Orthogonal Space-time Block Codes Research Articles

Robust rateless space time block coding for mmWave massive MIMO system

In the vast growing wireless communication system creating robust encoding mechanisms using space-time block codes (STBC) for mmWave massive MIMO to overcome uncertainty and ensure reliability is a critical concept to be covered. This article reviews the core concepts behind MIMO communication, Massive MIMO communication, space-time block codes, and rateless codes in the context of wireless communication systems. Building on the foundational concepts of information theory and mmWave massive MIMO, we developed space-time block codes that maintain orthogonality for real-valued symbols. Following a thorough analysis of these codes, we successfully extended them into rateless orthogonal space-time block codes (ROSTBC) for massive MIMO. This extension enables the codes to adapt their rates dynamically based on the unknown channel conditions at the transmitter. The research work was concluded by comparing static G4 encoded Tarokah work and other Orthogonal Space Time Block Codes (OSTBCs) with Rateless Space Time Block Codes (ROSTBCs). The results show that as the number of blocks used to encode the message in Rateless Space Time Block Code (ROSTBC) increases, this coding scheme can outperform static Rateless Space Time Block Codes (OSTBC) in very low SNR values by a minimum of 8.5 %. This work can enhance wireless communication by creating reliable communication over inherently unreliable systems.

A Deep Learning-based Approach for Channel Estimation in Multi-access Multi-antenna Systems

This paper studies estimating the channel state information at the end of receiver (CSIR) for multiple transmitters communicating with only one receiver so that the latter can decode the incoming signal more efficiently. The transmitters and the receiver are all equipped with multi-antennas and using orthogonal space-time block codes (OSTBC). An algorithm is developed based on deep learning for estimating multi-user multiple-input multiple-output (MU-MIMO) channels. The algorithm could estimate the CSIR using a single pilot block. The proposed convolutional neural network (CNN) architecture designed for this task begins with an input layer that accepts grayscale images, followed by six convolutional blocks for feature extraction and processing. The network concludes with a fully connected layer to output the estimated channel information. It is trained using a regression loss function to map input images to accurate channel information accurately. The performance of the proposed method is compared with classical methods like least square and subspace-based methods, including Capon and rank revealing QR (RRQR) methods. CNN achieved better performance in comparison with the reference. Computer simulations are included to validate the proposed method.

Performance Enhancement for B5G/6G Networks Based on Space Time Coding Schemes Assisted by Intelligent Reflecting Surfaces with Higher Modulation Orders.

Intelligent Reflecting Surfaces (IRS) and Multiple-Input Single-Output (MISO) technologies are essential in the fifth generation (5G) networks and beyond. IRS optimizes the signal propagation and the coverage and is a viable approach to address the issues caused by fading channels that limits the spectral efficiency, while MIMO enhances data rates, reliability, and spectral efficiency by using multiple antennas at both transmitter and receiver ends. This paper proposes an IRS-assisted MISO system using the Orthogonal Space-Time Block Code (OSTBC) scheme to enhance the channel reliability and reduce the Bit Error Rate (BER) in wireless communication systems. The proposed system exploits the benefits from the transmit diversity gain of the OSTBC scheme as well as from the bit energy to noise power spectral density (Eb/No) improvement of the IRS technology. The presented work explores these combined technologies across different modulation schemes. The obtained results outperform the similar previously published works by considering higher-order modulation schemes as well as the deployment of rate ¾ OSTBC-assisted IRS. Moreover, the obtained results demonstrate that the integration of OSTBC with IRS can yield significant performance improvements in terms of Eb/No by 7 dB and 13 dB when using 16 reflecting elements and 64 reflecting elements, respectively.

Hard Successive Interference Cancellation for M-QAM MIMO Links in the Presence of Rayleigh Deep-Fading.

