N-Dimensional FFT Computation Without Transpose Operations
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Current methods for computing n-dimensional Fast Fourier Transform (FFT) require sequential one-dimensional FFT computations, leading to bottlenecks in data flow and increased overhead due to transpose operations, especially for large data sets.
Innovation Solution
The method extends the Cooley-Tukey 1-D FFT algorithm to perform n-dimensional FFT directly from input data, eliminating the need for one-dimensional FFT computations and transpose operations, and utilizes n-D butterflies, n-D quad-flies, or n-D hybrid flies for maximum parallelism and in-place computations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If sequential one-dimensional FFT computations are used for n-dimensional FFT, then the computation can be performed using well-established algorithms, but data flow bottlenecks and overhead from transpose operations occur
Solution Approach 1:
The patent segments the n-dimensional FFT computation into independent n-D butterfly operation blocks that can be executed in parallel. Each butterfly block processes a specific subset of data points independently, eliminating the sequential dependency inherent in 1-D FFT approaches and removing the need for transpose operations between dimensions.
Solution Approach 2:
The patent transitions from computing FFT sequentially along one dimension at a time to performing parallel computations across multiple dimensions simultaneously using n-D butterfly operations. This dimensional approach allows all n dimensions to be processed in parallel without requiring data transposition, fundamentally changing the computational paradigm from sequential to parallel hypercube-based processing.
2Adaptability or versatility
If transpose operations are performed between successive one-dimensional FFT computations, then n-dimensional FFT can be computed using 1-D FFT algorithms, but computation overhead increases exponentially with dimensionality and data size
Solution Approach 1:
The patent extracts and eliminates the transpose operation step from the traditional 1-D FFT-based n-D FFT computation pipeline. By formulating a direct n-D FFT algorithm using n-D butterflies, the method removes the intermediate data reordering step that causes exponential overhead, keeping only the essential computational operations.
Solution Approach 2:
The patent performs preliminary reorganization of data into hypercube structures before computation, allowing all n-D butterfly operations to proceed in parallel without subsequent transpose operations. The initial data arrangement enables direct parallel processing across all dimensions, preventing the exponential time loss that would otherwise occur with repeated transpositions.
3Quantity of substance
If in-place computation is used, then memory usage is minimized, but computation dependencies must be carefully managed to maintain correctness
Solution Approach 1:
The patent implements nested butterfly operations where smaller-dimensional butterflies are nested within larger-dimensional structures. This nested organization allows in-place computation by systematically reusing the same memory locations at different computational stages, with each nested level operating on progressively larger data subsets while maintaining proper dependency ordering.
Data Source
AI summary
A method and apparatus for calculating multidimensional Fast Fourier Transforms (FFTs) efficiently without transpose data flow and with in-place computations. If higher throughput computations are desired, computations are done in pipelined stages with parallel computing devices. A wide range of trade-offs can be made between the computation speed and the hardware complexity. This is based on an extension of Cooley-Tuckey algorithm to n-dimensional data. A mathematical derivation of the algorithm has been provided. This invention makes it possible to perform n-dimensional FFTs, n>1 without relying on one-dimensional FFT computations as in the prior art.