Hermitian Matrix Decomposition Circuit Using Jacobi Rotations
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Solution Overview
Problem
Fast Eigenvalue decomposition is challenging in MIMO communication networks due to the need for high precision operations, which result in increased complexity, logic requirements, memory usage, and costs.
Innovation Solution
A decomposition circuit that performs Jacobi rotations and uses table lookups to efficiently reduce off-diagonal terms in channel response matrices, reducing logic, memory, and cost requirements while maintaining fast computation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If high precision floating point computations are used for Eigenvalue decomposition, then computation accuracy is improved, but device complexity and cost increase
Solution Approach 1:
The patent segments the Eigenvalue decomposition problem into multiple iterative steps, where each step processes a subset of matrix elements. The Jacobi rotation method divides the N×N matrix into 2×2 submatrices that are processed independently in successive iterations, reducing the computational burden on any single processing unit while maintaining overall accuracy.
Solution Approach 2:
The patent employs fixed-point arithmetic instead of floating-point operations, using simpler, less resource-intensive computational objects. Fixed-point numbers require less logic and memory resources compared to floating-point representations, effectively replacing complex computational objects with cheaper alternatives that still achieve sufficient precision for the application.
2Measurement precision
If conventional floating point operations are used, then computation precision is maintained, but processing speed decreases
Solution Approach 1:
The patent substitutes complex floating-point mechanical operations with simpler fixed-point arithmetic operations. By replacing the floating-point computation mechanism with fixed-point arithmetic, the system achieves faster processing speeds while maintaining adequate precision through iterative refinement of the decomposition process.
Solution Approach 2:
The patent uses iterative Jacobi rotations that periodically update matrix elements until convergence is achieved. This periodic action allows the system to progressively improve precision through multiple passes, with each iteration bringing the decomposition closer to the final accurate result while maintaining fast per-iteration processing.
3Reliability
If full precision Eigenvalue decomposition is implemented, then receiver performance is optimized, but implementation cost increases
Solution Approach 1:
The patent performs partial decomposition iterations until a convergence criterion is met, rather than completing the full mathematical decomposition. By performing just enough iterations to achieve the required precision threshold, the system obtains sufficient accuracy for receiver performance optimization without incurring the full cost of complete high-precision Eigenvalue decomposition.
Solution Approach 2:
The patent changes the precision parameter from floating-point to fixed-point representation, and dynamically adjusts the number of iteration steps based on convergence requirements. This parameter change allows the system to adapt the computation precision to the actual needs of the application, reducing resource requirements while maintaining adequate receiver performance.
Data Source
AI summary
Methods and apparatus for fast Eigenvalue decomposition of Hermitian matrices are disclosed. In an exemplary embodiment, a method is provided for performing a decomposition iteration that includes identifying a largest off-diagonal term of a channel response matrix X, generating a 2×2 Hermitian matrix X2 that includes the largest off-diagonal term, and generating a 2×2 Unitary matrix Ū2 from the 2×2 Hermitian matrix X2. The decomposition iteration also includes multiplying the 2×2 Unitary matrix Ū2 with the 2×2 Hermitian matrix X2 to generate an updated largest off-diagonal term and updating the channel response matrix X with the updated largest off-diagonal term. The method also includes performing one or more additional decomposition iterations until all off-diagonal terms of the channel response matrix X are less than a target value.