained. Similarly, the other optimization algorithms used in flat fading MIMO signal detection can also be leaded into the MIMO-OFDM system. MIMO-OFDM detection methods consist of linear and nonlinear detection test.
Zero Forcing algorithm is regard the signal of each transmitting antenna output as the desired signal, and regard the remaining part as a disturbance, so the mutual interference between the various transmitting antennas can be completely neglected. The specific algorithm is as follows:
For k = 0, 1, 2,………….,K-1, so that,
R(k) = [R1(k),R2(k),…….……..,RN(k)]T (14)
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S(k) = [S1(k),S2(k),…………….,SM(k)]T (15)
N(k) = [N1(k),N2(k),……...……,NN(k)]T (16)
H(k) = (17)
Here R(k), S(k), N(k) respectively express output signal, the input signal and noise vector of the k sub-channels in MIMO-OFDM system, for M transmitting antennas and N receiving antennas, H(k) expresses channel matrix of the k sub-channels, mathematical expression of sub- channel in the MIMO-OFDM system is as follows:
R(k) = H(k)S(k) + N(k) (18)
There is a linear relationship between input signal S(k) and output signal R(k), that is similar to the flat fading channel for each subcarrier channel in MIMO-OFDM system. Its equivalent block diagram is shown in Figure
5. Therefore, signal detection can be transformed into K
sub-channels in their signal detection to complete in
MIMO-OFDM system and each sub-channel detection of
the above can be used flat fading MIMO channel to
achieve the detection algorithm.
Zero-forcing (ZF) detection algorithm for MIMO detection algorithm is the most simple and basic algorithms, and the basic idea of zero forcing algorithm is get rid of MIMO-channel interference by multiplying received signal and the inverse matrix of channel matrix. Zero-Forcing solution of MIMO-OFDM system is as follows:
SZF = H-1 R = S + H-1 N (19)
In which H-1 is the channel matrix for the generalized inverse matrix, the type is obtained for hard-decision demodulation after that to be the source signal estimates:
ZF = E(SZF) (20)
S1(k) R1(k) (k)
S2(k) R2(k) (k)
Sub- carrier channel H(k)
Sub- channel detection
SM(k) RN(k) (k)
Channel
Estimation
Figure 4: Baseband block diagram of k subcarrier channel in MIMO-OFDM system
The system discussed above has been designed and
results are shown in the form of SNR vs BER plot for different modulations and different channels. Here different antenna configurations such as 2x2, 4x4 and
8x8 are used to show the advantage in term of SNR of
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using 8x8 antenna configuration over the other
configurations. The analyses have been done for three
channels AWGN, and Rayleigh channel.
BER vs SNR Plot for MIMO-OFDM for 128-PSK for AWGN channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1
10
BER vs SNR Plot for MIMO-OFDM using 32-PSK for AWGN channel
0
10
Tx = Rx = 2
Tx = Rx = 4 -2
Tx = Rx = 8 10
-1
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
-2
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
BER vs SNR Plot for MIMO-OFDM using 256-PSK for AWGN channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1
10
BER vs SNR Plot for MIMO-OFDM using 64-PSK for AWGN channel
0
10
Tx = Rx = 2 -2
Tx = Rx = 4
Tx = Rx = 8
-1
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
-2 Figure 5(d)
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
BER vs SNR Plot for MIMO-OFDM using 512-PSK for AWGN channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1
10
-2
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
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BER vs SNR Plot for MIMO-OFDM using 1024-PSK for AWGN channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1
10
PSK, with the increase in the level of the modulation the
BER will also increase. In order to mitigate that effect we
have to increase the SNR values for higher level modulation.
