Windowed Sliding Transformer Reduces DFT Complexity
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Solution Overview
Problem
Conventional methods require computing a full DFT for each portion of an input signal after applying a window function, leading to computational inefficiency in sliding transform applications.
Innovation Solution
The implementation of a windowed sliding transformer that computes a sliding DFT and applies a kernel to the transformed representation, allowing for efficient computation of windowed transforms by reusing previous frequency-domain representations and reducing the complexity of transform operations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If a full DFT is computed for each portion of the input signal after applying a window function, then the transform accuracy is maintained, but the computational complexity increases to O(N^2)
Solution Approach 1:
The patent pre-computes and stores the DFT of the window function (kernel) in frequency domain before the sliding transform operation. This preliminary action allows the windowed sliding DFT to be computed efficiently by combining the pre-stored kernel with the sliding DFT results, avoiding the need to re-compute the full windowed DFT for each position and thus reducing computational complexity from O(N^2) to O(N) while maintaining transform accuracy
Solution Approach 2:
The patent merges the window function application with the sliding DFT operation by using the convolution property in frequency domain. Instead of applying the window function in time domain and then computing a full DFT for each position, the method combines the pre-computed window kernel with the sliding DFT results through efficient frequency domain multiplication, achieving both windowing and transformation in a unified computationally efficient process
2Loss of information
If a full DFT is computed for each starting sample, then the complete frequency information is obtained, but the processing time increases significantly
Solution Approach 1:
The patent pre-computes the DFT of the window function and stores it as a kernel in frequency domain. This preliminary computation is performed only once, and the pre-stored kernel is then reused for all sliding window positions, significantly reducing the total processing time while ensuring that complete frequency information is obtained for each position through efficient combination with the sliding DFT results
Solution Approach 2:
The patent implements a continuous sliding DFT approach where the transform for each position is computed based on the previous position's results, maintaining continuous processing without interruption. The sliding DFT updates incrementally as the window moves through the signal, ensuring that frequency information is continuously obtained for all positions without the need to re-compute from scratch, thus reducing processing time while maintaining information completeness
Data Source
AI summary
Methods and apparatus to perform windowed sliding transforms are disclosed. An example apparatus includes a transformer to transform a first block of time-domain samples of an input signal into a first frequency-domain representation based on a second frequency-domain representation of a second block of time-domain samples of the input signal, and a windower to apply a third frequency-domain representation of a time-domain window function to the first frequency-domain representation.