A novel approach for solving linear estimation problem in multi-user massive MIMO systems is
proposed. In this approach, the difficulty of matrix inversion is attributed to the incomplete definition of the
dot product. The general definition of dot product implies that the columns of channel matrix are always
orthogonal whereas, in practice, they may be not. If the latter information can be incorporated into dot
product, then the unknowns can be directly computed from projections without inverting the channel
matrix. By doing so, the proposed method is able to achieve an exact solution with a 25% reduction in
computational complexity as compared to the QR method. Proposed method is stable, offers an extra
flexibility of computing any single unknown, and can be implemented in just twelve lines of code.
Network-on-Chip (NoC) is a new approach for designing the communication subsystem among IP cores in a System-on-Chip (SoC). NoC applies networking theory and related methods to on-chip communication and brings out notable improvements over conventional bus and crossbar interconnections. NoC offers a great improvement over the issues like scalability, productivity, power efficiency and signal integrity challenges of complex SoC design. In an NoC, the communication among different nodes is achieved by routing packets through a pre-designed network fabric according to some routing algorithm. Therefore, architecture and related routing algorithm play an important role to the improvement of overall performance of an NoC. A Diametrical 2D Mesh routing architecture has the facility of having some additional diagonal links with simple 2D Mesh architecture. In this work, we have proposed a Modified Extended 2D routing algorithm for this architecture, which will ensure that a packet always reaches the destination through the possible shortest path, and the path is always deadlock free.
MODIFIED LLL ALGORITHM WITH SHIFTED START COLUMN FOR COMPLEXITY REDUCTIONijwmn
Multiple-input multiple-output (MIMO) systems are playing an important role in the recent wireless
communication. The complexity of the different systems models challenge different researches to get a good
complexity to performance balance. Lattices Reduction Techniques and Lenstra-Lenstra-Lovàsz (LLL)
algorithm bring more resources to investigate and can contribute to the complexity reduction purposes.
In this paper, we are looking to modify the LLL algorithm to reduce the computation operations by
exploiting the structure of the upper triangular matrix without “big” performance degradation. Basically,
the first columns of the upper triangular matrix contain many zeroes, so the algorithm will perform several
operations with very limited income. We are presenting a performance and complexity study and our
proposal show that we can gain in term of complexity while the performance results remains almost the
same.
Macromodel of High Speed Interconnect using Vector Fitting Algorithmijsrd.com
At high frequency efficient macromodeling of high speed interconnects is all time challenging task. We have presented systematic methodologies to generate rational function approximations of high-speed interconnects using vector fitting technique for any type of termination conditions and construct efficient multiport model, which is easily and directly compatible with circuit simulators.
Influence of channel fading correlation on performance of detector algorithms...csandit
This paper analyzes the impact of fading correlation and cross polarization coupling on the
error performance of V-BLAST MIMO system that employs detector algorithms like ZF, MMSE
and ML with ordering and successive cancellation. Simulation results show the BER
performance of these detectors for different modulation schemes. It is observed that lesser the
channel fading correlation and cross polarization coupling values better is the performance of
these detectors. Study is extended to see the effect of transmit and receive antenna correlation
on Ergodic MIMO capacity.
A SEMI BLIND CHANNEL ESTIMATION METHOD BASED ON HYBRID NEURAL NETWORKS FOR UP...ijwmn
The paper describes how to improve channel estimation in Single Carrier Frequency Division Multiple
Access (SC-FDMA) system, using a Hybrid Artificial Neural Networks (HANN). The 3rd Generation
Partnership Project (3GPP) standards for uplink Long Term Evolution Advanced (LTE-A) uses pilot based
channel estimation technique. This kind of channel estimation method suffers from a considerable loss
ofbitrate due to pilot insertion; all data frame sent contains reference signal. The HANN converts data
aided channel estimator to semi blind channel estimator. To increase convergence speed, HANN uses some
channel propagation Fuzzy Rules to initialize Neural Network parameters before learning instead of a
random initialization, so its learning phase ismore rapidly compared to classic ANN.HANN allows more
bandwidth efficient and less complexity. Simulation results show that HANN has better computational
efficiency than the Minimum Mean Square Error (MMSE) estimator and has faster convergence than
classic Neural Networks estimators.
An Efficient Clustering Method for Aggregation on Data FragmentsIJMER
Clustering is an important step in the process of data analysis with applications to numerous fields. Clustering ensembles, has emerged as a powerful technique for combining different clustering results to obtain a quality cluster. Existing clustering aggregation algorithms are applied directly to large number of data points. The algorithms are inefficient if the number of data points is large. This project defines an efficient approach for clustering aggregation based on data fragments. In fragment-based approach, a data fragment is any subset of the data. To increase the efficiency of the proposed approach, the clustering aggregation can be performed directly on data fragments under comparison measure and normalized mutual information measures for clustering aggregation, enhanced clustering aggregation algorithms are described. To show the minimal computational complexity. (Agglomerative, Furthest, and Local Search); nevertheless, which increases the accuracy.
Network-on-Chip (NoC) is a new approach for designing the communication subsystem among IP cores in a System-on-Chip (SoC). NoC applies networking theory and related methods to on-chip communication and brings out notable improvements over conventional bus and crossbar interconnections. NoC offers a great improvement over the issues like scalability, productivity, power efficiency and signal integrity challenges of complex SoC design. In an NoC, the communication among different nodes is achieved by routing packets through a pre-designed network fabric according to some routing algorithm. Therefore, architecture and related routing algorithm play an important role to the improvement of overall performance of an NoC. A Diametrical 2D Mesh routing architecture has the facility of having some additional diagonal links with simple 2D Mesh architecture. In this work, we have proposed a Modified Extended 2D routing algorithm for this architecture, which will ensure that a packet always reaches the destination through the possible shortest path, and the path is always deadlock free.
MODIFIED LLL ALGORITHM WITH SHIFTED START COLUMN FOR COMPLEXITY REDUCTIONijwmn
Multiple-input multiple-output (MIMO) systems are playing an important role in the recent wireless
communication. The complexity of the different systems models challenge different researches to get a good
complexity to performance balance. Lattices Reduction Techniques and Lenstra-Lenstra-Lovàsz (LLL)
algorithm bring more resources to investigate and can contribute to the complexity reduction purposes.
In this paper, we are looking to modify the LLL algorithm to reduce the computation operations by
exploiting the structure of the upper triangular matrix without “big” performance degradation. Basically,
the first columns of the upper triangular matrix contain many zeroes, so the algorithm will perform several
operations with very limited income. We are presenting a performance and complexity study and our
proposal show that we can gain in term of complexity while the performance results remains almost the
same.
Macromodel of High Speed Interconnect using Vector Fitting Algorithmijsrd.com
At high frequency efficient macromodeling of high speed interconnects is all time challenging task. We have presented systematic methodologies to generate rational function approximations of high-speed interconnects using vector fitting technique for any type of termination conditions and construct efficient multiport model, which is easily and directly compatible with circuit simulators.
Influence of channel fading correlation on performance of detector algorithms...csandit
This paper analyzes the impact of fading correlation and cross polarization coupling on the
error performance of V-BLAST MIMO system that employs detector algorithms like ZF, MMSE
and ML with ordering and successive cancellation. Simulation results show the BER
performance of these detectors for different modulation schemes. It is observed that lesser the
channel fading correlation and cross polarization coupling values better is the performance of
these detectors. Study is extended to see the effect of transmit and receive antenna correlation
on Ergodic MIMO capacity.
A SEMI BLIND CHANNEL ESTIMATION METHOD BASED ON HYBRID NEURAL NETWORKS FOR UP...ijwmn
The paper describes how to improve channel estimation in Single Carrier Frequency Division Multiple
Access (SC-FDMA) system, using a Hybrid Artificial Neural Networks (HANN). The 3rd Generation
Partnership Project (3GPP) standards for uplink Long Term Evolution Advanced (LTE-A) uses pilot based
channel estimation technique. This kind of channel estimation method suffers from a considerable loss
ofbitrate due to pilot insertion; all data frame sent contains reference signal. The HANN converts data
aided channel estimator to semi blind channel estimator. To increase convergence speed, HANN uses some
channel propagation Fuzzy Rules to initialize Neural Network parameters before learning instead of a
random initialization, so its learning phase ismore rapidly compared to classic ANN.HANN allows more
bandwidth efficient and less complexity. Simulation results show that HANN has better computational
efficiency than the Minimum Mean Square Error (MMSE) estimator and has faster convergence than
classic Neural Networks estimators.
