62 results about "QR decomposition" patented technology

The invention discloses a tire longitudinal force and lateral force prediction method based on vehicle state parameter estimation, which adopts a seven-order spherical radius criterion to replace the original three-order spherical radial volume criterion on the basis of the traditional volume Kalman filter (CKF), and improves the estimation precision by optimizing sampling point distribution and weight configuration. And the stability of numerical calculation is ensured in combination with QR decomposition. A vehicle-mounted sensor is used for collecting real-time data such as a front wheel turning angle and longitudinal / lateral acceleration under a complex working condition, a time sequence feature set is built by combining longitudinal speed, lateral speed, a side slip angle and yaw velocity estimated by an algorithm, and meanwhile tire force calculated by a Dugoff tire model serves as target output. The TCN-GRU-based tire force prediction method is designed, the calculation complexity can be reduced while key time sequence information is kept, and the real-time prediction performance of the tire force is improved.

The invention provides an image restoration method based on adaptive weighted tensor completion, and relates to the technical field of image processing and application, and the method comprises the steps: obtaining to-be-restored image data, and carrying out the tensor of the to-be-restored image data, and obtaining input tensor data; constructing a tensor completion model based on an adaptive weighted tensor nuclear norm; wherein a weight matrix in the tensor completion model can be adaptively updated along with input tensor data; and based on an alternating direction multiplier method or an approximate singular value decomposition method based on tensor QR decomposition, solving the tensor completion model, and outputting restored tensor data to realize image restoration. According to the scheme, the image restoration quality can be improved.

The invention relates to the technical field of telephone communication, and discloses a robust recursive least square adaptive filter based on inverse QR decomposition and a storage medium, and the method comprises the steps: calculating an inner product of an adaptive weight vector at a moment and an input signal vector, and obtaining an output signal at the moment of the adaptive filter; calculating a difference value between the expected signal and the output signal at the moment, obtaining an estimation error signal, and constructing a nonlinear weighting function; recursively updating and acquiring an inverse correlation matrix of the moment input signal vector based on the moment input signal vector and transpose thereof, a nonlinear weighting function, the inverse correlation matrix of the input signal vector and the covariance of the estimated error signal, and constructing a block matrix containing the inverse correlation matrix of the moment input signal vector; inverse QR matrix decomposition and unitary rotation operation are sequentially carried out on the block matrix, an array before inverse QR decomposition and an array after inverse QR decomposition at the moment are obtained, a gain vector at the moment is constructed, and a self-adaptive weight vector at the moment is updated and obtained.

In order to solve the problems of high calculation complexity and low efficiency in hydraulic fracturing numerical simulation, the invention provides a method for parallel simulation of hydraulic fracturing crack propagation. A three-dimensional fracture grid is constructed by adopting a displacement discontinuity method (DDM) and a finite volume method (FVM), and stress and fluid pressure distribution in fractures are calculated. Crack expansion is driven by optimizing DDM stress coefficient matrix generation and combining FVM to calculate crack tip pressure. OpenMP multi-thread scheduling and Intel MKL and Eigen library optimization are adopted, LU or QR decomposition is dynamically selected according to the matrix scale, and therefore the calculation efficiency is improved. According to the method, by initializing a three-dimensional fracture grid, optimizing a stress matrix, constructing a fluid-solid coupling equation set, carrying out iterative solution, checking convergence and generating a three-dimensional fracture expansion model, efficient fracture expansion simulation is achieved, and an efficient and accurate numerical tool is provided for fracture design of unconventional oil and gas reservoirs.

The invention relates to the technical field of compressed sensing, and particularly discloses a sparse adaptive system identification method and system based on inverse QR decomposition, and the method comprises the steps: a receiving signal obtaining step: transmitting a transmitting signal to a detected target, and then obtaining a receiving signal, the receiving signal comprises a direct coupling interference signal and a target signal; an estimation signal obtaining step of obtaining an estimation signal corresponding to the transmitting signal based on the transmitting signal through an adaptive filtering model based on inverse QR decomposition; and an identification step: eliminating the direct coupling interference signal through the estimation signal to obtain the target signal.

