Sign-Symmetric Algorithm for Dyadic Fraction Multiplication
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Solution Overview
Problem
Arithmetic right shifts in signed integer multiplication or division introduce rounding errors due to their non-linear nature, complicating signal processing algorithms like DCT and IDCT, which rely on approximating irrational constants with dyadic rational fractions to minimize complex operations.
Innovation Solution
A sign-symmetric algorithm is developed to approximate the product of an integer and an irrational value using a series of addition, subtraction, and right shift operations, minimizing metrics such as mean asymmetry, mean error, and magnitude of error, thereby simplifying processing and reducing rounding errors.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If arithmetic right shifts are used to perform division by powers of two in fixed-point signal processing, then computational complexity is reduced, but rounding errors are introduced due to non-linear operation
Solution Approach 1:
The patent applies asymmetry by using different shift amounts for positive and negative numbers in the sign-symmetric algorithm. Specifically, positive numbers are shifted right by s1 bits while negative numbers are shifted right by s2 bits, where s1 and s2 are different. This asymmetric treatment of signs allows the algorithm to minimize rounding errors while maintaining computational efficiency, as the different shift amounts compensate for the non-linear rounding behavior of arithmetic right shifts on signed integers.
Solution Approach 2:
The patent changes the parameter of shift amount based on the sign of the input value. By dynamically adjusting the shift parameter (s1 for positive, s2 for negative), the algorithm optimizes the approximation of irrational constants while using only simple shift and add operations. This parameter change strategy allows the system to maintain low computational complexity while reducing the harmful rounding errors inherent in uniform arithmetic shifts.
2Device complexity
If dyadic rational fractions are used to approximate irrational constants in DCT/IDCT algorithms, then multiplication operations are simplified to additions and shifts, but approximation errors increase
Solution Approach 1:
The patent segments the approximation process by treating positive and negative constants separately with different dyadic rational approximations. Instead of using a single approximation for all constants, the algorithm divides the computation into segments based on sign, allowing each segment to be optimized independently. This segmentation enables the use of sign-symmetric algorithms that minimize overall approximation error while maintaining the simplicity of addition and shift operations.
Solution Approach 2:
The patent changes the dyadic rational approximation parameters based on the sign of the constant being approximated. By selecting different numerator and denominator values for positive and negative constants, the algorithm optimizes the approximation accuracy for each case. This parameter optimization, combined with sign-symmetric algorithm design, reduces the cumulative approximation error in DCT/IDCT computations while maintaining operational simplicity.
3Measurement precision
If sign-symmetric algorithms are used to minimize rounding errors, then error balance is improved, but algorithm complexity increases
Solution Approach 1:
The patent extracts the sign information from the input values and uses it to select different processing paths. By separating the sign handling from the magnitude processing, the algorithm achieves sign-symmetry without significantly increasing complexity. The extraction of sign-based control logic allows the main computational path to remain simple while still achieving balanced error compensation through different shift amounts for positive and negative numbers.
Data Source
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AI summary
A product of an integer value and an irrational value may be determined by a sign-symmetric algorithm. A process may determine possible algorithms that minimize metrics such as mean asymmetry, mean error, variance of error, and magnitude of error. Given an integer variable x and rational dyadic constants that approximate the irrational fraction, a series of intermediate values may be produced that are sign-symmetric. The intermediate values may include a sequence of addition, subtraction and right shift operations the when summed together approximate the product of the integer and irrational value. Other operations, such as additions or subtractions of Os or shifts by 0 bits may be removed.