In our paper, we propose a generalized version of the Alternating Projections Digital Hard Successive Interference Cancellation (AP-HSIC) algorithm that is capable of decoding any order of constellation M in an M-Quadrature Amplitude Modulation (QAM) system. Our approach applies to Rayleigh deep-fading Multiple-Input Multiple-Output (MIMO) channels with high-level Additive White Gaussian Noise (AWGN). It can handle various destructive phenomena without restricting the number of antenna arrays in the transmitter/receiver. Importantly, it does not rely on closed-loop MIMO feedback or the need for Channel-State Information Transmission (CSIT). We have demonstrated the effectiveness of our approach and provided a Bit Error Rate (BER) analysis for 16-, 32-, and 64-QAM modulation systems. Real-time simulations showcase the differences and advantages of our proposed algorithm compared to the Multi-Group Space-Time Coding (MGSTC) decoding algorithm and the Lagrange Multipliers Hard Successive Interference Cancellation (LM-HSIC) algorithm, which we have also developed here. Additionally, our paper includes a mathematical analysis of the LM-HSIC algorithm. The AP-HSIC algorithm is not only effective and fast in decoding, including interference cancellation computational feedback, but it can also be integrated with any Linear Processing Complex Orthogonal Design (LPCOD) technique, including Complex Orthogonal Design (COD) schemes such as high-order Orthogonal Space-Time Block Code (OSTBC) with high-order QAM symbols.

Orthogonal Space–Time Block Code Based Over-the-Air Computation in Backscatter Sensor Networks

Orthogonal Space–Time Block Code Based Over-the-Air Computation in Backscatter Sensor Networks

Application of STBC in MIMO-OFDM Broadband Wireless Communications

The Space Time Block Coding (STBC) is applied in multiple transmit antennas using data transfer for fading channels. After being encoded, the data is broken into n streams of continuously broadcast strings over n transmit antennas using STBC. The received signal is the combination of n broadcast signals and it becomes erroneous due to the entry of noise at receiver end. The probabilistic decoding approach is applied instead of joint detection approach for recovering data via decoupling of signals delivered from various antennas. The maximum likelihood decoding scheme employs the orthogonal structure of STBC called as OSTBC (Orthogonal Space Time Block Code) for providing the maximum likelihood decoding algorithm depending upon linear computation at receiver. In the present work, a MATLAB/SIMULINK environment of OSTBC has been used in order to acquire the highest dimension of variability for specific number of broadcasts as well as receive antennas utilizing a simple decoding techniques. With or without Grey Coding, OSTBC / STBC is used in MATLAB / SIMULINK blocs. The OSTBC algorithm calculates the highest achievable transmission rate for every amount of transmit antennas in any constellations including M-PSK array. M-PSK STBCs are used for different complex constellations to achieve ½ as well as ¾ of highest allowable transmission rate for Multiple Input Multiple Output (MIMO) transmit antennas utilizing various constellations

Enhancing the performance of LiFi communication with OSTBC, QAM, and OFDM: High-capacity, low-complexity transceiver design

Light fidelity (LiFi) is one of the most promising areas within wireless optical communication system. It provides many benefits, including high capacity, improved security and enhanced cost-effectiveness. However, there is still challenges which are link blockages, attenuation, interference and bandwidth restriction of modulation techniques. Orthogonal space–time block coding (OSTBC) offers a suitable solution to address link blockage by providing an alternative communication channel between the transmitter and the receiver. OSTBC has more tolerate to the practical channel impact limitation such as attenuation and interference that may occur in overlapping regions. In addition, the use of the OSTBC technique enhances the system’s ability to transfer data as the transmission rate is duplicated by the number of transmitters and receivers. This paper proposes the utilisation of a LiFi transceiver-based OSTBC that employs quadrature amplitude modulation (QAM) in conjunction with orthogonal frequency-division multiplexing (OFDM). Co-simulation tools have been done by integrated Matlab with Optisystem, were used in the design of LiFi transceiver with all assumption required. The findings reveal the capability of LiFi transceiver of transmitting rate up to 15 Gbps with maintaining a satisfactory bit error rate (BER) of 10−4 over a link range of 200 m evenin the presence of realistic channel effects. Moreover, the LiFi transceiver’s structure used does not incorporate optical amplifier components, resulting in reduced computational complexity compared to conventional LiFi transceivers. The proposed system proved its progress over the previous work under the influence of the same input effects.