The invention belongs to the field of intelligent manufacturing, and discloses a component state real-time sensing method based on parameterized loads. According to the method, parameterized representation is carried out on the load, an overall stiffness matrix is obtained in combination with finite element analysis, and a displacement data set of a component under the parameterized load is constructed through Monte Carlo statics simulation. On the basis of POD, in combination with parameterized loads, a parameterized intrinsic orthogonal decomposition method is provided, then a projection base vector used for expressing a component displacement deformation mode is obtained, sensor layout optimization is conducted on a component deformation sensitive area through QR decomposition, and finally a real-time and accurate displacement state sensing result is obtained. The calculation amount of real-time sensing of the displacement state of the component can be effectively reduced, and the operation efficiency is improved; and by optimizing the layout of the sensor, the displacement sensing precision of the component is improved.

The present invention discloses a method for minimal-delay fine-grained parallel Givens rotation of ultra-large-scale matrix QR decomposition, relating to the technical field of signal and information processing. The method comprises: an outer loop and an inner loop, wherein the outer loop is responsible for calculating a row vector of a Q matrix and an R matrix, and the inner loop is responsible for the Givens rotation of the two row vectors; the outer loop is decomposed into multiple loop blocks and a multi-level PE unit array is used to realize parallel operation of the outer loop slices, while the parallel calculation of the inner loop slices is realized by a step-by-step decreasing element update module inside the PE unit; the collaborative design of the inner and outer loop slices significantly reduces the idle time of hardware resources and optimizes the generation and transmission of key data, so that the system starts calculation with minimal delay; the method improves the utilization efficiency of hardware accelerators and is suitable for ultra-large-scale matrix decomposition tasks in the fields of signal processing, image processing, and scientific computing.

The application discloses a low-complexity deep learning MIMO detection method based on partial MAP, comprising the following steps: obtaining multiple groups of training and test samples, and performing hierarchical and data preprocessing on the training and test data samples through QR decomposition to obtain K training sublayers and calculate the input and output labels of each training sublayer; training a DNN for generating local LLR in each detection stage of the K training sublayers based on a partial MAP method; performing layer-by-layer inspection from the Kth layer to the first layer, for each detection stage, calculating the local LLR by using the trained DNN, updating the LLR result of the current layer in combination with the prior information transmitted by the previous layer, and transmitting the LLR result to the next layer as prior information until the final LLR result is obtained; inputting the final LLR result into an FEC decoder to perform error correction and obtain a final signal estimation result; and the partial MAP directly outputs log-likelihood ratio soft information, and the combination of the partial MAP and a forward error correction code can obtain better detection performance.

This invention discloses a method and device for reconstructing the deformation of mesh antenna nodes, belonging to the field of spaceborne mesh antennas. Based on the geometric position and topological connection relationship of the mesh antenna, a mapping model between node displacement and cable element strain is established. RBF radial basis interpolation is introduced, and the displacement information of a small number of control points is used to characterize the deformation field of the entire reflector surface, establishing an explicit relationship between cable element strain and control point deformation. Singular value decomposition is performed on this mapping matrix to extract the dominant strain characteristics; then, combined with column-pivot QR decomposition, the most linearly independent cable elements with the most significant response to the dominant strain are automatically identified as the placement positions for strain sensors. This invention achieves high-precision reconstruction of mesh antenna node displacements with a limited number of sensors, effectively reducing the number of sensors required while ensuring reconstruction accuracy, thus solving the problems of large sensor count and high system complexity in on-orbit deformation monitoring of spaceborne mesh antennas.

The application discloses a radar target single-station RCS fast scanning angle method, which comprises the following steps: firstly, a triangle grid is used to divide a created radar target geometric model; secondly, an ACA algorithm is used to perform low-rank compression processing on an excitation matrix; thirdly, a double-layer mixed compression method combining an ACA algorithm and QR decomposition is used; fourthly, QR decomposition is performed on the compressed matrix, and ACA algorithm is used for re-compression; fifthly, the number of excitations required for iterative solving in the solving process is further reduced; and finally, a multilevel fast multipole method is used to combine a near-zone block diagonal preconditioning technology to construct a matrix equation, so that the number of iterative solving times of the matrix equation is greatly reduced, the solving of the matrix equation is accelerated, and the solving efficiency of the radar target single-station RCS under the excitation of a plane wave at multiple angles is improved.

