Estimation of bit error rate in 2×2 and 4×4 multi-input multi-output-orthogonal frequency division multiplexing systems

Multiple-input, multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) systems with multiple input antennas and multiple output antennas in dynamic environments face the challenge of channel estimation. To overcome this challenge and to improve the performance and signal-to-noise ratio, in this paper we used the Kalman filter for the correct estimation of the signal in dynamic environments. To obtain the original signal at the receiver end bit error rate factor plays a major role. If the signal to noise ratio is high and the bit error rate is low then signal strength is high, the signal received at the receiver end is almost similar to the i th transmitted signal. The dynamic tracking characteristic of Kalman filter is used to establish a dynamic space-time codeword and a collection of orthogonal pilot sequences to prevent interference among transmissions in this paper. Using the simulation, the Kalman filter method can be compared to the other channel estimation method presented in this paper that can track time-varying channels rapidly.


PROPOSED METHOD 2.1. MIMO
A system with single input and output (SISO), in which one transmission antenna transfers data to a particular reception antenna, is featured in most wireless communication systems. To enhance the receiver's efficiency, additional sending and receiving antennas can be added [24]. The above situation is bound to affect in the future as communication networks with multiple input and output become more prominent. MIMO techniques are commonly divided into three categories. The very first category comprises developing spatial variety in an attempt to improve performance and efficiency. Some other category employs a multilayer technique to increase the capacity [24]. Hence, by analyzing the transmission channel's characteristics MIMO wireless systems [21], [22], which utilize the same bandwidth spectrum as a SISO system and boost capacity utilization by spatial modulation, frequently include multiple transmitters and receiver antennas. Unlike classic SISO systems, using unpredictable decaying and crosstalk latency spread, the MIMO system uses mm wave to almost double the flow of data. Furthermore, MIMO enables spatial variety at both the transmitter and receiver, resulting in higher error rate transmission quality [24]. Spatial multiplexing is a technique for improving the capacity of a MIMO system by transmitting multiple independent streams of data near one another.
Future standards [24] should establish maximum throughput including the mode of transmission that accomplishes the data rate required for the lowest level of quality. Spatial multiplexing has been utilized to enhance the capacity of a MIMO link by transmitting independent data streams in the same time slot and frequency range concurrently from each transmit antenna and multiple streams of data at the receiver which uses channel information for each transmission path. Figure 1 represents the basic architecture of a M×N MIMO communication model [24]. The channel is a N×M matrix with MN sub-channels, as can be shown. The MIMO system can also be considered as a collection of many transmit beamformers, each of which transmits to one of the m Rx antennas. As per the authors, the MIMO system is a great way to maximize channel and network performance.
The multiple antenna receiver splits and demodulates the stream of data using sophisticated spatial coding and selects the most appropriate processing method [24]. Since N sub-streams are transmitted to the channel around the same time and every delivered information comprises the same frequency band, the frequency band is not enhanced. MIMO system can generate many parallel space channels if the channels are independent. The transmission of data will be improved by using these channels to broadcast information at the same time.

. MIMO channel capacity
The bandwidth in the case of MIMO will be estimated by (1),

Orthogonal frequency division multiplexing
Orthogonal frequency-division multiplexing (OFDM) [25] is an extremely versatile and effective modulation mechanism that is utilized in the development of all major wireless and wired standards today. In 4G technology, OFDM is a multicarrier and parallel transmission system which is a variant of the multicarrier modulation system frequency-division multiplexing (FDM) [25]. With its high-speed data transmission capability, high information measuring capacity, and resistance to multipath attenuation and latency, OFDM is broadly utilized in wireless communications systems.
The main schematic diagram of an OFDM transmitter to the receiver can be in Figure 1. It is a modulation and multiplexing technique. Multiplexing, in general, refers to the transmission of independent signals generated by various sources [24]. Multiplexing is a mechanism that is capable of integrating several communication signals for one of them to transmit an otherwise single signal communication method sequentially, that is used to differentiate signals in the system. The signal is divided into distinct channels in OFDM, modulated with data, and then re-multiplexed to generate the OFDM carrier [24]. To manage multimedia services, orthogonal frequency division multiplexing can support large data rates in mobile wireless networks. To estimate the channel study of the OFDM technology is necessary by improving system performance [24]. OFDM technology may be utilized efficiently in mobile networks to mitigate the disadvantages of frequency-selective fading and narrowband interference from parallel, closely spaced frequencies. If there is no orthogonality between two overlapping channels, internal channel interference would occur between the signals. Due to these important advantages, OFDM technology has been widely utilized by several wireless protocols, especially wireless local area network (WLAN), wireless metropolitan area network (WMAN), and digital video broadcasting (DVB). Complex filters are used in the modulation scheme [24].