Coding techniques for diversity enhancement of dense wavelength division multiplexing MIMO‐FSO fault protection protocols systems over atmospheric turbulence channels

AbstractAn enhanced transmission is presented in a multiple‐input‐multiple‐output (MIMO) dense‐wavelength division multiplexed (DWDM) free‐space‐optical (FSO) communication link using diversity coding techniques under the effect of turbulent weather phenomenon. The findings show good performance with an (8 channels × 2.5 Gbps data rate/channel) 20 Gbps 1500 m transmission distance. The bit‐error‐rate (BER), outage probability (OP), and signal‐to‐noise ratio (SNR) of the diversity combining techniques using maximum‐ratio combining (MRC), selection combining (SC), and equal‐gain combining (EGC) technique are evaluated in this work. The obtained results illustrate that Alamouti, space‐time coding (STC), space‐time block coding (STBC), space‐time trellis code (STTC), orthogonal STBC (O‐STBC), and quasi‐orthogonal STBC (QO‐STBC) on the minimum mean‐square‐error, and MRC are worth implementing on the DWDM‐FSO wireless communication systems. The mitigation of atmospheric turbulence is achieved using MIMO diversity combining techniques coding. The simulation results for diversity coding techniques using QO‐STBC/STTC and SC/MRC in the MIMO‐DWDM FSO communication system can improve BER performance, OP, and SNR. The MRC exhibits the lowest OP and BER when compared with the SC and EGC. The numerical results demonstrate that the FSO communication link using DWDM QO‐STBC/STTC improves the power penalty at both BER values under varying atmospheric turbulence conditions for ST, MT, and WT, in comparison to FSO systems without DWDM QO‐STBC/STTC diversity coding techniques.

Spatial diversity processing mechanism based on the distributed underwater acoustic communication system.

To address the problem of unreliable single-link underwater acoustic communication caused by large signal delays and strong multipath effects in shallow water environments, this paper proposes a distributed underwater acoustic diversity communication system (DUA-DCS). DUA-DCS employs a maneuverable distributed cross-medium buoy network to form multiple distributed, non-coherent, and parallel communication links. In the uplink, a receiving diversity processing mechanism of joint decision feedback equalizer embedded phase-locked loop and maximum signal-to-interference ratio combining (DFE-PLL-MSIRC) is proposed to achieve waveform-level diversity combining of underwater signals. A phase-locked loop module is embedded in each branch of the decision feedback equalizer to eliminate the residual frequency and phase errors after Doppler compensation. Meanwhile, the combining coefficients are determined based on the maximum signal-to-interference ratio criterion, taking into account the residual inter-symbol interference after equalization, resulting in efficient and accurate computation. Additionally, the combined decision values are fed back to the feedback filters in each branch to ensure more accurate feedback output. Simulation and lake experiment results demonstrate that, compared to the single-link communication system, DFE-PLL-MSIRC can achieve a diversity gain of more than 5.2 dB and obtain about 3 dB more diversity gain than the comparison algorithm. And the BER of DFE-PLL-MSIRC can be reduced by at least one order of magnitude, which is lower by at least 0.6 order of magnitude compared to the comparison algorithm. In the downlink, a transmitting diversity processing mechanism of complex orthogonal space-time block coding (COSTBC) is proposed. By utilizing a newly designed generalized complex orthogonal transmission matrix, complete transmission diversity can be achieved at the coding rate of 3/4. Compared to the single-link communication system, the system can achieve a diversity gain of more than 6 dB.

Orthogonal Space-Time Block Coding for Double Scattering V2V Links with LOS and Ground Reflections

This work presents the performance analysis of space-time block codes (STBCs) for vehicle-to-vehicle (V2V) fast-fading channels in scenarios with modified line-of-sight (LOS). The objective is to investigate how the V2V MIMO (multiple-input multiple-output) system performance is influenced by two important impairments: deterministic ground reflections and an increased Doppler frequency (time-variant channels). STBCs of various coding rates (using an approximation model) are evaluated by assuming antenna elements distributed over the surface of two contiguous vehicles. A multi-ray model is used to study the multiple constructive/destructive interference patterns of the transmitted/received signals by all pairs of Tx–Rx antenna links considering ground reflections. A double scattering model is used to include the effects of stochastic channel components that depend on the Doppler frequency. The results show that STBCs are capable of counteracting fades produced by destructive self-interference components across a range of inter-vehicle distances and for a range of Doppler frequency values. Notably, the effectiveness of STBCs in deep fades is shown to outperform schemes with exclusive receive diversity, despite the interference created by the loss of orthogonality in time-varying channels with a moderate increase of Doppler frequency (mainly due to higher vehicle speeds, higher frequency or shorter time slots). Higher-order STBCs with rate losses are also evaluated using an approximation model, showing interesting gains even for low coding rate performance, particularly when accompanied by a multiple antenna receiver. Overall, these results can shed light on how to exploit transmit diversity in time-varying vehicular channels with modified LOS.