The application relates to a QR decomposition-based matrix-vector multiplication implementation method based on a vector processor, and the QR decomposition algorithm is a vsip_cqrd_f function in a VSIP library, the VSIP library is a vector, signal and image processing library launched by a GE intelligent platform, and is an open standard application programming interface specially provided for developers who need to use intensive signal and image processing. The technical scheme of the application mainly comprises the following steps: step one, judging whether input matrix C needs transposition conjugate processing and whether the matrix C is on the left or right in matrix multiplication, so as to enter different branch processing; step two, pre-processing the A matrix; step three, adopting different processing schemes according to the quantity size to realize matrix-vector multiplication operation. The application is designed and realized for the QR decomposition-based matrix-vector multiplication in the light of the advantages of the vector processor, such as large storage space, on-chip cache and efficient and fast data transmission, and can play a great advantage in complex matrix operation.

The application discloses a spherical decoding detection method and device, electronic equipment and a readable storage medium, and belongs to the technical field of wireless communication. The application obtains a Q matrix and an R matrix by performing orthogonal triangular QR decomposition on a channel response matrix, wherein each column in the Q matrix is arranged according to the size of channel energy; determines a target equalization signal according to the Q matrix and the R matrix; performs an ML path search process according to the target equalization signal to select N branches with minimum metrics as survivor paths; performs ML complement path search on the survivor paths to obtain ML complement paths; takes the survivor paths with minimum metrics as ML paths, and obtains the likelihood ratio information of each bit of each symbol at each layer according to the ML paths and the ML complement paths. The application can reduce the operation complexity of the spherical decoding algorithm while maintaining the decoding detection performance.

The invention discloses a joint sorting QR decomposition method and related device for multi-antenna detection, and belongs to the technical field of multi-input multi-output signal detection, and the method comprises the steps: initializing an orthogonal matrix and an upper triangular matrix; acquiring a channel matrix H; extracting candidate columns from residual columns in the channel matrix H based on the sum of residual energy and self energy joint weighting, updating an orthogonal matrix and an upper triangular matrix according to the candidate columns until all columns in the channel matrix H are traversed, and obtaining a final orthogonal matrix and a final upper triangular matrix; and taking the final orthogonal matrix and the upper triangular matrix as a joint sorting QR decomposition result for multi-antenna detection, the method and the related device can shear out the nodes with the super radius as early as possible, reduce access nodes, and improve the speed of multi-antenna detection while ensuring the detection performance.

The invention discloses a QR decomposition matrix inversion optimization method and system. The method comprises the following steps: receiving signal data, and carrying out FFT (Fast Fourier Transform) preprocessing and data downsampling; double-buffer optimization parallel processing is adopted, and FFT signal data are stored based on a double-buffer FIFO data storage structure; calculating a covariance matrix of the FFT signal data; adopting Givens rotation to gradually eliminate off-diagonal elements of the covariance matrix, carrying out Givens rotation calculation, and adopting an optimized CORDIC calculation method to calculate a Givens rotation factor; a calculation process of Givens rotation is accelerated by adopting a Systolic array architecture; and outputting a final QR decomposition result according to a Givens rotation calculation result. According to the method, the data flow is optimized, the calculation throughput rate is improved, hardware resource occupation is reduced, a low-power-consumption and high-performance hardware implementation scheme is provided for efficient matrix calculation, and the method has wide engineering application value.

The invention relates to a TSN (Time Sensitive Network) timestamp recovery method based on an FPGA (Field Programmable Gate Array), which belongs to the technical field of TSNs and comprises the following steps of: S1, collecting synchronous messages of different TSN nodes in parallel by multiple ports of the FPGA, and analyzing timestamp data by taking a message serial number as an anchor point; s2, constructing an incomplete timestamp matrix and a mask matrix by adopting a time sliding window mechanism, and automatically marking missing data; s3, a random index set is generated through LFSR, and matrix sparse sampling is achieved through a bitmap judgment unit; and S4, based on an FPGA accelerated CUR matrix decomposition algorithm, solving a pseudo-inverse matrix through parallel MGS-QR decomposition, and reconstructing a complete timestamp matrix through block parallel matrix chain multiplication. The FPGA parallel architecture, ping-pong cache and fixed-point number processing are adopted, microsecond-level low-delay recovery is achieved, hardware resource occupation is greatly reduced, robustness in a severe network environment is improved, and the method is suitable for a high-precision clock synchronization scene of an industrial TSN node.

