Sampling in the light of Wigner distribution

Area of Science:

• Signal processing
• Applied mathematics
• Optical engineering

Background:

• Traditional Fourier domain analysis has limitations in understanding signal sampling.
• Perfect reconstruction of non-bandlimited signals remains a challenge.
• The Nyquist sampling criterion is fundamental but has limitations.

Purpose of the Study:

• To propose a new method for analyzing signal sampling and reconstruction using the Wigner domain.
• To demonstrate the Wigner domain's advantages over the Fourier domain for signal analysis.
• To generalize the Nyquist sampling criterion and develop adaptive preprocessing techniques.

Main Methods:

• Analysis of sampling and reconstruction conditions in the Wigner domain.
• Derivation of a generalized Nyquist sampling criterion based on Wigner domain observations.
• Demonstration of preprocessing operations for signal adaptation.

Main Results:

• The Wigner domain offers superior insight into signal sampling compared to the Fourier domain.
• A method is presented for perfect sampling and reconstruction of non-bandlimited complex functions.
• A generalized Nyquist criterion is derived and validated.
• Simple optical preprocessing methods are shown to adapt signals to the new criterion.

Conclusions:

• The Wigner domain provides a powerful framework for understanding and improving signal sampling and reconstruction.
• The generalized Nyquist criterion and adaptive preprocessing offer practical solutions for signal adaptation.
• The proposed methods, particularly optical implementations, have significant potential for real-world applications.