Modified Givens Rotation for Complex Matrix Nulling
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Solution Overview
Problem
Current systolic arrays require complex-to-real data reduction for processing complex matrices, adding overhead and complexity due to the need for phase factoring and conversion before performing Givens rotations.
Innovation Solution
A modified Givens rotation method that directly processes complex matrices by using a modified Givens rotation matrix with complex numbers, eliminating the need for complex-to-real data reduction by employing complex conjugates and avoiding phase factoring.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If complex-to-real data reduction is performed for processing complex matrices, then Givens rotation can be applied, but hardware and software complexity increases due to phase factoring and conversion overhead
Solution Approach 1:
The patent extracts and eliminates the unnecessary phase factoring step from the conventional complex matrix processing pipeline. By directly applying Givens rotation to complex matrices without converting to real numbers, the method removes the overhead of phase tracking and complex-to-real conversion, thereby reducing hardware and software complexity while maintaining processing capability
Solution Approach 2:
The patent changes the parameter representation by allowing complex numbers to be processed directly in their native form rather than converting them to real number pairs. This parameter change eliminates the need for phase angle calculations and conversions, simplifying the overall processing complexity while preserving the mathematical integrity of complex matrix operations
2Reliability
If complex-to-real data reduction is performed, then conventional Givens rotation can be used, but processing time increases due to additional conversion steps
Solution Approach 1:
The patent performs preliminary action by preparing the complex matrix data in a format that is directly compatible with the modified Givens rotation algorithm. By organizing the complex numbers and their conjugates in advance and establishing the direct rotation framework before processing begins, the method eliminates the need for time-consuming phase factoring and conversion steps during the actual rotation operations
Solution Approach 2:
The patent skips the intermediate conversion steps that conventional methods require. By rushing through the processing pipeline with a direct complex-to-complex rotation approach, the method eliminates the time spent on phase extraction, real-number conversion, and subsequent reconstruction, thereby significantly reducing total processing time while maintaining accuracy
3Ease of operation
If phase factoring is applied for complex matrix processing, then Givens rotation can be performed, but hardware resources increase due to additional phase tracking components
Solution Approach 1:
The patent extracts and removes the phase tracking components from the hardware architecture. By eliminating the need to calculate, store, and manipulate phase angles separately, the method reduces the quantity of hardware resources required, removing unnecessary adders, multipliers, and storage elements that would be needed for phase factoring and conversion operations
Solution Approach 2:
The patent makes the processing unit universal by designing it to handle complex numbers directly without requiring specialized phase tracking subsystems. The same hardware resources that perform the core Givens rotation operations also naturally handle the complex arithmetic, eliminating the need for dedicated phase processing components and reducing overall hardware resource requirements
Data Source
AI summary
Nulling a cell of a complex matrix is described. A complex matrix and a modified Givens rotation matrix are obtained for multiplication by a processing unit, such as a systolic array or a CPU, for example, for the nulling of the cell to provide a modified form of the complex matrix. The modified Givens rotation matrix includes complex numbers c*, c, −s, and s*, wherein the complex number s* is the complex conjugate of the complex number s, and wherein the complex number c* is the complex conjugate of the complex number c. The complex numbers c and s are associated with complex numbers of the complex matrix including the cell to be nulled. The modified form is then output by the processing unit. The modified Givens rotation matrix may be implemented as a systolic array or otherwise used for processing complex numbers or matrices.