This invention discloses an efficient QR matrix decomposition method based on FPGA for implementing systolic matrices, belonging to the field of software-defined radio signal transmission technology for airborne networks. It is based on the design of a multi-channel digital front-end implemented on an FPGA chip. The method includes: designing a general handshake coordination mechanism, designing a timing flow for systolic matrix multiplication, performing fast QR matrix decomposition single-loop operations by calculating the Householder mirror transformation matrix, designing a top-level loop pipeline for QR matrix decomposition, and finally verifying the hardware scheme of the fast QR decomposition algorithm. This invention converts resource consumption into timing consumption by designing the overall pipeline logic and loop operations, enabling fast processing and accurate decomposition of digital information in a multi-channel digital front-end on an FPGA, effectively saving on-chip resources. In addition, this invention has excellent scalability, supporting 64 to 128-bit high-precision fixed-point number formats, and the data matrix order can reach up to 128.

This application provides a method and system for ad push based on user browsing behavior, belonging to the field of data processing technology. First, this application simultaneously acquires the user's historical search terms and current page interaction behavior, including click weights and dwell time, and constructs an intent vector representing long-term preferences and a behavior vector reflecting current interests accordingly. Next, the two vectors are combined into an observation matrix, and a QR decomposition technique is used to extract an orthogonal basis vector representing the user's independent, core intent as a reference standard. Subsequently, the grey relational coefficient and correlation degree between this reference vector and the attribute vectors of each candidate ad are calculated to quantify the degree of matching between the two. Finally, the ads with the highest correlation are selected and accurately pushed to the user. This application achieves accurate mining of the user's core intent and matching and pushing highly relevant ads in complex page environments.

The present application relates to the field of digital image watermarking technology in information security, and provides a double color data image watermarking, image watermarking embedding, image watermarking extraction method and system. The double color data image watermarking method comprises an embedding process and an extraction process. In the embedding process, the color watermarking image W is encrypted by using HDNA, Mandelbrot graph and three-dimensional IPCM chaotic system to obtain an encrypted watermarking; the color carrier image is decomposed by QR decomposition, and then the encrypted watermarking is embedded into the color carrier image. In the extraction process, the encrypted watermarking information is extracted by QR decomposition, and then the original color watermarking information is extracted from the encrypted watermarking information by HDNA and Mandelbrot graph. The present application provides a safe, transparent and efficient double color image watermarking method, which has the advantages of high security, strong robustness, large copyright information capacity and fast operation speed, and has stronger application value.

The invention relates to the technical field of carbon emission calculation, and discloses an electric power carbon emission factor calculation method, system, equipment, medium and product based on QR matrix factorization. According to the method, a node power balance equation and a carbon emission conservation equation are determined by obtaining carbon emission information and tidal current power information of all generators in a power grid; the method comprises the following steps: constructing a carbon emission factor linear equation, solving the carbon emission factor linear equation based on QR matrix decomposition to obtain a carbon emission factor value of each node, so that efficient solving is realized by utilizing sparse matrix characteristics and QR decomposition, rapid response of multi-node carbon emission factor solving is realized, and the method does not depend on initial solution setting; the slow convergence or failure risk of an iteration method is avoided, the numerical stability is excellent, and the accuracy and reliability of carbon emission factor calculation are improved.

The present invention relates to a method for determining insulation resistance (R1, R2) and discharge capacitance (C1, C2) for ground (PE) of an ungrounded DC power supply system, wherein a linear difference equation is implemented as a function of a measurement voltage (UG(k)) and a grid voltage (UN(k)) with a measurement voltage (UM(k)), and a grid parameter (θi) formed from a shunt (RM), insulation resistance (R1, R2) to be determined, and discharge capacitance (C1, C2) to be determined. Using N linear differential equations, a measurement value equation system is implemented with a sample sequence ((UG(k), UN(k), UM(k))) and a grid parameter (θi) for measurement times from k = 1, 2 to N with a sampling period T. Another method step comprises a step (S6) of calculating the estimated grid parameter (θi) as an approximate solution to the measurement value equation system, wherein the sum of squared errors between the grid parameter (I) and the estimated grid parameter (θi) is minimized, a step (S7) of minimizing the sum of squared errors through QR decomposition of the measurement-value matrix (Ψ), wherein the QR decomposition is recursively calculated, and a step (S8) of calculating each insulation resistance (R1, R2) and each discharge capacitance (C1, C2) from the estimated grid parameter (I) and repeating the method step with the estimated grid parameter (I), each calculated to include samples available at the current measurement time.