BER vs SNR Plot for MIMO-OFDM using 32-PSK for RAYLEIGH channel
0
10
Tx = Rx = 2
-2 Tx = Rx = 4
Tx = Rx = 8
-1
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
-2
10
Figure 5 (a)-(f) : SNR vs BER plots for M-PSK over
AWGN channel for MIMO-OFDM system employing different antenna configurations (a) 32-PSK, (b) 64-PSK , (c) 128-PSK , (d) 256-PSK , (a) 32- PSK , (e) 512-PSK & (f) 1024-PSK
For MIMO-OFDM system SNR vs BER plots using M- PSK over AWGN channel employing different antenna configurations are shown in Figure 5. The graphs gives the clear idea that in MIMO-OFDM system as we goes on increasing the no. of Transmitters and Recievers the BER keeps on decreasing due to space diversity and the proposed system provide better BER performance as compared to the other antenna configurations.
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
BER vs SNR Plot for MIMO-OFDM using 64-PSK for RAYLEIGH channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1
10
-2
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
In table 5.1 the advantage of using higher order (8 X 8)
antenna configuration over lower order (4 X 4) antenna
configuration is shown in the form of SNR gain in dB for M-PSK. As, we goes on to higher order antenna configuration the BER will keeps on decreasing. For M-
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BER vs SNR Plot for MIMO-OFDM using 128-PSK for RAYLEIGH channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
BER vs SNR Plot for MIMO-OFDM using 1024-PSK for RAYLEIGH channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1 -1
10 10
-2 -2
10 10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
BER vs SNR Plot for MIMO-OFDM using 256-PSK for RAYLEIGH channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1
10
-2
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
BER vs SNR Plot for MIMO-OFDM using 512-PSK for RAYLEIGH channel
0
10
Tx = Rx = 2
Tx = Rx = 4
Tx = Rx = 8
-1
10
-2
10
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
-3
10
0 10 20 30 40 50 60 70 80 90
signal to noise ratio (dB)
Figure 5 (a)-(f) : SNR vs BER plots for M-PSK over Rayleigh channel for MIMO-OFDM system employing different antenna configurations (a) 32-PSK, (b) 64-PSK , (c) 128-PSK , (d) 256-PSK , (a) 32- PSK , (e) 512-PSK & (f) 1024-PSK
SNR vs BER plots for M-PSK over Rayleigh channel for MIMO-OFDM system employing different antenna configurations are presented in Figure 6. Here the graphs indicates that in MIMO-OFDM system the BER keeps on decreasing due to space diversity, when we increases the no. of Transmitters and Recievers and the proposed system provide better BER performance as compared to the other antenna configurations. Here the BER is higher than the BER for MIMO-OFDM with M- PSK in AWGN channel.
The superiority of using higher order (8 X 8) antenna configuration over lower order (4 X 4) antenna configuration is shown in the form of SNR gain in dB for M-PSK over Rayleigh channel in table 5.2. As, we goes on to higher order antenna configuration the BER will keeps on decreasing. The SNR gain varies from one modulation level to another due to random noise and fading effect.
Different Modulation levels | SNR improvement for Rayleigh Channel (db) |
32-PSK | 4 dB |
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64-PSK | 3.7 dB |
128-PSK | 6 dB |
256-PSK | 5.4 dB |
512-PSK | 7 dB |
1024-PSK | 5 dB |
In this paper, an idea about the performance of the MIMO-OFDM systems at higher modulation levels and for different antenna configurations is presented. MIMO-OFDM system can be implemented using higher order modulations to achieve large data capacity. But there is a problem of BER (bit error rate) which increases as the order of the modulation increases. The solution to this problem is to increase the value of the SNR so, that the effect of the distortions introduced by the channel will also goes on decreasing, as a result of this, the BER will also decreases at higher values of the SNR for high order modulations.
The motive of using high order antenna configuration (8x8) is to increase the space diversity, which will automatically lower the BER at given SNR as compared to lower order Antenna configuration (2x2, 4x4). By doing so, higher data capacity at any given SNR can be achieved. The MIMO-OFDM system with 8X8 antenna configuration provides better performance in terms of SNR as compared to the MIMO-OFDM system with 4X4 antenna configuration at a BER of 10-2, these results are shown in the table 1 and 2.
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