An Efficient Clustering Method for Aggregation on Data FragmentsIJMER
Clustering is an important step in the process of data analysis with applications to numerous fields. Clustering ensembles, has emerged as a powerful technique for combining different clustering results to obtain a quality cluster. Existing clustering aggregation algorithms are applied directly to large number of data points. The algorithms are inefficient if the number of data points is large. This project defines an efficient approach for clustering aggregation based on data fragments. In fragment-based approach, a data fragment is any subset of the data. To increase the efficiency of the proposed approach, the clustering aggregation can be performed directly on data fragments under comparison measure and normalized mutual information measures for clustering aggregation, enhanced clustering aggregation algorithms are described. To show the minimal computational complexity. (Agglomerative, Furthest, and Local Search); nevertheless, which increases the accuracy.
In this video from the 2015 HPC User Forum in Broomfield, Barry Bolding from Cray presents: HPC + D + A = HPDA?
"The flexible, multi-use Cray Urika-XA extreme analytics platform addresses perhaps the most critical obstacle in data analytics today — limitation. Analytics problems are getting more varied and complex but the available solution technologies have significant constraints. Traditional analytics appliances lock you into a single approach and building a custom solution in-house is so difficult and time consuming that the business value derived from analytics fails to materialize. In contrast, the Urika-XA platform is open, high performing and cost effective, serving a wide range of analytics tools with varying computing demands in a single environment. Pre-integrated with the Hadoop and Spark frameworks, the Urika-XA system combines the benefits of a turnkey analytics appliance with a flexible, open platform that you can modify for future analytics workloads. This single-platform consolidation of workloads reduces your analytics footprint and total cost of ownership."
Learn more: http://www.cray.com/products/analytics/urika-xa
Watch the video presentation: http://wp.me/p3RLEV-3yR
Sign up for our insideBIGDATA Newsletter: http://insidebigdata.com/newsletter
Wideband Modeling of Twisted-Pair Cables for MIMO ApplicationsLantiq
Computer modeling and simulation approach to speed development of broadband networking technologies. http://bit.ly/1853xoY
More on our expertise: http://www.lantiq.com
NEW BER ANALYSIS OF OFDM SYSTEM OVER NAKAGAMI-n (RICE) FADING CHANNELijcseit
Modern wireless communication systems support high speed multimedia services. These services require
high data rates with acceptable error rates. Orthogonal Frequency Division Multiplexing (OFDM) is a
capable candidate to solve this problem. In this paper, a new expression for the BER of OFDM system has
been derived over Nakagami–n (Rice) fading channels using characteristics function (CHF) approach. The
exact probability density function of first order of Nakagami-n (Rice) random vector is used to derive the
expression for the error rates of OFDM system. The BER derivation of Rician fading channel is slightly
more complex compared to the Nakagami–m distribution because the PDF of the Rician RV contains an
explicit term of a modified Bessel function of first kind. Earlier, this problem was solved by replacing the
Bessel function with its infinite series and exponential integral representation. Here we propose an integral
expression to remove the complexity of the expression.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A Weighted Duality based Formulation of MIMO SystemsIJERA Editor
This work is based on the modeling and analysis of multiple-input multiple-output (MIMO) system in downlink communication system. We take into account a recent work on the ratio of quadratic forms to formulate the weight matrices of quadratic norm in a duality structure. This enables us to achieve exact solutions for MIMO system operating under Rayleigh fading channels. We outline couple of scenarios dependent on the structure of eigenvalues to investigate the system behavior. The results obtained are validated by means of Monte Carlo simulations.
Performance Enhancement of MIMO-OFDM using Redundant Residue Number System IJECEIAES
Telecommunication industry requires high capacity networks with high data rates which are achieved through utilization of Multiple-Input-MultipleOutput (MIMO) communication along with Orthogonal Frequency Division Multiplexing (OFDM) system. Still, the communication channel suffers from noise, interference or distortion due to hardware design limitations, and channel environment, and to combat these challenges, and achieve enhanced performance; various error control techniques are implemented to enable the receiver to detect any possible received errors and correct it and thus; for a certain transmitted signal power the system would have lower Bit Error Rate (BER). The provided research focuses on Redundant Residue Number System (RRNS) coding as a Forward Error Correction (FEC) scheme that improves the performance of MIMO-OFDM based wireless communications in comparison with current methods as Low-Density Parity Check (LDPC) coders at the transmitter side or equalizers at receiver side. The Bit Error Rate (BER) performance over the system was measured using MATLAB tool for different simulated channel conditions, including the effect of signal amplitude reduction and multipath delay spreading. Simulation results had shown that RRNS coding scheme provides an enhancement in system performance over conventional error detection and correction coding schemes by utilizing the distinct features of Residue Number System (RNS).
A Broadband Wireless Access technology known as
Worldwide Interoperability for Microwave Access (WiMAX) is
based on IEEE 802.16 standards. It uses orthogonal frequency
division multiple accesses (OFDMA) as one of its multiple access
technique. Major design factors of OFDMA resource allocation are
scheduling and burst allocation. To calculate the appropriate
dimensions and location of each user’s data so as to construct the
bursts in the downlink subframe, is the responsibility of burst
allocation algorithm. Bursts are calculated in terms of number of
slots for each user. Burst Allocation Algorithm is used to overcome
the resource wastage in the form of unused and unallocated slots per
frame. It affects the Base station performance in mobile WiMAX
systems. In this Paper, HOCSA (Hybrid One Column Striping with
Non Increasing Area) algorithm is proposed to overcome frame
wastage. HOCSA is implemented by improving eOCSA algorithm
and is evaluated using MATLAB. HOCSA achieves significant
reduction of resource wastage per frame, leading to more
exploitation of the WiMAX frame.
At this present scenario, the demand of the system capacity is very high in wireless network. MIMO
technology is used from the last decade to provide this requirement for wireless network antenna
technology. MIMO channels are mostly used for advanced antenna array technology. But it is most
important to control the error rate with enhanced system capacity in MIMO for present-day progressive
wireless communication. This paper explores the frame error rate with respect to different path gain of
MIMO channel. This work has been done in different fading scenario and produces a comparative analysis
of MIMO on the basis of those fading models in various conditions. Here, it is to be considered that
modulation technique as QPSK to observe these comparative evaluations for different Doppler frequencies.
From the comparative analysis, minimum amount of frame error rate is viewed for Rician distribution at
LOS path Doppler shift of 0 Hz. At last, this work is concluded with a comparative bit error rate study on
the basis of singular parameters at different SNR levels to produce the system performance for uncoded
QPSK modulation.
Study on transmission over Nakagami-m fading channel for multiple access sche...TELKOMNIKA JOURNAL
In this paper, a downlink performance in non-orthogonal multiple access (NOMA) system is considered. With regard to different priority for two NOMA users, we exploit the closed-form expressions of outage probability over wireless fading channel following Nakagami-m fading. The fixed power allocation factor scheme is examined to reduce the complexity in computation regarding performance analysis. In our analysis, perfect successive interference cancellation (SIC) is applied in order to achieve perfect signal decoding operation. Simulation results show that the considered NOMA downlink scheme is affected by transmit SNR, power allocation factors, fading parameters.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Data detection method for uplink massive MIMO systems based on the long recu...IJECEIAES
Although the mean square error (MMSE) approach is recognized to be near optimal for uplinking large-scale multiple-input-multiple-output (MIMO) systems, there are certain difficulties in the procedure related to matrix inversion. The long recurrence enlarged conjugate gradient (LRE-CG) approach is proposed in this study as a way to iteratively realize the MMMS algorithm while avoiding the complications of matrix inversion. In addition, a diagonal-approximate starting solution to the LRE-CG approach was used to speed up the conversion rate and reduce the complications required. It has been discovered that the LRE-CG-based approach has the ability to significantly reduce computational complexity. By comparing simulation results, it is clear that this new methodology surpasses well-established ways like the Neumann series approximation-based method and the Gauss-Siedel iterative method. With a small number of iterations, the suggested approach achieves near-optimal performance of a standard MMSE algorithm.
Optimum Network Reconfiguration using Grey Wolf OptimizerTELKOMNIKA JOURNAL
Distribution system Reconfiguration is the process of changing the topology of the distribution
network by opening and closing switches to satisfy a specific objective. It is a complex, combinatorial
optimization problem involving a nonlinear objective function and constraints. Grey Wolf Optimizer (GWO)
is a recently developed metaheuristic search algorithm inspired by the leadership hierarchy and hunting
strategy of grey wolves in nature. The objective of this paper is to determine an optimal network
reconfiguration that presents the minimum power losses, considering network constraints, and using GWO
algorithm. The proposed algorithm was tested using some standard networks (33 bus, 69 bus, 84 bus and
118 bus), and the obtained results reveal the efficiency and effectiveness of the proposed approach.