Orthogonality's importance
The core aspect of the orthogonal system is the orthogonality of the subcarriers [26]. The "orthogonal" aspect of the OFDM title emphasizes a mathematical association between the system's carriers' frequencies. It is possible to rearrange the carriers in an OFDM signal when only similar sidebands of the individual carriers overlap. The information, on the other hand, can be received without disturbance from the other transmitters. The transmitters must be theoretically orthogonal to effectively carry out this operation. If the carrier's separation is a multiple of 1/Ts, the carriers are linearly independent [24]. Ts represents the symbol duration. If the OFDM signal is defined using Fourier transform algorithms, the component's orthogonality can be retained. The OFDM transmitter transmits out a large number of tightly packed bandpass carriers. It is important to remember that there is no crosstalk from other subchannels at the central frequency of each subchannel.

Fading
The capacity of OFDM to deal with a multi-path disturbance at the recipient is one of the reasons for the growth. This phenomenon is called multipath [27], [28]. It provides two effects: i) frequency selective fading and ii) intersymbol interference (ISI). Frequency-selective fading can be overcome mostly by observed "uniformity" of a narrow-band channel [29], [30]. On the other hand, the symbols modulation at a very low rate enables the symbols to be significantly longer than the channel response, minimizing the interference.

Orthogonal pilot symbols
Pilot symbols are reference symbols that are used to compare the transmitted symbols with them to estimate the errors. Pilot sequences are ideal sequences [31], [32]. In the orthogonal pilot symbols, the pilot symbols of the antenna do not influence the pilot symbols of another antenna in the network system in Figure 1. The transmission of pilot symbols at four antennas could be formulated as (2). Every column shows the pilot codes that are sent out by every antenna. t is the time when the first antenna transmits pilot symbols Hp and the remaining values in that column are zero. The second columns represent the transmitted pilot symbols at time t+1. By flowing through the multipath channel, the OFDM without additive white Gaussian noise can be formulated as (3).
The anticipated CFR will be given in terms using the LS network prediction model as (4).
By using the above designed orthogonal pilot symbols, we can remove interference from the antennas.

Kalman filter
The KF is used to reduce noise and to upgrade the quality of the signal that is transmitted. MIMO systems are used in 4G technology. Multiple data streams are delivered through multiple antennas in MIMO systems, causing interference between the symbols and increasing the BER. The KF is operated to analyze and upgrade the errors. It is a quick procedure that is similar to other channel estimation methods like least squares and minimum mean squares. The averaging method is used in the LS and minimum mean square error (MMSE) procedures, but the KF is an iterative process that distinguishes it from all other channel estimating techniques [33]- [36].
The KF is analogous to a hidden Markov model having Gaussian distributions for state variables. The multiplication of one Gaussian distribution with another Gaussian distribution generates another Gaussian distribution. The KF argument predicts that the system's state at time t progressed from its former state at time t-1. The system state can be formulated as (5): where Zt is a state vector at time t containing information about states. At is transition matrix of the state which put on each system state at time t-1 has an influence on the state of the system at time t. St stands for exterior control systems. Bt is the external control matrix. Qt is a covariance matrix. The system's measurement can be done in a variety of ways as (6): where Xt is the measurement vector Dt is the transformation matrix of measurements. Rt is the vector containing the AGWN for every observation in the measurement matrix and its covariance is Vt. In the case of a well-designed, single-dimensional linear system with an estimation of errors drawn from a zero-mean Gaussian distribution, the KF will be contemplated as a perfect estimator. The KF algorithm consists of two steps: prediction and measurement update. The equations for the KF projection part can be constructed as (7) and (8): where Zt|t−1 is the state information of the predicted system at time t; Pt|t−1 is the predicted error matrix correlate to Zt|t−1. The measurement revised expressions for KF can be evolved as (9)-(11): where Kt is Kalman gain, Xt|t is state information of updated system, and Pt|t is updated error matrix. KF creates a statement framework and a statistical model for a linear system for the estimation of system state information. Using projection and data enrichment, the estimated uncertainty of the system state is decreased. As a result, KF is given more importance for better-estimated performance.    Table 1, it is observed that the normalized mean square error of the least square method has more errors compared to minimum mean square error method. When the signal-to-noise ratio increases the occurrence of errors decreases and at the receiver, we receive better quality of the signal. In Figure 3, the BER performance for 4×4 antennas are better. When the number of antennas at the transmitter and receiver sides increases, the less number of data bits get corrupted. Hence the BER is less for 44 antennas than for 2×2 antennas. In Figures 3(a)

Comparisons of BER performance in 2×2 and 4×4 STBC MIMO-OFDM systems without KF
Even though 4×4 MIMO antenna installation is complex but their high data rates, low latency, and high data capacity provide better performance. 16-PSK provides more errors but we can transmit 4 bits at a time, because of this we can save channel bandwidth, and data rate increases. Therefore, we prefer 44 MIMO-OFDM with 16-PSK for better throughput.
In Tables 2 and 3, it is seen that 4×4 antennas have fewer errors when compared to 2×2 antennas. When the number of antennas at both transmitter and receiver sides increases, the same data stream bits are transmitted multiple times because of this reason 4×4 antennas are preferable. A comparison of BER in 4-PSK, 8-PSK, and 16-PSK modulation techniques is done in Tables 2 and 3. In 2×2 and 4×4 MIMO-OFDM systems 16-PSK modulation has more errors when compared to 4-PSK and 8-PSK modulation techniques, but 16-PSK is preferable because it transmits 4-bits at a time which improves data rate and data capacity and saves channel bandwidth.

