Fast Convolution Using One-Sided Exponential Functions
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Solution Overview
Problem
Existing methods for calculating convolution of signals with symmetric functions, such as Gaussian or exponential functions, are inefficient and unsuitable for real-time processing due to the need for forward and reverse calculations, which are not feasible for infinite signals like speech or motion pictures.
Innovation Solution
A method for fast convolution using one-sided exponential functions, approximating symmetric functions, allowing for recursive calculations that reduce computational complexity and enable processing of both definite and indefinite signals by mirroring and truncating functions to manage infinite signal lengths.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If direct numerical computation is used for convolution with symmetric functions, then calculation accuracy is maintained, but computational time becomes excessive for real-time processing
Solution Approach 1:
The patent transforms the convolution calculation from direct numerical computation to a recursive algorithm that changes the computational parameters. By using the exponential function's mathematical properties (specifically that e^(x+y) = e^x * e^y), the computation shifts from O(m×n) complexity to O(n) complexity, dramatically reducing computational time while maintaining accuracy through controlled approximation
2Measurement precision
If forward and reverse calculations are performed to approximate symmetric functions, then convolution accuracy is improved, but the entire signal must be known in advance making real-time processing impossible
Solution Approach 1:
The patent applies asymmetry by using one-sided exponential functions instead of symmetric functions. This allows the convolution to be computed in a single forward direction using recursive relations, eliminating the need for reverse calculations and enabling real-time processing of indefinite signals while maintaining acceptable approximation accuracy
Solution Approach 2:
The patent segments the symmetric function approximation into two separate one-sided exponential function calculations (forward and backward), then combines them. This segmentation allows each part to be computed independently using simple recursive relations, improving both accuracy and real-time capability
3Measurement precision
If the convolution kernel size is increased to improve approximation accuracy, then measurement precision improves, but device complexity and computational load increase
Solution Approach 1:
The patent changes the fundamental parameter of the calculation method from direct convolution with large kernels to recursive calculation with small effective kernels. The recursive approach maintains high accuracy equivalent to large kernel convolutions but achieves it through parameter transformation rather than increasing computational resources
Data Source
AI summary
A method and an apparatus embodying the method for fast convolution of a signal with a one-sided exponential function is disclosed. Additionally, a method and a system embodying the method for fast convolution of a signal with complex exponential function localized around an origin by an exponential function envelope utilizing the method and a system embodying the method for fast convolution of a signal with a one-sided exponential function is disclosed.