In this video from the 2015 HPC User Forum in Broomfield, Barry Bolding from Cray presents: HPC + D + A = HPDA?
"The flexible, multi-use Cray Urika-XA extreme analytics platform addresses perhaps the most critical obstacle in data analytics today — limitation. Analytics problems are getting more varied and complex but the available solution technologies have significant constraints. Traditional analytics appliances lock you into a single approach and building a custom solution in-house is so difficult and time consuming that the business value derived from analytics fails to materialize. In contrast, the Urika-XA platform is open, high performing and cost effective, serving a wide range of analytics tools with varying computing demands in a single environment. Pre-integrated with the Hadoop and Spark frameworks, the Urika-XA system combines the benefits of a turnkey analytics appliance with a flexible, open platform that you can modify for future analytics workloads. This single-platform consolidation of workloads reduces your analytics footprint and total cost of ownership."
Learn more: http://www.cray.com/products/analytics/urika-xa
Watch the video presentation: http://wp.me/p3RLEV-3yR
Sign up for our insideBIGDATA Newsletter: http://insidebigdata.com/newsletter
Wideband Modeling of Twisted-Pair Cables for MIMO ApplicationsLantiq
Computer modeling and simulation approach to speed development of broadband networking technologies. http://bit.ly/1853xoY
More on our expertise: http://www.lantiq.com
NEW BER ANALYSIS OF OFDM SYSTEM OVER NAKAGAMI-n (RICE) FADING CHANNELijcseit
Modern wireless communication systems support high speed multimedia services. These services require
high data rates with acceptable error rates. Orthogonal Frequency Division Multiplexing (OFDM) is a
capable candidate to solve this problem. In this paper, a new expression for the BER of OFDM system has
been derived over Nakagami–n (Rice) fading channels using characteristics function (CHF) approach. The
exact probability density function of first order of Nakagami-n (Rice) random vector is used to derive the
expression for the error rates of OFDM system. The BER derivation of Rician fading channel is slightly
more complex compared to the Nakagami–m distribution because the PDF of the Rician RV contains an
explicit term of a modified Bessel function of first kind. Earlier, this problem was solved by replacing the
Bessel function with its infinite series and exponential integral representation. Here we propose an integral
expression to remove the complexity of the expression.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A Weighted Duality based Formulation of MIMO SystemsIJERA Editor
This work is based on the modeling and analysis of multiple-input multiple-output (MIMO) system in downlink communication system. We take into account a recent work on the ratio of quadratic forms to formulate the weight matrices of quadratic norm in a duality structure. This enables us to achieve exact solutions for MIMO system operating under Rayleigh fading channels. We outline couple of scenarios dependent on the structure of eigenvalues to investigate the system behavior. The results obtained are validated by means of Monte Carlo simulations.
Performance Enhancement of MIMO-OFDM using Redundant Residue Number System IJECEIAES
Telecommunication industry requires high capacity networks with high data rates which are achieved through utilization of Multiple-Input-MultipleOutput (MIMO) communication along with Orthogonal Frequency Division Multiplexing (OFDM) system. Still, the communication channel suffers from noise, interference or distortion due to hardware design limitations, and channel environment, and to combat these challenges, and achieve enhanced performance; various error control techniques are implemented to enable the receiver to detect any possible received errors and correct it and thus; for a certain transmitted signal power the system would have lower Bit Error Rate (BER). The provided research focuses on Redundant Residue Number System (RRNS) coding as a Forward Error Correction (FEC) scheme that improves the performance of MIMO-OFDM based wireless communications in comparison with current methods as Low-Density Parity Check (LDPC) coders at the transmitter side or equalizers at receiver side. The Bit Error Rate (BER) performance over the system was measured using MATLAB tool for different simulated channel conditions, including the effect of signal amplitude reduction and multipath delay spreading. Simulation results had shown that RRNS coding scheme provides an enhancement in system performance over conventional error detection and correction coding schemes by utilizing the distinct features of Residue Number System (RNS).
A Broadband Wireless Access technology known as
Worldwide Interoperability for Microwave Access (WiMAX) is
based on IEEE 802.16 standards. It uses orthogonal frequency
division multiple accesses (OFDMA) as one of its multiple access
technique. Major design factors of OFDMA resource allocation are
scheduling and burst allocation. To calculate the appropriate
dimensions and location of each user’s data so as to construct the
bursts in the downlink subframe, is the responsibility of burst
allocation algorithm. Bursts are calculated in terms of number of
slots for each user. Burst Allocation Algorithm is used to overcome
the resource wastage in the form of unused and unallocated slots per
frame. It affects the Base station performance in mobile WiMAX
systems. In this Paper, HOCSA (Hybrid One Column Striping with
Non Increasing Area) algorithm is proposed to overcome frame
wastage. HOCSA is implemented by improving eOCSA algorithm
and is evaluated using MATLAB. HOCSA achieves significant
reduction of resource wastage per frame, leading to more
exploitation of the WiMAX frame.
At this present scenario, the demand of the system capacity is very high in wireless network. MIMO
technology is used from the last decade to provide this requirement for wireless network antenna
technology. MIMO channels are mostly used for advanced antenna array technology. But it is most
important to control the error rate with enhanced system capacity in MIMO for present-day progressive
wireless communication. This paper explores the frame error rate with respect to different path gain of
MIMO channel. This work has been done in different fading scenario and produces a comparative analysis
of MIMO on the basis of those fading models in various conditions. Here, it is to be considered that
modulation technique as QPSK to observe these comparative evaluations for different Doppler frequencies.
From the comparative analysis, minimum amount of frame error rate is viewed for Rician distribution at
LOS path Doppler shift of 0 Hz. At last, this work is concluded with a comparative bit error rate study on
the basis of singular parameters at different SNR levels to produce the system performance for uncoded
QPSK modulation.
Study on transmission over Nakagami-m fading channel for multiple access sche...TELKOMNIKA JOURNAL
In this paper, a downlink performance in non-orthogonal multiple access (NOMA) system is considered. With regard to different priority for two NOMA users, we exploit the closed-form expressions of outage probability over wireless fading channel following Nakagami-m fading. The fixed power allocation factor scheme is examined to reduce the complexity in computation regarding performance analysis. In our analysis, perfect successive interference cancellation (SIC) is applied in order to achieve perfect signal decoding operation. Simulation results show that the considered NOMA downlink scheme is affected by transmit SNR, power allocation factors, fading parameters.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Data detection method for uplink massive MIMO systems based on the long recu...IJECEIAES
Although the mean square error (MMSE) approach is recognized to be near optimal for uplinking large-scale multiple-input-multiple-output (MIMO) systems, there are certain difficulties in the procedure related to matrix inversion. The long recurrence enlarged conjugate gradient (LRE-CG) approach is proposed in this study as a way to iteratively realize the MMMS algorithm while avoiding the complications of matrix inversion. In addition, a diagonal-approximate starting solution to the LRE-CG approach was used to speed up the conversion rate and reduce the complications required. It has been discovered that the LRE-CG-based approach has the ability to significantly reduce computational complexity. By comparing simulation results, it is clear that this new methodology surpasses well-established ways like the Neumann series approximation-based method and the Gauss-Siedel iterative method. With a small number of iterations, the suggested approach achieves near-optimal performance of a standard MMSE algorithm.
Optimum Network Reconfiguration using Grey Wolf OptimizerTELKOMNIKA JOURNAL
Distribution system Reconfiguration is the process of changing the topology of the distribution
network by opening and closing switches to satisfy a specific objective. It is a complex, combinatorial
optimization problem involving a nonlinear objective function and constraints. Grey Wolf Optimizer (GWO)
is a recently developed metaheuristic search algorithm inspired by the leadership hierarchy and hunting
strategy of grey wolves in nature. The objective of this paper is to determine an optimal network
reconfiguration that presents the minimum power losses, considering network constraints, and using GWO
algorithm. The proposed algorithm was tested using some standard networks (33 bus, 69 bus, 84 bus and
118 bus), and the obtained results reveal the efficiency and effectiveness of the proposed approach.
