Causal Impulse Response Calculation via Rational Extrapolation
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Solution Overview
Problem
Conventional methods for calculating causal impulse response from a band-limited spectrum often result in non-causality due to truncation, leading to significant discrepancies in the spectrum of the impulse response.
Innovation Solution
A computer-implemented method that removes the non-causal portion of the time-domain impulse response and modifies the spectrum by adding a causal signal to reduce the difference between the original and modified spectra, ensuring improved causality through rational extrapolation and spectrum compensation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional spectrum extrapolation methods (linear extrapolation, window function) are used to obtain causal impulse response from band-limited spectrum, then the impulse response becomes causal, but the spectrum of the impulse response shows significant discrepancy from the original spectrum
Solution Approach 1:
The patent applies rational function extrapolation instead of conventional linear extrapolation or window functions. This changes the mathematical parameters and functional form used to extend the band-limited spectrum, achieving both causality preservation and spectrum accuracy. The rational function approach modifies the extrapolation parameters to satisfy causality constraints while minimizing spectral discrepancy.
Solution Approach 2:
The patent employs an iterative optimization process where the impulse response is calculated, its spectrum is compared with the original band-limited spectrum, and the rational function parameters are adjusted based on this feedback. This closed-loop approach continues until the spectrum discrepancy is minimized, ensuring both causality and spectral fidelity.
2Ease of operation
If inverse fast Fourier transform is applied to truncated band-limited spectrum, then time-domain impulse response is obtained, but the impulse response becomes non-causal
Solution Approach 1:
The patent applies rational function extrapolation to the band-limited spectrum before performing the inverse fast Fourier transform. This preliminary action extends the spectrum in a causality-preserving manner, ensuring that the subsequent IFFT produces a causal impulse response. By preparing the spectrum with appropriate extrapolation beforehand, the causality requirement is satisfied from the outset.
Solution Approach 2:
The patent changes the mathematical approach from direct IFFT of truncated spectrum to IFFT of rationally extrapolated spectrum. This parameter and method change ensures that the frequency domain representation satisfies causality constraints before transformation to time domain, resulting in a causal impulse response.
Data Source
AI summary
A computer-implemented method for calculating a time-domain impulse response with improved causality based on a first spectrum in a frequency domain is disclosed. The first spectrum may be band-limited. The method may calculate a first time-domain impulse response from the first spectrum. The method may remove a non-causal portion of the first system time-domain impulse response to obtain a second time-domain impulse response, and calculate a second spectrum of the second time-domain impulse response. The method may further modify the second spectrum by adding a causal signal such that a difference between the second spectrum and the first spectrum is reduced. The method may also calculate the time-domain impulse response with improved causality from the modified second spectrum.