Non-Orthogonal Multiple Access Based on Orthogonal Space-Time Block Codes for Mobile Communications

Non-orthogonal multiple access (NOMA), which combines multiple user signals and transmits the combined signal over one channel, can achieve high spectral efficiency for mobile communications. However, combining the multiple signals can lead to degradation of bit error rates (BERs) of NOMA under severe channel conditions. In order to improve the BER performance of NOMA, this paper proposes a new NOMA scheme based on orthogonal space-time block codes (OSTBCs). The proposed scheme transmits several multiplexed signals over their respective orthogonal time-frequency channels, and can gain diversity effects due to the orthogonality of OSTBC. Furthermore, the new scheme can detect the user signals using low-complexity linear detection in contrast with the conventional NOMA. The paper focuses on the Alamouti code, which can be considered the simplest OSTBC, and theoretically analyzes the performance of the linear detection. Computer simulations under the condition of the same bit rate per channel show that the Alamouti code based scheme using two channels is superior to the conventional NOMA using one channel in terms of BER performance. As shown by both the theoretical and simulation analyses, the linear detection for the proposed scheme can maintain the same BER performance as that of the maximum likelihood detection, when the two channels have the same frequency response and do not bring about any diversity effects, which can be regarded as the worst case.

Design and performance evaluation of a hybrid LiFi transceiver using OSTBC-based spatial multiplexing and transmit diversity

The need for data transmission systems that are high-speed, secure, and reliable increases as wireless communication technologies continue to advance rapidly. Light fidelity (LiFi) can meet this need through its several advantages, including high transmission rates, low electromagnetic interference, and indoor localization capabilities. This study proposes a hybrid LiFi transceiver, which utilizes LiFi technology using orthogonal space–time block coding (OSTBC), based on the spatial multiplexing (SM) and transmit diversity (Tx). The suggested hybrid LiFi transceiver aims to integrate the advantages of OSTBC, Tx diversity, and SM techniques to enhance the performance of the overall transceiver. The suggested transceiver plan to implement one of the diversity techniques, which are either Tx diversity or SM at a time. The selected technique will depends on the channel quality, optimizing the transceiver's throughput and ensuring effective utilization of the available spectrum. Through simulation results, this research aims to demonstrate that the proposed hybrid LiFi transceiver surpasses conventional LiFi transceivers that lack Tx diversity or SM. Anticipated outcomes of this research include superior data rates, enhanced reliability, and increased resilience against real channel impairments. These findings hold substantial promise for applications that requires high-speed and reliable communication, including smart homes, the Internet of Things, and forthcoming 5 G/6 G communication networks.

Design and Simulation of MIMO Systems over Rayleigh Fading Channel and Using Near-Maximum-Likelihood Detection

While coding is the main element for the success of the MIMO system implementation, the communication channel is the main component that ascertains this. This paper describes the design and simulation of a 2x2, 4x4, and 8x8 MIMO system which is used exclusively in 5G mobile systems. The detection performed in such a system is near-maximum-likelihood detection. The noise added at the front end of the receiver is additive white Gaussian noise with zero mean and variance which depends upon the signal-to-noise ratio (SNR) while the signal is transmitted over a frequency-selective Rayleigh fading channel. The MIMO system operates at a much higher level of complexity to exploit the channel space-time resources. Orthogonal space-time block codes are used at the transmitter. An Image is transmitted and the received image is compared with the transmitted image and BER is calculated. This is done for varying SNRs. The whole project is performed over a front-end application where the various parameters are populated and run. This makes the simulation extremely easy for the 2x2 or 4x4 or 8x8 MIMO application with varying channel parameters. The modulation scheme used in this work is BPSK.