Low-complex Bayesian estimator for imperfect channels in massive muti-input ...IJECEIAES
Motivated by the fact that the complexity of the computations is one of the main challenges in large multiple input multiple output systems, known as massive multiple-input multiple-output (MIMO) systems, this article proposes a low-complex minimum mean squared error (MMSE) Bayesian channel estimator for uplink channels of such systems. First, we have discussed the necessity of the covariance information for the MMSE estimator and how their imperfection knowledge can affect its accuracy. Then, two reduction phases in dimension and floating-point operations have been suggested to reduce its complexity: in phase 1, eigenstructure reduction for channel covariance matrices is implemented based on some truncation rules, while in phase 2, arithmetic operations reduction for matrix multiplications in the MMSE equation is followed. The proposed procedure has significantly reduced the complexity of the MMSE estimator to the first order O(M), which is less than that required for the conventional MMSE with O(M3 ) in terms of matrix dimension. It has been shown that the estimated channels using our proposed procedure are asymptotically aligned and serve the same quality as the full-rank estimated channels. Our results are validated by averaging the normalized mean squared error (NMSE) over a length of 500 sample realizations through a Monte Carlo simulation using MATLAB R2020a.
Transceiver Design for MIMO Systems with Individual Transmit Power ConstraintsIJECEIAES
This paper investigate the transceiver design for single-user multiple-input multiple- output system (SU-MIMO). Joint transceiver design with an improper modulation is developed based on the minimum total mean-squared error (TMSE) criterion under two different cases. One is equal power allocation (EPA) and other is the power constraint that jointly meets both EPA and total transmit power constraint (TTPC) (i.e ITPC). Transceiver is designed based on the assumption that both the perfect and imperfect channel state information (CSI) is available at both the transmitter and receiver. The simulation results show the performance improvement of the proposed work over conventional work in terms of bit error rate (BER).
Channel Estimation in MIMO OFDM Systems with Tapped Delay Line ModelIJCNCJournal
The continuous increase in the user demands fornew-generation communication systems, is making the wireless channel more complex and challenging for estimation, developing a simulation model for the channel,and evaluating the performance of different MIMO systems. In this work, a simulation model for multipath fading channels in wireless communication is performed. The model includes a selection of typical Tapped-Delay-Line channel models that can be implemented to reproduce the effects of representative channel distortion and interference. Based on the simulation results, the proposed method exhibits accurate channel estimation performance for frequency-selective fading channels. The proposed work employed LS, MMSE, and ML methods for channel estimation, using 16 and 32 pilots and fixed pilot locations in each frame. Results are obtained for 4x4, 8x8, 16x16, 16x8, and 16x4 MIMO systems and tapped delay line systems.
Channel Estimation in MIMO OFDM Systems with Tapped Delay Line ModelIJCNCJournal
The continuous increase in the user demands fornew-generation communication systems, is making the wireless channel more complex and challenging for estimation, developing a simulation model for the channel,and evaluating the performance of different MIMO systems. In this work, a simulation model for multipath fading channels in wireless communication is performed. The model includes a selection of typical Tapped-Delay-Line channel models that can be implemented to reproduce the effects of representative channel distortion and interference. Based on the simulation results, the proposed method exhibits accurate channel estimation performance for frequency-selective fading channels. The proposed work employed LS, MMSE, and ML methods for channel estimation, using 16 and 32 pilots and fixed pilot locations in each frame. Results are obtained for 4x4, 8x8, 16x16, 16x8, and 16x4 MIMO systems and tapped delay line systems.
DESIGN OF DELAY COMPUTATION METHOD FOR CYCLOTOMIC FAST FOURIER TRANSFORMsipij
In this paper the Delay Computation method for Common Sub expression Elimination algorithm is being implemented on Cyclotomic Fast Fourier Transform. The Common Sub Expression Elimination algorithm is combined with the delay computing method and is known as Gate Level Delay Computation with Common Sub expression Elimination Algorithm. Common sub expression elimination is effective
optimization method used to reduce adders in cyclotomic Fourier transform. The delay computing method is based on delay matrix and suitable for implementation with computers. The Gate level delay computation method is used to find critical path delay and it is analyzed on various finite field elements. The presented algorithm is established through a case study in Cyclotomic Fast Fourier Transform over finite field. If Cyclotomic Fast Fourier Transform is implemented directly then the system will have high additive complexities. So by using GLDC-CSE algorithm on cyclotomic fast Fourier transform, the additive
complexities will be reduced and also the area and area delay product will be reduced.
The efficient interleaving of digital-video-broadcasting-satellite 2nd genera...TELKOMNIKA JOURNAL
The DVB-S2 system is designed as a toolbox to permit the execution of the satellite programs. Interleaver is an essential part of the DVB-S2 system. The current general block interleaver in DVB-S2 is not best, which leads to high BER and maybe not satisfy the requirements of the system. The purpose of this paper is to study the several interleaver types and comparative analyses are done between them to find which of these give better performance. Simulations results obtained prove that the 2D interleavers minimize BER more than other interleavers of DVB-S2. Further, the performance of 2D interleaver is better on a system that required a low SNR.
Newton-raphson method to solve systems of non-linear equations in VANET perfo...journalBEEI
Nowadays, Vehicular Ad-Hoc Network (VANET) has got more attention from the researchers. The researchers have studied numerous topics of VANET, such as the routing protocols of VANET and the MAC protocols of VANET. The aim of their works is to improve the network performance of VANET, either in terms of energy consumption or packet delivery ratio (PDR) and delay. For this research paper, the main goal is to find the coefficient of a, b and c of three non-linear equations by using a Newton- Raphson method. Those three non-linear equations are derived from a different value of Medium Access Control (MAC) protocol's parameters. After that, those three coefficient is then will be used in optimization of the VANET in terms of energy, PDR, and delay.
Dynamic RWX ACM Model Optimizing the Risk on Real Time Unix File SystemRadita Apriana
The preventive control is one of the well advance controls for recent security for protection of data
and services from the uncertainty. Because, increasing the importance of business, communication
technologies and growing the external risk is a very common phenomenon now-a-days. The system
security risks put forward to the management focus on IT infrastructure (OS). The top management has to
decide whether to accept expected losses or to invest into technical security mechanisms in order to
minimize the frequency of attacks, thefts as well as uncertainty. This work contributes to the development
of an optimization model that aims to determine the optimal cost to be invested into security mechanisms
deciding on the measure component of UFS attribute. Our model should be design in such way, the Read,
Write & Execute automatically Protected, Detected and Corrected on RTOS. We have to optimize the
system attacks and down time by implementing RWX ACM mechanism based on semi-group structure,
mean while improving the throughput of the Business, Resources & Technology.
False Node Recovery Algorithm for a Wireless Sensor NetworkRadita Apriana
This paper proposes a fault node recovery algorithm to enhance the lifetime of a wireless sensor
network when some of the sensor nodes shut down. The algorithm is based on the grade diffusion algorithm
combined with the genetic algorithm. The algorithm can result in fewer replacements of sensor nodes and
more reused routing paths. In our simulation, the proposed algorithm increases the number of active nodes
up to 8.7 times, reduces 98.8%, and reduces the rate of energy consumption by approximately 31.1%.
ESL Reading Research Based on Eye Tracking TechniquesRadita Apriana
Reading is a complex cognitive process. Through eye tracking techniques the cognitive features
can be observed in depth, and therefore a better grasp of the instant cognitive processing of language can
be achieved. Compared to traditional methods, the greatest advantage of eye tracking is that is that the
reader proceeds with reading coherently and fluently without interrupt and interference, which secures the
results in higher ecological validity. This paper is mainly about studies on eye movement features in ESL
reading, and particularly illustrates eye movement patterns, the relationship between eye movement and
perceptual processing, and eye movement control, aiming at providing some basic evidences for
explaining the relationship between eye movement behavior and cognitive processing thus helping to
solve some theoretical problems in reading.
Internet Access Using Ethernet over PDH Technology for Remote AreaRadita Apriana
There was still is gap among people living in city and in remote area to get information access,
especially who lived in the Eastern part of Indonesia. People living in such remote area usually were
isolated from town by natural condition like rivers, valleys, hills and so on. Therefore, telecommunication
infrastructure for remote area using cooper was not effective and efficient way to build. The issue was how
information and communication technology could penetrate such areas. This research aimed to propose
technology that could be implemented to overcome the difficulties. Ethernet over Plesiochronous Digital
Hierarchy (EoPDH) was one of many techniques that provided Ethernet connectivity over non-Ethernet
networks. EoPDH was a standardized methodology for transporting native Ethernet frames over the
existing established PDH transport technology. To provide last milefor the local people, use of Mesh
Wireless Local Area Network was made and connected to internet gateway via Ethernet over PDH based
microwave radio link. The test showed that The Ethernet frames were successfully transported to remote
area with good quality of service such as throughput, response time, and transaction rate.
Laboratory design of an ozone concentration regulator which is build on the theory of continuity
equation for gas flow in parallel pipes in conjunction with the ozone elimination potentials of an ozone
destructor is presented. At an initial oxygen flow rate of 33.33 cm3s-1, ozone concentration was generated
and varied between 429.30 parts per million (ppm) to 3826.60 ppm. Similarly at an initial oxygen flow rate
of 25 cm3s-1, ozone concentration was generated and varied between 387.30 ppm to 4435.20 ppm. Effect
of flow meter when the ozone concentration was set to approximate 1000 ppm were investigated and
reported. Fine tuning of the regulator is necessary to ensure concentration stability for long duration
experimental work.