Comparisons of BER performance in 2×2 and 4×4 STBC MIMO-OFDM systems with KF
From Figure 4, KF is an iterative process. KF predicts and estimates errors. Other techniques like averaging methods. KF reduces errors fast when compared with other techniques. Because of this we are getting low latency. From Figures 4(a) to 4(c), BER implementation for 4×4 antennas are preferable. When the number of antennas at the transmitter and receiver side increases, the number of data bits gets corrupted. Hence the BER is less for 4×4 antennas than for 2×2 antennas. In Figures 4(a) to 4(c), 4-PSK, 8-PSK, and 16-PSK are used. When the number of bits transmitted per sec in PSK modulation increases the BER values are increased. When number bits transmitted per sec increases interface increases. So, BER value increases. Even though 4x4 MIMO antenna installation is complex but their high data rates, low latency, and high data capacity provide better performance. 16-PSK provides more errors but we can transmit 4 bits at a time, because of this we can save channel bandwidth, and data rate increases. Therefore, we prefer 4×4 MIMO-OFDM with 16-PSK for better throughput.  KF is an iterative process. It predicts and estimates errors. It resembles the hidden Markov model because of its current and previous state matrixes. It has two stages prediction and estimation. KF estimates the current state from the previous state. KF is a faster process when compared with other techniques because of its iterative nature. It provides fewer errors because of its dynamic channel tracking property.

Comparison of SNR, BER in 2×2 and 4×4 in both cases with and without filter
From Tables 6 and 7, in conventional MIMO-OFDM, BER Performance is worst compared to MIMO-OFDM with KF. In MIMO-OFDM multiple antennas are used at both transmitters and receivers and several data streams are transmitted at once. This causes interference among adjacent data symbols, and errors occur. KF is an iterative process. It uses dynamic channel estimation and tracking property to update and estimate the errors.  Figure 5 analyzes the effect of MSE with respect to SNR in 2×2 and 44 with and without the filter. In view of MSE, the ideal case (MIMO-OFDM) without a filter offers very high MSE even for a small change in SNR changes. The MSE is more against the least-squares method against differential change in the signal-to-noise ratio frequency. Accordingly, the least-squares complex exponential (LSCE) is the least preferable for MIMO applications. Similarly, 2×2 KF CE offers less MSE in comparison with the LSCE. However, the changes in the frequency of SNR are improvable in comparison with 44 MIMO-OFDM structure. As a result, the different resultant values of MSE and SNR show the effectiveness of the proposed 44 MIMO-OFDM with STBC using the KF approach applied to STBC using the KF.

CONCLUSION
In this paper, the Kalman channel estimation method is applied to 2×2, 4×4 MIMO-OFDM antennas. KF is an iterative process that reduces the errors which occur due to interference. When the number of bits transmitted per second increases in the PSK modulation technique, then errors also increase. Even though errors are increased, the more bit PSK modulation is preferable because, when the number of bits transmitted per second increases, the data rate also increases, and channel bandwidth is reduced. KF dynamic tracking property makes it unique from others in predicting and estimating errors. The 4×4 MIMO-OFDM system has better performance when correlated to 2×2 MIMO-OFDM systems. When the number of transmitting and receiving antennas increases, multiple data streams are transmitted at once. So, the implementation of a 4×4 MIMO-OFDM system with STBC using the KF is preferable. STBC provides high diversity gains.

Mokkapati Ravi Kumar
completed his Ph.D. from KoneruLakshmaiah Education Foundation in the area of Communications. His significant research interests are ionospheric scintillation studies, satellite communications, and radio wave propagation. Ionospheric scintillation studies over low latitude regions are conducted using GPS/IRNSS navigation systems. Various algorithms have been proposed for mitigating the ionospheric scintillation effects. Complimentary ensemble empirical mode decomposition (CEEMD) technique is used along with multifractal detrended fluctuation analysis (MF-DFA) for detecting the noise due to scintillations at K L University. Ionospheric gradient studies were carried out during the solar maximum period. The correlation analysis has been performed between the scintillations and spread F parameters using GPS and Ionosonde measurements. He is involved with the research work carried out at K L University using GPS and IRNSS/NAVIC receivers. He can be contacted atravikumar@kluniversity.in.

Jagupilla Lakshmi Prasanna
received her B.Tech. in Electronics and Computers Engineering from affiliated JNT University, Kakinada, India in 2013. She was awarded the M.Tech., Degree in VLSI Design from affiliated JNTU Kakinada, India in 2015, and she is Pursuing Ph.D., Degree from National Institute of Technology, Warangal, India, in the area of photovoltaics for solar cell applications. Currently, she is in the position of Assistant Professor in the Department of ECE, KoneruLakshmaiah Education Foundation, Vaddeswaram, Guntur Dist., AP, India. Her research interests include VLSI, solar cells, photovoltaics, and IoT. She can be contacted at lakshmiprasannanewmail@kluniversity.in.