The application provides a sorting QR decomposition method for a MIMO system, comprising: determining an initial matrix to be sorted according to a channel matrix of a MIMO system to be detected; determining column vectors to be sorted of the matrix and a column number thereof, denoted as a target number; if the target number is greater than a preset value, performing a first sorting strategy: in a first round of iteration, exchanging column orders of a column vector to be sorted with a smallest norm value and a first column vector to be sorted, and in each subsequent round of iteration, exchanging column orders of a column vector to be sorted with a second smallest norm value and a column vector to be sorted in a current round, to obtain a first target matrix; and a last column vector of the first target matrix corresponds to a highest layer traversal constellation point of an FSD algorithm. The above scheme can obtain an optimized target matrix. The application also provides an FSD-based signal detection method, which performs FSD signal detection by using the above optimized target matrix, so that the number of highest layer traversal constellation points is reduced, and the detection accuracy and reliability are still ensured.

This invention relates to the field of digital signal processing technology, and discloses a DSP multi-core parallel QR decomposition method and apparatus based on tile algorithm. The method includes: dividing the matrix to be decomposed into blocks in the horizontal direction according to the number of cores of the DSP processor and setting tile blocks; performing QR decomposition on the main diagonal tile blocks; and updating all tile blocks to the right of the main diagonal tile blocks in the horizontal direction; pairwise coupling of the main diagonal tile blocks with each tile block in the vertical direction of the same column; performing QR decomposition on the coupling matrix formed by each pair of couplings; and updating all coupling matrices to the right of the pair of coupling matrices in the horizontal direction; and cyclically executing the decomposition and horizontal update of the main diagonal tile blocks, the coupling decomposition of the tile blocks in the same column, and the update of the right coupling matrix until all main diagonal tile blocks in the matrix to be decomposed have been processed, resulting in an upper triangular matrix. This invention can fully utilize the parallel processing advantages of multi-core processors to achieve inter-core parallel data processing.

Embodiments of the present disclosure provide a visual inertia fused localization method and a device. The method comprises: acquiring target observation data corresponding to a target state vector; constructing a first residual model according to the target observation data; projecting the residual model according to a first-order partial derivative matrix, to obtain a second residual model, the second residual model comprising a product term of a second-order partial derivative square matrix and a system state matrix, and a covariance term corresponding to a first observation noise term; and eliminating a three-dimensional coordinate error vector of the second residual model, to obtain a target residual model, and performing EKF updating on the target state vector according to the target residual model. The embodiments do not involve left null space calculation and projection, and QR decomposition dimension reduction calculation, thereby reducing the amount of calculation, and improving positioning efficiency.

The application provides a multi-base station two-dimensional positioning method based on a pseudo-TDMA mechanism, which requests access by sending a Blink frame by a tag, a main base station allocates a communication time slot and issues an offset, and the tag is aligned to a specified time slot to realize pseudo-TDMA orderly communication; the tag sends a Poll frame in the time slot, and multiple base stations reply a Response frame in turn according to a preset gradient delay to avoid air signal collision; the system monitors time slot drift in real time and corrects through a Response frame feedback; multi-base station ranging data is collected to construct an over-determined equation, QR decomposition is used to solve the least square to obtain a two-dimensional coordinate of the tag, and abnormal base station data is removed and redundant positioning is supported. The application has the beneficial effects that the communication collision and strict network synchronization problems in a high-density scene are effectively overcome, the application has the advantages of low collision rate, high channel utilization rate, high positioning precision and strong fault tolerance, and is particularly suitable for large-scale personnel positioning in complex environments such as underground.

The present application belongs to the technical field of wireless signal positioning, and discloses a ground-based radiated source TDOA positioning method based on the constraint of the earth ellipsoid. First, the TDOA observation quantity about the ground-based radiated source is obtained by using multiple adjacent space observation platforms, and is converted into a distance difference observation quantity. Then, a scalar product matrix is constructed by using the distance difference observation quantity, and a pseudo-linear observation equation is constructed based on the same. Next, the influence of the distance difference observation error and the prior observation error of the observation platform position vector on the pseudo-linear observation equation is quantitatively analyzed by using a first-order error analysis method, and the singularity problem of the pseudo-linear observation error covariance matrix is avoided by using matrix QR decomposition, so that an optimal weighting matrix is obtained. Then, by using the earth ellipsoid constraint to which the ground-based radiated source is subjected and the algebraic characteristics of the auxiliary variable, a positioning optimization model containing double quadratic equality constraints is constructed. The present application utilizes the earth ellipsoid constraint to which the ground-based radiated source is subjected, and effectively improves the positioning precision.