Identifying of the Cielab Space Color for the Balinese Papyrus CharactersRadita Apriana
Balinese papyrus is one of the media to write the ideas from minstrels in ancient times. Currently,
many ancient literatures which are written in the papyrus, that also very difficult to identify because the
writings were beginning to rot or fade influenced by age. In this study takes the Balinese papyrus
characters as its object. The improvement of the image quality in image processing of the characters here
refers to the change for the quality of the image before its next development. There is the process of
identifying the color space to spot the shape of the papyrus characters. The difference of the colors is so
small so that it needs CIELab to process the identification out of its background and noise.
Filtering Based Illumination Normalization Techniques for Face RecognitionRadita Apriana
The main challenge experienced by the present face recognition techniques and smooth filters
are the difficulty in managing illumination. The differences in face images that are created by illumination
are normally bigger compared to the differences in inter-person that is utilized to differentiate identities.
However, face recognition over illumination has more uses in a lot of applications that deal with subjects
that are not cooperative where the highest potential of the face recognition as a non-intrusive biometric
feature can be executed and utilized. A lot of work has been put into the research and development of
illumination and face recognition in the present era and a lot of critical methods have been introduced.
Nevertheless, there are some concerns with face recognition and illumination that require further
considerations which include the deficiencies in comprehending the sub-spaces in illumination pictures,
problems with intractability in face modelling and complicated mechanisms of face surface reflections.
Analysis and Estimation of Harmonics Using Wavelet TechniqueRadita Apriana
The paper develops an approach based on wavelet technique for the evaluation and estimation of
harmonic contents of power system waveform. The proposed algorithm decomposes the signal waveforms
into the uniform frequency sub-bands corresponding to the odd harmonic components of the signal. The
proposed implementation of algorithm determines the frequency bands of harmonics which retain both the
time and frequency relationship of the original waveforms and uses a method to suppress those
harmonics.Thewaveletalgorithm is selected to obtain compatible output bands with the harmonic groups
defined in the standards for power-supply systems. A comparative analysis will be done with the input and
the results obtained from the wavelet transform (WT) for different measuring conditions and Simulation
results are given.
A Novel Method for Sensing Obscene Videos using Scene Change DetectionRadita Apriana
Video scene change detection has great importance of managing and analyzing large amount of
videos. Traditionally this technique used for indexing, segmenting and categorizing different types of
videos. Very few works addressed to classify obscene using scene change detection method. In this
research we proposed a simple approach for sensing objectionable videos by observing scene changes
into different video genres. Video scenes are grouped into set of key frames. After analyzing duration of
each scene and counting the number of key frames of designated scene, it has been shown that obscene
videos have infrequent scene changing nature. While in sports, dramas, music and action films have large
number of scene changes. We used six types of video genres and the decision has been made by setting
a threshold based on extracted key frames. Experimental result showed that the accuracy is 83.33% and
false positive rate is 16.67%.
Robust SINS/GNSS Integration Method for High Dynamic ApplicationsRadita Apriana
As high dynamic movement is always accompanied by colored noise which lacks of mathematical
model, traditional Kalman filtering based on an assumption of white Gaussian noise always faces serious
divergence. To enhance the performance in high dynamic environment with uncertain colored noise, a kind
of robust filtering based on H-infinity technology is developed. State model of the algorithm is derived from
SINS error propagation. Both position and velocity errors are used as the measurements. A simulation
system which includes a tri-axial turntable and a GNSS signal simulator is used to verify the integration
design under high dynamic environment. Simulation results proved that both the accuracy and robustness
of the integration design have been improved significantly.
Study on Adaptive PID Control Algorithm Based on RBF Neural NetworkRadita Apriana
Aim at the limitation of traditional PID controller has certain limitation, the traditionalPID control is
often difficult to obtain satisfactory control performance, and the RBF neural networkis difficult to meet the
requirement ofreal-time control system.To overcome it, an adaptive PID control strategy based on (RBF)
neural network isproposed in this paper. The resultsshow that the proposed controller is practical and
effective, because of the adaptability, strong robustness and satisfactory controlperformance.It is also
revealed from simulation results that the proposed control algorithm is valid for DC motor and also
provides the theoretical and experimental basis.
Solving Method of H-Infinity Model Matching Based on the Theory of the Model ...Radita Apriana
People used to solve high-order H model matching based on H control theory, it is too
difficult. In this paper, we use model reduction theory to solve high-order H model matching problem, A
new method to solve H model matching problem based on the theory of the model reduction is
proposed. The simulation results show that the method has better applicability and can get the expected
performance.
A Review to AC Modeling and Transfer Function of DCDC ConvertersRadita Apriana
In this paper, AC modeling and small signal transfer function for DC-DC converters are
represented. The fundamentals governing the formulas are also reviewed. In DC-DC converters, the
output voltage must be kept constant, regardless of changes in the input voltage or in the effective load
resistance. Transfer function is the necessary knowledge to design a proper feedback control such as PID
control to regulate the output voltage as linear PID and PI controllers are usually designed for DC-DC
converters using standard frequency response techniques based on the small signal model of the
converter.
DTC Method for Vector Control of 3-Phase Induction Motor under Open-Phase FaultRadita Apriana
Three-phase Induction Motor (IM) drives are widely used in industrial equipments. One of the
essential problems of 3-phase IM drives is their high speed and torque pulsation in the fault conditions.
This paper shows Direct Torque Control (DTC) strategy for vector control of a 3-phase IM under opencircuit
fault. The objective is to implement a solution for vector control of 3-phase IM drives which can be
also used under open-phase fault. MATLAB simulations were carried out and performance analysis is
presented.
Impact Analysis of Midpoint Connected STATCOM on Distance Relay PerformanceRadita Apriana
This paper presents the impact of the Static Synchronous Compensator (STATCOM) on the
performance of distance protection of EHV transmission lines. A 400kV transmission system having
midpoint connected STATCOM with its control circuit is modeled using MATLAB/SIMULINK software. The
impact of STATCOM on distance relay for different fault conditions and different fault locations is analyzed.
Simulation results indicate that the presence of the STATCOM in the transmission system significantly
changes the line impedance seen by the distance relay to be lower or higher than the actual line
impedance. Due to this the performance of the distance relay changes, either overreaches or under
reaches.
Optimal Placement and Sizing of Distributed Generation Units Using Co-Evoluti...Radita Apriana
Today, with the increase of distributed generation sources in power systems, it’s important to
optimal location of these sources. Determine the number, location, size and type of distributed generation
(DG) on Power Systems, causes the reducing losses and improving reliability of the system. In this paper
is used Co-evolutionary particle swarm optimization algorithm (CPSO) to determine the optimal values of
the listed parameters. Obtained results through simulations are done in MATLAB software is presented in
the form of figure and table in this paper. These tables and figures, show how to changes the system
losses and improving reliability by changing parameters such as location, size, number and type of DG.
Finally, the results of this method are compared with the results of the Genetic algorithm (GA) method, to
determine the performance of each of these methods.
Similarity and Variance of Color Difference Based DemosaicingRadita Apriana
The aim of the project is to find the missing color samples at each pixel location by the
combination of similarity algorithm and the variance of colour difference algorithm. Many demosaicing
algorithms find edges in horizontal and vertical directions, which are not suitable for other directions.
Hence using the similarity algorithm the edges are found in different directions. But in this similarity
algorithm sometimes the horizontal and vertical directions are mislead. Hence this problem can be rectified
using the variance of colour difference algorithm. It is proved experimentally that this new demosaicing
algorithm based on similarity and variance of colouyr difference has better colour peak signal to noise ratio
(CPSNR). It has better o0bjective and subjective performance. It is an analysis study of both similarity and
colour variance algorithms.
Energy Efficient RF Remote Control for Dimming the Household ApplaincesRadita Apriana
During recent years there is a strong trend towards radio frequency (RF) remote controls as it is
delivering even more comfort to the users and increased usability with high robustness of the RF links.
Lower power consumption with new features and security make RF remote control systems more
competitive with widely used IR remote control systems. In this paper we propose, a RF module based
real time system, which is an integrated system designed to control the dimming or speed of the
appliances. Zigbee module is used as RF module to establish wireless link between Remote and appliance
section having 9600kbs baud rate and range of 100m. Dimming control circuits is part of appliances
section which is used to control the appliances corresponding to signals generated by the remote section.