Enhancing MIMO Spatial-Multiplexing and Parallel-Decoding under Interference by Computational Feedback

In this paper, we propose a new digital Hard-Successive-Interference-Cancellation (HSIC), the Alternating Projections-HSIC (AP-HSIC), an innovative fast computational feedback algorithm that deals with various destructive phenomena from different types of interferences. The correctness and convergence of the proposed algorithm are provided, and its complexity is given. The proposed algorithm possesses the functionality of canceling digital interference without the aid of physical feedback between the receiver and the transmitter or the loading of learning information about the state of the Multiple Input–Multiple Output (MIMO) channel to the transmitter. The proposed AP-HSIC algorithm enables a parallel decoding process from the parallel transmission of Orthogonal- Space–Time-Block-Coding (OSTBC) under a complex and challenging wireless environment to facilitate the Dynamic Spectrum Sharing (DSS) capability. We present a performance comparison of the proposed algorithm with the algorithm for Multi-Group-Space–Time-Coding (MGSTC) under MIMO fading channels and general interference or high-level Additive White Gaussian Noise (AWGN). Mathematical analysis and real-time simulations show the advantages of the proposed algorithm compared to the MGSTC decoding algorithm.

Application of maximum rank distance codes in designing of STBC-OFDM system for next-generation wireless communications

Space-Time Block Coded (STBC) Orthogonal Frequency Division Multiplexing (OFDM) satisfies higher data-rate requirements while maintaining signal quality in a multipath fading channel. However, conventional STBCs, including Orthogonal STBCs (OSTBCs), Non-Orthogonal (NOSTBCs), and Quasi-Orthogonal STBCs (QOSTBCs), do not provide both maximal diversity order and unity code rate simultaneously for more than two transmit antennas. This paper targets this problem and applies Maximum Rank Distance (MRD) codes in designing STBC-OFDM systems. By following the direct-matrix construction method, we can construct binary extended finite field MRD-STBCs for any number of transmitting antennas. Work uses MRD-STBCs built over Phase-Shift Keying (PSK) modulation to develop an MRD-based STBC-OFDM system. The MRD-based STBC-OFDM system sacrifices minor error performance compared to traditional OSTBC-OFDM but shows improved results against NOSTBC and QOSTBC-OFDM. It also provides 25% higher data-rates than OSTBC-OFDM in configurations that use more than two transmit antennas. The tradeoffs are minor increases in computational complexity and processing delays.

Constant Weight Polar Coded Orthogonal Space-Time Block Codes for Dimmable Indoor MIMO-VLC Systems

The concatenation of constant weight polar code (CWPC) with orthogonal space-time block code (OSTBC), known as CWPC-OSTBC scheme is proposed for the indoor multiple-input multiple output visible light communication (MIMO-VLC) systems in this letter. The CWPC helps to mitigate against flickering, provides desired dimming targets, and coding gain while the OSTBC achieves MIMO diversity gain. Flicker mitigation and dimming control are obtained based on the modified Knuth balancing method with enhanced prefix coding. This method does not utilise lookup table and thus, it is practicable in the real-time. Numerical results are presented to discuss the channel capacity and bit-error-rate (BER) performance of the proposed CWPC-OSTBC scheme considering different parameters. The results show that increasing the transmission antennas of the OSTBC system provides diversity gains. More so, the CWPC-OSTBC scheme yields an improved coding gain compared to polar code without balancing and other state-of-the-art methods.

Information Rate Optimization for the Non-Regenerative Linear MIMO Relay Channel With a Direct Link and Variable Duty Cycle

We consider the optimization of a two-hop linear relay channel based on an amplify-and-forward Multiple-Input Multiple-Output (MIMO) relay. The relay is assumed to derive the output signal by applying a Relay Transform Matrix (RTM) applied to the input signal. Assuming perfect channel state information about the channel at the relay and iid transmitted symbols, the RTM is optimized according to two different criteria: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i)</i> MIMO information rate; <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ii)</i> information rate based on Orthogonal Space–Time Block Codes. The two assumptions have been addressed in part in the literature. The optimization problem is reduced to a manageable convex form, whose KKT equations are explicitly solved. Then, a parametric solution is given, which yields the power constraint and the information rate achieved with uncorrelated transmitted symbols as functions of a positive indeterminate. The solution for a given average power constraint at the relay is amenable to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">water-filling-like</i> algorithm, and extends earlier literature results addressing the case without the direct link. The duty cycle of the two-hop relaying process is also addressed in the general form of the achievable rate. Simulation results are reported, which are relevant to a Rayleigh fading MIMO relay channel and the role of the direct link SNR is precisely assessed. Duty cycle optimization is also considered by a numerical example.

