This paper focuses on the analysis and modeling of jitter and phase noise in ring oscillators, which are widely used in digital and communication systems. The authors develop a parallel treatment of frequency-domain phase noise and time-domain clock jitter for ring oscillators, using a phase-noise model based on impulse sensitivity functions (ISF). The paper covers the following key aspects: 1. **Introduction**: Ring oscillators are essential in various applications, including clock recovery circuits, disk-drive read channels, and integrated frequency synthesizers. Recent work has focused on modeling jitter and phase noise in these oscillators. 2. **Phase Noise**: The output of a practical oscillator is described by a periodic function with amplitude and phase fluctuations. The phase noise is analyzed using the ISF, which represents the sensitivity of the oscillator to perturbations. The paper derives expressions for phase noise and jitter, considering both single-ended and differential ring oscillators. 3. **Jitter**: Timing jitter is discussed, including its statistics and the impact of correlated and uncorrelated noise sources. The paper provides expressions for phase jitter and shows that it is influenced by the number of stages in the oscillator. 4. **Calculation of ISF**: An approximate expression for the rms value of the ISF is derived, which is useful for practical simulations. The paper also explores the relationship between the number of stages and the ISF, showing that the ISF decreases with an increasing number of stages. 5. **Expressions for Jitter and Phase Noise**: Detailed expressions for phase noise and jitter are derived for single-ended and differential CMOS ring oscillators, considering both long- and short-channel regimes. The paper highlights the differences in performance between single-ended and differential oscillators. 6. **Design Implications**: The paper discusses the design considerations for minimizing jitter and phase noise, including the impact of symmetry, substrate and supply noise, and the number of stages. 7. **Experimental Results**: Theoretical predictions are compared with experimental measurements, demonstrating good agreement. The paper includes detailed measurements and analysis of phase noise and jitter for different types of ring oscillators. 8. **Conclusion**: The paper concludes with a summary of the key findings, emphasizing the importance of the ISF model in understanding and optimizing the performance of ring oscillators. The paper provides a comprehensive analysis of jitter and phase noise in ring oscillators, offering valuable insights for both theoretical and practical applications.This paper focuses on the analysis and modeling of jitter and phase noise in ring oscillators, which are widely used in digital and communication systems. The authors develop a parallel treatment of frequency-domain phase noise and time-domain clock jitter for ring oscillators, using a phase-noise model based on impulse sensitivity functions (ISF). The paper covers the following key aspects: 1. **Introduction**: Ring oscillators are essential in various applications, including clock recovery circuits, disk-drive read channels, and integrated frequency synthesizers. Recent work has focused on modeling jitter and phase noise in these oscillators. 2. **Phase Noise**: The output of a practical oscillator is described by a periodic function with amplitude and phase fluctuations. The phase noise is analyzed using the ISF, which represents the sensitivity of the oscillator to perturbations. The paper derives expressions for phase noise and jitter, considering both single-ended and differential ring oscillators. 3. **Jitter**: Timing jitter is discussed, including its statistics and the impact of correlated and uncorrelated noise sources. The paper provides expressions for phase jitter and shows that it is influenced by the number of stages in the oscillator. 4. **Calculation of ISF**: An approximate expression for the rms value of the ISF is derived, which is useful for practical simulations. The paper also explores the relationship between the number of stages and the ISF, showing that the ISF decreases with an increasing number of stages. 5. **Expressions for Jitter and Phase Noise**: Detailed expressions for phase noise and jitter are derived for single-ended and differential CMOS ring oscillators, considering both long- and short-channel regimes. The paper highlights the differences in performance between single-ended and differential oscillators. 6. **Design Implications**: The paper discusses the design considerations for minimizing jitter and phase noise, including the impact of symmetry, substrate and supply noise, and the number of stages. 7. **Experimental Results**: Theoretical predictions are compared with experimental measurements, demonstrating good agreement. The paper includes detailed measurements and analysis of phase noise and jitter for different types of ring oscillators. 8. **Conclusion**: The paper concludes with a summary of the key findings, emphasizing the importance of the ISF model in understanding and optimizing the performance of ring oscillators. The paper provides a comprehensive analysis of jitter and phase noise in ring oscillators, offering valuable insights for both theoretical and practical applications.