This system provides energy efficient solution for household uses.
A Novel and Advanced Data Mining Model Based Hybrid Intrusion Detection Frame...Radita Apriana
An Intrusion can be defined as any practice or act that attempt to crack the integrity,
confidentiality or availability of a resource. This may contain of a deliberate unauthorized attempt to access
the information, manipulate the data, or make a system unreliable or unusable. With the expansion of
computer networks at an alarming rate during the past decade, security has become one of the serious
issues of computer systems.IDS, is a detection mechanism for detecting the intrusive activities hidden
among the normal activities. The revolutionary establishment of IDS has attracted analysts to work
dedicatedly enabling the system to deal with technological advancements. Hence, in this regard, various
beneficial schemes and models have been proposed in order to achieve enhanced IDS. This paper
proposes a novel hybrid model for intrusion detection. The proposed framework in this paper may be
expected as another step towards advancement of IDS. The framework utilizes the crucial data mining
classification algorithms beneficial for intrusion detection. The Hybrid framework would hence forth, will
lead to effective, adaptive and intelligent intrusion detection.
For some kinds of products, such as car, aircraft, government acquisition, the consumers have
strict requirements to the reliability of these products. Then the manufacturer is inclined to provide the twodimensional
preventive maintenance policy to take the usage degree of the product into account. As a
result, two-dimensional preventive maintenance policy in the warranty period has recently obtained
increasing attention from manufacturers and consumers. In this paper, we focused on the optimization of
based warranty cost and proposed a new expected cost model of the two-dimensional imperfect
preventive maintenance policy from the perspective of the manufacture. The optimal two-dimensional
preventive maintenance was obtained by minimizing based warranty cost. And asymmetric copula function
was applied to modeling the failure function of the product. At last, numerical examples are given to
illustrate the proposed models, of which the results prove the model effective and validate.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
2. ISSN: 2302-4046
TELKOMNIKA Vol. 13, No. 2, February 2015 : 337 – 351
338
Therefore, there is a need to eliminate the outright inversion. Various methods for
approximate matrix inversion have been proposed in literature: including Cayley-Hamilton
theorem method, Neumann series expansion method, QR decomposition method, random
matrix methods, and methods based on polynomial and truncated polynomial filters [13-21].
These methods still require a lot of computational effort and some of them have proven to be
even more complex than the direct inversion. However, it is important to note that the challenge
is fundamentally different in at least three possible ways. To begin with, the channel matrix does
not have a structure. Secondly, instead of being deterministic, it is random. Finally, there is no
guarantee of sparsity. So we can dispense with the traditional methods for solving the linear
estimation problem [22].
With this in view, we propose a novel method to solve such large linear systems. The
proposed method treats the linear estimation problem as a coordinate transformation problem.
By doing so, the method is able to achieve zero norms in error and residue in contrast to the
state of the art iterative methods for large systems – LSQR, LSMR, and Kaczmarz that achieve
much higher norms - and a 25% reduction in computation complexity when compared to the QR
method, the leading protagonist in exact methods [23-25]. Also by employing the proposed
method, a particular unknown, which may be the only information required by a single user, can
be computed without evaluating the entire solution vector while no such flexibility is available in
the traditional methods. Furthermore, the method is stable and light on computation as well as it
only relies on multiplication with simple rank-one projection matrices for obtaining the solution.
Finally, the method can be programmed in just ten lines of codes.
Rest of the article is organized as follows. Section 2 begins with a system model and
defines the underlying problem. Basic idea behind the proposed method is explored in Section
3. The method is then explained in Section 4. Section 5 contains its complete proof. Section 6
provides the proof of the choice of reflection matrices. A complete algorithm for step by step
solution of the linear estimation problem according to the proposed method is outlined in
Section 7. This section also includes a user-friendly code for the algorithm. Comparison of the
proposed method with QR decomposition is carried out in section 8. The process of generation
of inverse vectors is also discussed in this section. Computational complexity of the proposed
algorithm is analyzed in Section 9 which is followed by its stability analysis in Section 10.
Section 11 concludes the article in which the comparison of the proposed method is carried out
with the state of the art methods available in literature for solving large systems.
2. Problem
A narrowband memoryless MIMO channel with transmit and receive antennas can
be modeled as a system of linear equations [13].
(1)
With,
⋮ ⋮ ⋮
(2)
is a large, random and non-sparse matrix. is the input vector. is the output vector. Zero
Forcing (ZF) estimate of the input vector is:
(3)
Recovery of requires the inversion of a potentially large matrix .
3. TELKOMNIKA ISSN: 2302-4046
A New Approach to Linear Estimation Problem in Multi-user … (Muhammad Ali Raza Anjum)
339
3. Basic Idea
Linear estimation problem presented by Equation (3) can be viewed as a coordinate
transformation problem. can be envisaged as the vector that undergoes the coordinate
transformation, represents its coordinates in the new coordinate system, and contains the
basis vectors for the new coordinate system [26]. The basis vectors can be orthogonal or non-
orthogonal. For the first case, we are lead to the classic Least Squares (LS) solution depicted in
Figure 1.
Figure 1. Computation of unknowns in an orthogonal coordinate system
In this case, the estimates , , … , are the projections of on the respective basis vectors.
⋮
(4)
When the basis vectors are not orthogonal, is computed by forming the inverse of
matrix. This is precisely what we desire to avoid. We would like to compute from the direct
estimate of the projections. In order to do so, we focus our attention on Figure 2 for the case of
non-orthogonal basis vectors. As can be observed from figure, employing the definition of dot
product in Equation (4) will yield much longer estimates than the actual ones. Reason for these
incorrect estimates can be attributed to the presence of identity matrix in the general definition
of the dot product. We demonstrate this fact by re-writing the first term in Equation (4).
Figure 2. Computation of first unknown in a non-orthogonal coordinate system
4. ISSN: 2302-4046
TELKOMNIKA Vol. 13, No. 2, February 2015 : 337 – 351
340
(5)
Identity matrix depicts a set of orthogonal basis vectors. As they are not, we can
replace the matrix in Equation (5) by another matrix, say .
(6)
To solve for , we re-arrange Equation (6) a bit.
(7)
Or,
(8)
Equation (8) requires the vectors and to have equal projections on a plane
spanned by the column space of . We observe from Figure 2 that and have equal
projections on a plane orthogonal to the second basis vector . and can be projected on
such plane by a projection matrix of the form:
(9)
We can substitute in Equation (5) to solve for .
(10)
Similarly for ,
(11)
will project and on a plane orthogonal to . Using Equation (10) and (11), we can
directly solve for without inverting the matrix . Equation (10) and (11) also have a
connotation that can be solved independently of .
4. Method
Now we present the method for solving the general problem of the form . We
begin by expanding Equation (3),
… (12)
Multiplying Equation (12) by a matrix ,
… (13)
In order for to be zero,
(14)
5. TELKOMNIKA ISSN: 2302-4046
A New Approach to Linear Estimation Problem in Multi-user … (Muhammad Ali Raza Anjum)
341
Equation (13) becomes,
… (15)
Multiplying Equation (15) by a matrix ,
… (16)
In order for to be zero,
(17)
Equation (16) becomes,
… (18)
Continuing in the same fashion,
… … … (91)
or,
… … (20)
Multiplying both sides by ,
… … (21)
Or,
(22)
With,
… (23)
For the -th unknown,
(24)
With,
… … ∃ (25)
5. Proof
In this section, we provide the proof of the proposed method. We begin the proof by
writing the basic least squares problem.
(26)
6. ISSN: 2302-4046
TELKOMNIKA Vol. 13, No. 2, February 2015 : 337 – 351
342
Expanding Equation (26),
⋮ … ⋮
(27)
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (28)
If columns of are orthonormal, then Equation (28) becomes,
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮
(29)
⋮ (30)
If columns of are orthogonal, then Equation (28) becomes,
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮
(31)
⋮ (32)
If columns of are neither orthonormal nor orthogonal, then the off-diagonal terms in
Equation (28) prevent the direct solution. If we can somehow eliminate these terms, then a
direct solution of the form (32) is possible. In order to do that, we insert a block matrix matrix
in Equation (26) and re-write it as:
(33)
Expanding (33),
7. TELKOMNIKA ISSN: 2302-4046
A New Approach to Linear Estimation Problem in Multi-user … (Muhammad Ali Raza Anjum)
343
⋮ ⋮ … ⋮ ⋮ (34)
⋮
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ ⋮ (35)
If only,
∃ (36)
Then Equation (35) takes the form,
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (37)
And the solution becomes,
⋮ (38)
For Equation (38) to hold, we can choose to be:
… … ∃ (39)
is a projection matrix.
