Gradient-Ascent Eigen Decomposition for Fast MIMO Beamforming
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Solution Overview
Problem
Existing eigen-decomposition methods for beamforming in massive MIMO systems require a large number of iterations and high computational complexity, especially when eigenvalues are clustered close together, which is exacerbated by increasing antenna counts and user numbers.
Innovation Solution
Employing gradient ascent with adaptive step size optimization for eigenvalue and eigenvector determination, utilizing matrix deflation to reduce computational overhead and achieve convergence in fewer iterations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If the power method is used for eigen-decomposition, then the per-iteration complexity is low, but the number of iterations required to converge is large, especially when eigenvalues are clustered close together
Solution Approach 1:
The patent changes the fundamental parameter of the iterative approach by switching from the power method to gradient ascent optimization. This parameter change allows the algorithm to directly optimize the objective function (Rayleigh quotient) rather than repeatedly applying matrix multiplication, thereby reducing the number of iterations needed for convergence while maintaining low per-iteration complexity suitable for hardware implementation.
Solution Approach 2:
The patent substitutes the mechanical iterative multiplication process of the power method with a gradient-based optimization mechanism. By using gradient ascent with analytically derived update rules, the algorithm replaces the brute-force iterative approach with a more efficient mathematical mechanism that converges faster while keeping each iteration computationally simple.
2Productivity
If Lanczos iteration or partial SVD is used for eigen-decomposition, then convergence can be achieved, but the per-iteration complexity becomes cubic or higher
Solution Approach 1:
The patent changes the complexity parameter by adopting gradient ascent with a carefully designed update rule that requires only O(N^2) operations per iteration. This is achieved by deriving closed-form expressions for the gradient and using efficient matrix-vector multiplications, avoiding the O(N^3) complexity of Lanczos or SVD methods while maintaining fast convergence.
Solution Approach 2:
The patent extracts and utilizes only the essential computational components needed for eigen-decomposition, avoiding the full overhead of Lanczos iteration or partial SVD. By focusing on the gradient of the Rayleigh quotient and using deflation techniques, the method extracts the necessary information with minimal computational burden, achieving cubic or better complexity.
3Reliability
If user-specific eigen-beamforming is implemented to increase beamforming gains, then the signal strength and spectral efficiency improve, but the computational burden increases significantly with more antennas and users
Solution Approach 1:
The patent changes the computational parameter by implementing an optimized gradient ascent algorithm with deflation that reduces the complexity of eigen-decomposition. This allows user-specific eigen-beamforming to be maintained with improved beamforming gains while significantly reducing the computational burden, making it feasible for systems with large numbers of antennas and users.
Solution Approach 2:
The patent applies segmentation by using matrix deflation to decompose the eigen-decomposition problem into a sequence of simpler subproblems. After finding the first eigenvector, the algorithm deflates the covariance matrix to remove the contribution of the first eigenvector, then finds the second eigenvector, and so on. This segmentation reduces the overall computational burden compared to solving the full eigen-decomposition problem at once.
Data Source
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AI summary
This specification relates to systems and methods for determining eigenvalues and eigenvectors of a covariance matrix, for example for use in eigen-beamforming in MIMO systems. According to a first aspect of this specification, there is describes a method comprising: receiving an input covariance matrix; and determining one or more eigenvalues and/or eigenvectors of the matrix iteratively using gradient ascent, wherein each iteration of the gradient ascent has a step size, µ, that maximises a Rayleigh quotient along the gradient.