Multichannel Sampling of Pulse Streams at the Rate of Innovation (1004.5070v5)

Abstract: We consider minimal-rate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of infinite pulse streams was treated in previous works, either the rate of innovation was not achieved, or the pulse shape was limited to Diracs. In this paper we propose a multichannel architecture for sampling pulse streams with arbitrary shape, operating at the rate of innovation. Our approach is based on modulating the input signal with a set of properly chosen waveforms, followed by a bank of integrators. This architecture is motivated by recent work on sub-Nyquist sampling of multiband signals. We show that the pulse stream can be recovered from the proposed minimal-rate samples using standard tools taken from spectral estimation in a stable way even at high rates of innovation. In addition, we address practical implementation issues, such as reduction of hardware complexity and immunity to failure in the sampling channels. The resulting scheme is flexible and exhibits better noise robustness than previous approaches.

• The paper proposes a multichannel sampling architecture that samples streams of pulses with arbitrary shapes at their rate of innovation, generalizing prior methods limited to Dirac pulses.
• The architecture allows flexible waveform design for practical implementation and demonstrates improved resilience to noise compared to existing techniques.
• This research has implications for signal processing theory and practical applications in fields like bio-imaging, ultra-wideband communications, and neuronal activity monitoring.

Multichannel Sampling of Pulse Streams at the Rate of Innovation

The paper "Multichannel Sampling of Pulse Streams at the Rate of Innovation" by Kfir Gedalyahu, Ronen Tur, and Yonina C. Eldar addresses the challenge of sampling continuous pulse streams at their minimal required sampling rate, known as the rate of innovation. This rate corresponds to the number of degrees of freedom per unit time in a given signal. The authors propose a novel multichannel sampling architecture designed to achieve this rate for infinite streams of pulses having arbitrary shapes, thereby generalizing beyond previous works that were limited to sampling Dirac pulses or did not attain the rate of innovation.

Key Contributions

1. Multichannel Architecture: The proposed multichannel sampling system modulates the input signal using a set of carefully crafted waveforms, followed by integration in each channel. This generalization permits sampling of pulse streams with non-Dirac shapes and achieves a stable reconstruction process, even at high rates of innovation.
2. Waveform Design and Flexibility: By adjusting waveform parameters, the system offers flexible implementation options. Periodic waveforms, such as rectangular pulse sequences or cosine and sine patterns, can serve as modulation signals, facilitating practical circuit design. This flexibility extends to scenarios where single or multiple sampling channels may experience failure, with the architecture supporting continued signal recovery.
3. Theoretical and Practical Considerations: The paper provides theoretical guarantees for perfect reconstruction from the sampled data under conditions that assure full column rank of the mixing matrix. Practical concerns, including hardware complexity and noise robustness, are addressed. The proposed system demonstrates improved resilience to noise compared to prior methods employing integrators or exponential filters.
4. Comparison with Existing Methods: The system's performance surpasses existing single-channel and specific multichannel techniques, particularly in the presence of noise. Moreover, the architecture supports signal models where pulse streams exhibit shift-invariant properties, enabling potential reductions in sampling requirements.

Implications and Future Directions

The implications of this research are significant both theoretically and practically. Theoretically, it delineates the viability of achieving the rate of innovation for a broad class of signals beyond Diracs, potentially influencing how sampling paradigms are approached in signal processing. Practically, the method's modular design eases integration into existing signal processing frameworks, providing a versatile tool for applications in bio-imaging, ultra-wideband communications, and neuronal activity monitoring.

Future developments may explore hybrid systems optimized for specific application scenarios, leveraging advances in hardware miniaturization and digital signal processing algorithms to further enhance system robustness and efficiency. Investigation into extending the framework for signals with more complex structures, such as those exhibiting temporal or spectral nonstationarities, could also expand its applicability.

In conclusion, the proposed multichannel sampling architecture offers a cogent solution for pulse stream sampling at the rate of innovation, offering a promising direction for both theoretical exploration and practical application in various domains of digital signal processing.