The invention relates to a key technology in the field of computer image processing and data recovery, in particular to an asynchronous calculation method for a matrix singular value and a singular vector of the matrix singular value. The matrix singular value pure CPU asynchronous calculation method specifically comprises the following steps: uniformly partitioning an input matrix into a plurality of sub-matrixes according to rows, carrying out asynchronous parallel QR decomposition iterative calculation on each sub-matrix on a multi-core CPU platform, and extracting a column orthogonal basis and a row orthogonal basis of each sub-matrix; recombining the row orthogonal basis of each sub-matrix and then carrying out QR decomposition iteration again to generate an intermediate matrix of which the dimensionality is obviously reduced; performing approximate singular value decomposition on the intermediate matrix; and finally obtaining a plurality of first singular values and singular vectors of the original matrix through matrix multiplication recombination. According to the method, the parallel computing capacity of the multi-core CPU is fully utilized, GPU hardware support is not needed, and compared with a traditional SVD algorithm, the computing speed is obviously increased.

The embodiment of the invention provides a large-scale matrix QR decomposition single-core serial calculation method, device and equipment and a storage medium, block calculation is performed on a matrix according to the fast memory capacity to realize multi-stage matrix QR decomposition, and compared with a single-core serial QR decomposition method, the method has the advantages that the matrix is always accessed in a rectangular block mode when the matrix is accessed, so that the matrix access efficiency is improved. In the calculation step, the serial QR decomposition step is executed in the diagonal matrix blocks, and the Level-3 BLAS is used when the non-diagonal matrix blocks are updated, so that the calculation complexity and memory access ratio in a single core are improved, and the calculation efficiency of the single core is improved.

The application discloses a kind of geometric guidance and task self-adapting multithreaded spherical decoding method and related device, belong to wireless communication technical field, including: the channel matrix of system model is obtained, the channel matrix of system model is carried out QR decomposition, and the result of QR decomposition is obtained;Search tree is constructed based on the result of QR decomposition;The search radius and optimal solution of search tree are dynamically updated, and the geometric threshold value is determined;According to the search radius and optimal solution of updated search tree and the geometric threshold value determined, the search tree is searched by multithreading according to SE order, and the symbol vector of transmitting antenna in system model is obtained, and the geometric guidance and task self-adapting multithreaded spherical decoding is completed, the method and related device can improve search efficiency, shorten search time simultaneously, reduce the cost of search task scheduling and synchronization.

A method for multi-objective vector fitting includes the following steps: 1) processing a given dataset to construct a single-objective fitting dataset; 2) determining initial poles and using the initial poles and frequency sampling points to establish a fundamental matrix; 3) separating the real and imaginary parts of the fundamental matrix to construct a real fundamental matrix; 4) performing a simplified QR decomposition on the real fundamental matrix to obtain the first M columns of the complete Q matrix, denoted as Q. s Matrix, and for Q s The matrix is ​​stored; 5) Construct and solve the pole coefficient equation to obtain the pole coefficients; 6) Construct the pole characteristic equation using the pole coefficients and the initial poles to calculate the new poles; 7) Calculate the residues of each matrix element corresponding to the new poles to complete the vector fitting. This invention provides a method for multi-objective vector fitting that significantly reduces the computational complexity and improves computational efficiency.

Various aspects of the present disclosure generally relate to wireless communication. In some aspects, a user equipment (UE) may transmit an indication of a dynamic precoding resource block group (PRG) size UE capability. The UE may receive, in accordance with the dynamic PRG size UE capability, a dynamic PRG size configuration. The UE may receive, in accordance with the dynamic PRG size configuration, an indication of a dynamic PRG size of a PRG that contains a plurality of resource elements (REs) associated with a common Q matrix derived from a QR decomposition. The UE may receive data using the dynamic PRG size. Numerous other aspects are described.