Dealiased Real-Data FFT Convolution Through Implicit Zero-Padding
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Solution Overview
Problem
Conventional zero-padding techniques for FFT-based convolutions incur high memory and computational costs, especially in multidimensional scenarios, while traditional dealiasing methods lead to aliasing errors.
Innovation Solution
Implement implicit zero-padding by embedding real-valued input data into a complex array, separating even- and odd-indexed terms, and performing FFT operations on these subsets to reduce memory requirements and computational complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If conventional zero-padding techniques are used for FFT-based convolutions, then convolution accuracy is maintained, but memory requirements and computational costs increase significantly
Solution Approach 1:
The patent segments the convolution process into distinct phases: (1) embedding real-valued input data into a complex array, (2) performing FFT operations on the complex array, (3) executing pointwise operations on intermediate results, and (4) extracting convolution results via inverse FFT. This segmentation allows efficient memory management by processing data in structured stages rather than requiring full zero-padded arrays throughout the entire computation.
Solution Approach 2:
The patent transitions from real-valued data to complex-valued data, effectively adding an imaginary dimension to the computation. By embedding real input data into complex arrays and utilizing the properties of complex FFT, the method achieves accurate convolution results without requiring explicit zero-padding, thereby reducing memory requirements while maintaining computational precision.
2Reliability
If conventional zero-padding techniques are used for FFT-based convolutions, then aliasing errors are avoided, but computational complexity increases
Solution Approach 1:
The patent changes the parameter domain from real-valued to complex-valued data representation. By performing FFT operations on complex arrays embedded with real input data and applying appropriate pointwise operations, the method inherently prevents aliasing errors while avoiding the computational overhead of explicit zero-padding. The complex arithmetic naturally handles the spectral leakage and aliasing issues that plague real-valued FFT convolutions.
3Reliability
If traditional dealiasing methods are used, then aliasing errors are prevented, but memory buffer storage needs increase
Solution Approach 1:
The patent creates a unified complex array structure that serves multiple functions simultaneously: it stores the real-valued input data, enables FFT transformation, prevents aliasing through complex arithmetic properties, and facilitates efficient memory management. This multi-functional approach eliminates the need for separate buffer allocations required by traditional dealiasing methods, thereby reducing overall buffer storage requirements while maintaining aliasing error prevention.
Data Source
AI summary
Techniques, systems and methods are provided for performing convolution operations on real-valued input data using implicit zero-padding for dealiasing. A set of real-valued input data intended for convolution is embedded into a complex array, reorganizing the real-valued input into a format amenable to fast Fourier transform (FFT) processing. FFT operations are performed on the complex array to generate intermediate frequency-domain results. Pointwise operations on these intermediate results are then performed to integrate the intermediate results in a format that unifies the earlier-separated real-valued input data. The convolution output results matrix is extracted via inverse FFT operations performed on the results of these pointwise operations.