Reverse Non-Uniform DFT for Spectrometer Signal Interpolation
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Solution Overview
Problem
Time-based spectrometers face challenges in achieving uniform sampling due to scanning perturbations, leading to errors in interpolation and increased computational time, especially when using techniques like Brault's method which require significant oversampling and memory for pre-computed interpolation functions.
Innovation Solution
The implementation of a fast reverse non-uniform discrete Fourier Transform (NFFT) algorithm in conjunction with time sampling methods, allowing for proper Dirichlet kernel interpolation on an irregular grid, which approximates the non-uniform discrete Fourier transform using an FFT algorithm, thereby addressing interpolation function and scanning perturbation issues.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional interpolation methods (e.g., Brault's method) are used to correct scanning perturbations, then measurement precision is improved, but device complexity and computational time increase significantly due to required oversampling and pre-computed interpolation functions
Solution Approach 1:
The patent replaces complex mechanical interpolation function pre-computation and storage systems with a mathematical transformation approach using the non-uniform discrete Fourier transform. This substitutes the mechanical/system-level complexity of pre-computing and storing interpolation functions with a more elegant mathematical operation that processes data in real-time without requiring extensive memory resources for pre-computed tables.
Solution Approach 2:
The patent changes the parameter domain from time-domain interpolation to frequency-domain transformation. By applying the non-uniform discrete Fourier transform, the method converts the problem of time-domain interpolation into a frequency-domain operation, which simplifies the computational requirements and eliminates the need for complex pre-computed interpolation functions while maintaining measurement precision.
2Measurement precision
If traditional interpolation methods with significant oversampling are used, then measurement precision is improved, but loss of time increases due to extended computational processing time
Solution Approach 1:
The patent substitutes time-consuming time-domain interpolation calculations with a frequency-domain Fourier transform approach. This substitution dramatically reduces computational processing time while maintaining interpolation accuracy, as the Fourier transform can be computed efficiently using fast Fourier transform algorithms without requiring extensive oversampling or pre-computed tables.
Solution Approach 2:
The patent transitions the problem from the time domain to the frequency domain by applying the discrete Fourier transform. This dimensional change allows the interpolation to be performed through spectral operations rather than temporal interpolation, which is computationally much faster and does not require significant oversampling, thereby reducing processing time while preserving accuracy.
3Measurement precision
If time-based spectrometers use reference channels for uniform sampling, then measurement precision is improved, but device complexity increases due to additional calibration systems
Solution Approach 1:
The patent extracts the essential function of the reference channel (providing timing information for synchronization) while eliminating the need for complex calibration systems. By using the non-uniform discrete Fourier transform, the method achieves uniform sampling correction without requiring additional reference channels or complex calibration mechanisms, thereby reducing overall device complexity while maintaining sampling precision.
Solution Approach 2:
The patent replaces the mechanical/calibration-based reference channel system with a mathematical correction approach. Instead of using physical reference channels and complex calibration hardware to achieve uniform sampling, the method uses the non-uniform discrete Fourier transform to correct sampling irregularities mathematically, significantly reducing device complexity while preserving measurement precision.
Data Source
AI summary
A spectrometric system has a primary channel with a signal waveform and a reference channel with a signal waveform. A digital representation of the primary signal waveform and a digital representation of the reference signal waveform to provide a digital output representing the primary signal at datum points synchronized with the reference signal are processed by computing the Fourier transform of the primary signal waveform and using a fast reverse non-uniform discrete Fourier Transform technique to compute the reverse non-uniform discrete Fourier transform of the Fourier transform of the primary signal waveform to provide the digital output representing the primary signal at datum points synchronized with the reference signal.