(40)
projects a vector on an axis that is orthogonal to the plane defined by . An important
property of matrix is that the product is zero. This is because the projection of any vector
on an axis orthogonal to itself is always zero. Hence, the product is always zero.
(41)
This property of matrix can be can be used to eliminate the off-diagonal entries in
Equation (1).
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (42)
8. ISSN: 2302-4046
TELKOMNIKA Vol. 13, No. 2, February 2015 : 337 – 351
344
Starting with the last term in first row on the left hand side of Eq. (42), we multiply the
top row with .
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (43)
With,
(44)
Equation (43) becomes,
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (45)
Similarly, to eliminate we multiply the top row in Equation (45) with ,
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (46)
Such that,
(47)
We continue this multiplication in this manner until all the off diagonal entries in the first
row are removed.
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (48)
With ,
… … (49)
And,
… …
… …
(50)
Following the same procedure to remove off diagonal entries for the remaining rows of
Equation (48), it becomes:
9. TELKOMNIKA ISSN: 2302-4046
A New Approach to Linear Estimation Problem in Multi-user … (Muhammad Ali Raza Anjum)
345
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ (51)
With,
… … ∃ (52)
Multiplying both sides of Equation (52) with to revert back to our least squares solution as in
Equation (35),
⋮
…
…
⋮ ⋮ ⋱ ⋮ ⋮
…
…
⋮ ⋮ (53)
Solving for in Equation (53), we obtain Equation (38).
⋮ (Repeat)
6. Proof of the Choice of Matrix
In this section, we will provide a novel factorization of LS solution which confirms our
choice of matrix. For the sake of brevity, we will provide the proof for 2. The proof can be
similarly extended to any dimension. Substituting 2 in Equation (26) and expanding,
(54)
(55)
Proceeding directly for the solution,
(56)
This would yield,
(57)
(58)
10. ISSN: 2302-4046
TELKOMNIKA Vol. 13, No. 2, February 2015 : 337 – 351
346
(59)
The term can be in written two ways. Firstly, multiply
and divide it by ,
(60)
Secondly, multiply and divide it by ,
(61)
Opting Equation (60) and Equation (61) for first and second rows of Equation (59) respectively,
(62)
Terms and are scalars. Cancelling and simplifying Equation (62),
(63)
The factorization process reveals the same matrix we employed in Equation (38) as
the solutions obtained by Equation (38) and (63) are exactly same. This confirms our choice of
matrix in Equation (38).
7. Algorithm
In this section, we present an algorithm for proposed method. We will start with an
example, say computation of , and onwards build a generic algorithm. We begin by re-writing
Equation (15),
(64)
is generated from by removing its last column. This last column is used to construct
which is then multiplied w .
(65)
and are then updated.
11. TELKOMNIKA ISSN: 2302-4046
A New Approach to Linear Estimation Problem in Multi-user … (Muhammad Ali Raza Anjum)
347
(66)
(67)
This step can be achieved in a single line of code. It would eliminate the need of extra
variables to store their previous values of and . Similarly, is generated from by
removing its last column from which is constructed for subsequent multiplication by
and and then the update.
(68)
And,
(69)
(70)
(71)
Continuing in same fashion,
(72)
will just be a single column at this step so we have replaced it by .
(73)
To compute , it would be appear from Equation (24) and (25) that there would be a need to
store the original matrix.
(74)
But this would be unnecessary. In ordinary least squares,
0 (75)
is orthogonal to which makes the dot product in Equation (75) equal to zero. In our
case, it is the matrix that makes the dot product equal to zero.
0 (76)
Hence, multiplication by can be avoided. Here, ordinary summation will do the job.
∑
∑
(77)
Equation (77) represents the desired solution. A stepwise algorithm for computing any
arbitrary unknown is given in Table 1.
12. ISSN: 2302-4046
TELKOMNIKA Vol. 13, No. 2, February 2015 : 337 – 351
348
Table 1. Stepwise recursive algorithm for the proposed method
1. Initialize a variable that represents the -th unkown to be determined.
2. Initialize a variable such that and ensure .
3.
Generate from by separating and update the matrix using:
4.
Update the vector using:
5. Decrement .
6. Move back to step 3 and keep iterating until 1.
7.
Compute the unknown using:
∑
∑
8. Generation of Inverse Vectors and Comparison with Qr Decomposition
In this section, we will demonstrate that the first run of the algorithm described in
section 7 will generate the first inverse vector [27], the first unknown , and the complete
matrix,. We begin by considering Equation (65). In that equation, we removed the last column of
to form , constructed from the removed column, multiplied with , and then
finally disposed off that last column. We proceeded in the similar fashion until we arrived at in
Equation (72). But during the entire process, we disposed off the columns that have been used
to construct matrices at each stage. If these columns are kept instead, then they would
constitute the matrix and the last column will ultimately be the inverse vector [27]. This can
be explained as follows.
Removal of the -th column from will form as in Equation (65). Removed
column is kept as . Multiplying with removes the projections of the columns of
that lie along as described in Equation (65). Last column of the resulting matrix is removed to
generate in Equation (68). This column is kept as . will be orthogonal to . This
is because the projection of that lies along has been removed after multiplication with
. Continuing in the same fashion, projection of every removed column will be subtracted from
the remaining columns of matrix due to multiplication with matrix. In the end, we
obtain in Equation (72) from which all its components that lie parallel to separated columns
have been removed. As a result, will be orthogonal to all the remaining columns. This is
precisely the definition of inverse vector [27]. Hence, is the first inverse vector and the
desired solution to the first unknown .
Now we will explain the generation of and matrices. From , all its projections that
lie along the separated 1 columns have been removed. Hence, will generate 1)
zeros when multiplied with the matrix on the left hand side. This will form the first row of a
lower triangular matrix with 1 zeros. From , all the projections that lie along the last
2 columns have been removed. This will generate the second row of the lower triangular
matrix with 2 zeros. Continuing in the same fashion, we arrive at from which no
projection has been removed. This will generate the last row of the lower triangular matrix that is
full and does not contain any zeros. Given that the removed columns are stored row-wise in a
matrix, multiplying it with will produce a lower triangular matrix. As we have started from the
-the column and the moved backwards to the first column, the matrix generated will be lower
triangular. If instead we start from the first column and then move forward to the -th column,
the resulting matrix will be upper triangular. This matrix is termed as matrix in
13. TELKOMNIKA ISSN: 2302-4046
A New Approach to Linear Estimation Problem in Multi-user … (Muhammad Ali Raza Anjum)
349
decomposition. So will be equal to or . This completes the decomposition
process.
9. Computational Complexity
We begin the analysis of the computational complexity of the algorithm by examining
Equation (66) and (67).
Repeat
Repeat
The term will require the largest number of multiplications which is 1 . In
the next step, multiplication by will require 2 multiplications. Continuing in same
fashion to , there will be a total of 1 ⋯ 3 2 1 multiplications. The
series 1 ⋯ 3 2 1 can be summed up using Gauss’s formula for
the sum of first numbers. Since the last number is 1 , the sum of the series becomes
, resulting in multiplications. So the order of complexity for the computation of a
single unknown would be for a large system. It is tempting to think that the computation
of all -unknowns would require the order of complexity to rise to but this is not the case.
Once an unknown is computed, say , it can be substituted backwards which requires only
additional multiplications. Now for the second unknown, the reduced system does not have to
be solved all over again as all the ’s and ’s that are required for its calculation have been
computed in the first phase, i.e., the phase in which was calculated. Leaving aside the first
and the last unknowns which do not require any back substitution, a total of 2 extra
multiplications will be required for computing all unknowns. Therefore, the order of the
complexity will not rise above . Therefore, the total cost of the algorithm will be . Comparing
it with the QR algorithm which has the cost of [22],
.25
There is a 25% reduction in the total computational cost.
10. Stability
In this section we analyze the stability of the proposed method. For this purpose, we
determine its spectral radius. Spectral radius is defined as the absolute value of the largest
eigenvalue of the controlling matrix. It should be less than or equal to one for a stable operation
[22]. In the proposed method, matrices that control the solution are rank one-projection
matrices. These matrices have a single largest eigenvalue of one whereas all the remaining
eigenvalues are zero. Their spectral radius is 1. This means that if an arbitrary vector is
multiplied with these matrices, its length will not be inflated.
‖ ‖ (78)
Where,
When , Eq. (78) becomes ‖ ‖ ‖ ‖ . In case , ‖ ‖ . In general:
14. ISSN: 2302-4046
TELKOMNIKA Vol. 13, No. 2, February 2015 : 337 – 351
350
‖ ‖ ‖ ‖ (79)
Therefore, there is no risk of explosion in the length of after repeated multiplications.
