RRT: the regularized resolvent transform for high-resolution spectral estimation

Area of Science:

• Numerical analysis
• Signal processing
• Spectroscopy

Background:

• Accurate spectral estimation is crucial for analyzing complex time-domain signals.
• Existing methods may face limitations in resolution and computational efficiency for multidimensional data.

Purpose of the Study:

• Introduce a new numerical expression, the regularized resolvent transform (RRT).
• Demonstrate RRT's capability for high-resolution spectral estimation of multidimensional time signals.
• Highlight RRT's efficiency and stability in computational implementation.

Main Methods:

• Developed the regularized resolvent transform (RRT) as a direct transformation from time to frequency domain.
• Utilized Tikhonov regularization for handling singular matrices, with the regularization parameter q.
• Constructed a small data matrix R(s) from the time signal to compute the spectrum at each frequency s.

Main Results:

• RRT provides a direct transformation of truncated time-domain data into a frequency-domain spectrum.
• Under specific conditions, RRT is equivalent to the infinite time discrete Fourier transformation.
• Numerical implementation is computationally inexpensive, even for large datasets, due to small matrices involved.
• Demonstrated RRT's effectiveness on 1D model and 2D experimental NMR signals.

Conclusions:

• RRT is a powerful tool for high-resolution spectral estimation.
• The method is computationally efficient and numerically stable.
• RRT offers a viable alternative for analyzing complex multidimensional time signals, particularly in fields like NMR spectroscopy.