Physical layer secrecy of a cognitive radio with spatially correlated Alamouti OSTBC system

In this paper, an underlay spectrum-sharing system with Alamouti orthogonal space–time block coding (OSTBC) is considered to analyze and evaluate the physical layer security (PLS) performance of the cognitive radio system under a practical scenario with spatially correlated transmit antennas. It is assumed that there exists a passive eavesdropper and the cognitive channel and the wiretap channel follow Rayleigh fading distribution. To investigate and study the PLS performance of the cognitive system, first closed-form expressions are derived for three PLS metrics, namely: the probability of strictly-positive secrecy capacity (PSPSC), the secrecy outage probability (SOP), and the average secrecy capacity (ASC). Then numerical results obtained from the derived closed-form expressions are presented and validated by the computer simulations, to study the effects of spatial correlation on the PLS performance of the considered cognitive radio system under different parameters. It is shown that increasing the SNR of the cognitive system (Alice-to-Bob) channel yields an improvement in the PLS of the cognitive system. Moreover, a smaller value of the eavesdropping (Alice-to-Eve) channel SNR always leads to a better PLS for the cognitive system. It is also observed that the spatial correlation related to Alice-to-Bob channel degrades the PLS, and the spatial correlation related to Alice-to-Eve channel has less impact on the PLS performance.

Performance of an all-links MIMO-OSTBC mixed underlay cognitive RF/FSO system.

This paper presents a new model, to the best of our knowledge, of a mixed underlay cognitive radio frequency (RF)/free space optical (FSO) system in which both RF and FSO links consider multiple-input multiple-output (MIMO) orthogonal space-time block coding (OSTBC). In a dual-hop decode-and-forward configuration, the underlay cognitive radio network RF and FSO links experience κ-μ and Γ-Γ fading, respectively. For the above system model, a closed expression for the outage probability of the mixed underlay cognitive RF/FSO system with MIMO-OSTBC is derived, and the simulation results are verified using the Monte Carlo method. The results show that considering MIMO-OSTBC in all links of the mixed underlay cognitive RF/FSO system can effectively improve the communication performance of the mixed system and alleviate the degradation in the communication quality caused by the atmospheric turbulence. The communication performance of the MIMO-OSTBC mixed underlay cognitive RF/FSO system is further improved by changing the key parameters, such as peak transmit power, fading parameters, the number of secondary user transmit antennas and relay receive antennas, and relative speed of the primary user.

Error analysis of TAS‐OSTBC assisted downlink NOMA system over generalized η−μ fading Channel

SummaryIncorporation of multiple antennas in downlink non‐orthogonal multiple access (NOMA) system helps in achieving better performance by exploiting different diversity schemes. To this end, we propose a general framework of transmit antenna selection (TAS)‐assisted orthogonal space‐time block code (OSTBC) transmission for multiple NOMA users. We derive exact expressions of probability density function of the TAS‐OSTBC processed signal‐to‐noise ratio (SNR) at the output of maximum‐ratio combiner of the NOMA users. Using well‐known moment generating function approach, we evaluate the error performance of TAS‐OSTBC assisted NOMA users over generalized η−μ fading channel in presence of perfect and imperfect successive interference cancelation. We investigate the proposed system in different multi‐antenna scenarios and accordingly observe better error performance of the users by increasing the number of transmit and receive antennas. We perform their asymptotic analysis leading to full‐diversity order at high SNR. For numerical evaluation, we truncate the infinite series in derived results and examine the upper bound of truncation error. We evaluate nth moment of the output SNR and obtain channel quality estimation index as a performance measure of the proposed NOMA system. Moreover, we investigate the influence of power coefficients and fading parameters, η and μ, on the error performance of TAS‐OSTBC‐assisted NOMA users. We also compare the derived error performances with the existing schemes, wherein we observe that the proposed system shows remarkable improvement in the bit error rate (BER) of the NOMA users. For validation, we execute simulations which closely match with the derived analytical results.