11. Results/Discussion
We now present the simulation results of the algorithm. The components of the channel
matrix are chosen to be independent and identically distributed (IID) Gaussian random variables
with a zero mean and unity variance. is selected equal to as this refers to multi-user case in
M-MIMO systems because both the number of transmitting and receiving antennas become
very large. Also when , an exact solution is possible and the residue and hence the error
in the estimate can be zero. Various matrix sizes have been selected and the results obtained in
terms of the norm of residue, norm of the error in estimate and the computational time taken are
displayed in Table 2 against the state of the art algorithms available in literature. Codes required
for simulation of LSMR and LSQR algorithms have been adopted from the website of Stanford
University’s System Optimization Laboratory [28] and were used as is. As the proposed method
is exact, it achieves zero norms in all the cases. Its columns for residue and error norms are not
included in the table. Simulation results demonstrate that LSMR, LSQR, and Kaczmarz
algorithms yield much higher norms for both the residue and the error for large matrix sizes.
Due to these higher norms, the estimate becomes practically useless despite the fact that
LSMR and LSQR are much faster than the proposed algorithm. On the other hand, QR being an
exact method achieves zero error/residue norms. But it takes more computation time than the
proposed method. Hence, the proposed method is a much better choice in this scenario. It has
higher speed as compared to QR method and lower error/residue norms when compared to
LSMR, LSQR, and Kaczmarz algorithms.
Table 2. Comparison of the proposed algorithm with the state of the art methods
Proposed QR LSMR LSQR
Kaczmarz
100 iterations
time time ‖ ‖ ‖ ‖ Time ‖ ‖ ‖ ‖ time ‖ ‖ ‖ ‖ time
20
0 0 0.0804 0.6418 0 0.1596 1.3585 0 0.5191 1.2066 0.0630
40
0 0 0.1116 1.7462 0 0.1960 1.6275 0 1.3482 1.3463 0.1260
60
0.0156 0.0211 0.1589 2.0725 0 0.5782 1.8263 0 1.5294 2.3033 0.2200
80
0.0469 0.0627 0.1896 2.1318 0.0150 1.2171 2.3114 0 2.5787 3.0479 0.4070
100
0.0781 0.1045 0.2144 2.3678 0.0151 1.6671 3.0958 0 3.2469 4.3231 0.5010
120
0.1094 0.1459 0.2186 2.4746 0.0153 1.8697 3.5757 0 3.8631 5.2422 0.7050
140
0.1563 0.2083 0.2320 2.4962 0.0159 2.3005 3.6616 0 4.5234 9.3310 0.9870
160
0.2188 0.2916 0.2540 2.6058 0.0160 2.4816 3.7103 0 4.7671 11.7761 1.2980
180
0.3281 0.4375 0.3468 2.8010 0.0310 3.2124 3.7700 0.0320 5.6000 14.2675 1.6420
200
0.4375 0.5833 0.3862 2.8263 0.0780 3.4337 3.8408 0.1870 5.1865 23.5984 2.0490
Acknowledgements
Author is grateful to Professor Dianne O'Leary for her valuable comments about the
manuscript. Prof. O’Leary is currently serving as a Professor of Computer Science at the
Institute for Advanced Computer Studies, University of Maryland at College Park. Her
suggestions regarding the reduction in computational complexity of the algorithm were
especially invaluable for this manuscript.
15. TELKOMNIKA ISSN: 2302-4046
A New Approach to Linear Estimation Problem in Multi-user … (Muhammad Ali Raza Anjum)
351
References
[1] Mietzner J, Schober R, Lampe L, Gerstacker W and Hoeher P. Multiple-antenna techniques for
wireless communications - A comprehensive literature survey. IEEE Communications Surveys and
Tutorials. 2009; 11: 87-105.
[2] Tse D and Viswanath P. Fundamentals of wireless communication. Cambridge University Press
Marzetta TL. Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE
Transactions on Wireless Communications. 2005; 201(09): 3590-3600.
[3] Rusek F, Persson D, Buon Kiong L, Larsson EG, Marzetta TL, et al. Scaling Up MIMO: Opportunities
and Challenges with Very Large Arrays. IEEE Signal Processing Magazine. 2013; 30: 40-60.
[4] Vishwanath S, Jindal N, Goldsmith A. Duality, achievable rates, and sum-rate capacity of Gaussian
MIMO broadcast channels. IEEE Transactions on Information Theory. 2003; 49: 2658-2668.
[5] Weingarten H, Steinberg Y, Shamai S. The capacity region of the Gaussian multiple-input multiple-
output broadcast channel. IEEE Transactions on Information Theory. 2006; 52: 3936-3964.
[6] Larsson EG. MIMO detection methods: how they work. IEEE Signal Processing Magazine. 2009; 26:
91-95.
[7] Jalden J, Ottersten B. On the complexity of sphere decoding in digital communications. IEEE
Transactions on Signal Processing. 2005; 53: 1474-1484.
[8] Peel CB, Hochwald BM, Swindlehurst AL. A vector-perturbation technique for near-capacity
multiantenna multiuser communication-part I: channel inversion and regularization. IEEE Transactions
on Communications. 2005; 53: 195-202.
[9] Ryan DJ, Collings IB, Clarkson IVL, Heath RW Jr. Performance of vector perturbation multiuser MIMO
systems with limited feedback. IEEE Transactions on Communications. 2009; 57: 2633-2644.
[10] Windpassinger C, Fischer RFH, Huber JB. Lattice-reduction-aided broadcast precoding. ITG-
Fachbericht. 2003; 403-408.
[11] Liang YC, Cheu EY, Bai L, Pan G. On the relationship between MMSE-SIC and BI-GDFE receivers for
large multiple-input multiple-output channels. IEEE Transactions on Signal Processing. 2008; 56:
3627-3637.
[12] Couillet R, Debbah M. Random Matrix Methods for Wireless Communications. Cambridge University
Press. 2011.
[13] Edman F, Owall V. Implementation of a scalable matrix inversion architecture for triangular matrices.
14th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications
Proceedings. Piscataway, NJ, USA: IEEE. 2003; 2558-2562.
[14] Golub G, Van Loan C. Matrix Computations (Johns Hopkins Studies in Mathematical Sciences) (3rd
Edition). The Johns Hopkins University Press. 1996.
[15] Honig ML, Weimin X. Performance of reduced-rank linear interference suppression. IEEE
Transactions on Information Theory. 2001; 47: 1928-1946.
[16] Karkooti M, Cavallaro JR, Dick C. FPGA implementation of matrix inversion using QRD-RLS
algorithm. 39th Asilomar Conference on Signals, Systems and Computer. Piscataway, NJ, USA:
IEEE. 2005; 1625-1629.
[17] Ma L, Dickson K, McAllister J, McCanny J. QR decomposition-based matrix inversion for high
performance embedded MIMO receivers. IEEE Transactions on Signal Processing. 2011; 59: 1858-
1867.
[18] Muller RR, Verdu S. Design and analysis of low-complexity interference mitigation on vector channels.
IEEE Journal on Selected Areas in Communications. 2001; 19: 1429-1441.
[19] Stewart GW (1998) Matrix Algorithms: Volume 1: Basic Decompositions. SIAM.
[20] Wu D, Eilert J, Liu D, Wang D, Al-Dhahir N, et al. (2007) Fast complex valued matrix inversion for
multi-user STBC-MIMO decoding. IEEE Computer Society Annual Symposium on VLSI: Emerging
VLSI Technologies and Architectures, ISVLSI'07, March 9, 2007 - March 11, 2007. Porto Alegre,
Brazil: Inst. of Elec. and Elec. Eng. Computer Society. pp. 325-330.
[21] Strang G. Computational Science and Engineering. Wellesley-Cambridge Press. 2007.
[22] Fong D and Saunders M. LSMR: An iterative algorithm for sparse least-squares problems. SIAM
Juornal on Scientific Computing. 2011; 2950-2971.
[23] Paige CC and Saunders MA. LSQR: An Algorithm for Sparse Linear Equations and Sparse Least
Squares. ACM Trans Math Softw. 1982; 8: 43-71.
[24] Strohmer T, Vershynin R. A Randomized Kaczmarz Algorithm with Exponential Convergence. J
Fourier Anal Appl. 2009; 15: 262-278.
[25] Fleisch D. A Student's Guide to Vectors and Tensors. Cambridge University Press. 2011.
[26] Anjum MAR, Ahmed MM. A New Approach for Inversion of Large Random Matrices in Massive MIMO
Systems. PLoS ONE. 2014.
[27] http://www.stanford.edu/group/SOL/software/lsmr.html.