1. Presented by Fang Gong 1 University of California, Los Angeles, Los Angeles, USA 2 Nanyang Technological University, Singapore Fang Gong 1, Hao Yu 2 and Lei He 1 Fast, Non-Monte-Carlo Transient Noise Analysis for High-Precision Analog/RF Circuits by Stochastic Orthogonal Polynomials

2. Motivation Device noise can not be neglected for high-precision analog circuit anymore!  Signal-to-noise ratio (SNR) is reduced;  Has large impact on noise-sensitive circuits: PLLs (phase noise and jitter), ADCs (BER) … Device Noise Sources:  Thermal Noise: random thermal motion of the charge carriers in a conductor;  Flicker Noise (1/f Noise): random trapping and de-trapping of charge carriers in the traps located at the gate oxide interface.

3. Existing Work Monte Carlo method  Model the thermal noises as stochastic current sources attached to noise-free device components.  Sample the stochastic current sources to generate many traces. Non-Monte-Carlo method: [A. Demir, 1994]  Decouple the noisy system into a stochastic differential equation (SDE) and an algebraic constraint.  Use perturbation analysis and covariance matrix to solve for variance of transient noise in time domain. Examples of Commercial tools:  Transient noise analysis in HSPICE (Synopsys)  AFS transient noise analysis (Berkeley Design Automation), …

4. SDAE based Noise Analysis- primer slide stationary process with constant power spectral density (PSD) Stochastic differential algebra equation (SDAE) noise intensities Standard noise sources (White noise) Stochastic component deterministic component Integrate it to build Itô-Integral based SDAE Wiener process Modeling of Thermal Noise

5. Existing Solution to Itô-Integral based SDAE Stochastic Integral scheme for SDAE (e.g. backward differentiation formula (BDF) with fixed time-step) With piecewise linearization along nominal transient trajectory: Transient noise Nominal response  Sampled with Monte Carlo at each time step

6. New SOP based Solution 3σ boundary in time domain nominal response Stochastic Orthogonal Polynomials without Monte Carlo SoP expansions Expand random variables with SoP

7. Experimental Results Experiment Settings  Consider both thermal and flicker noise for all MOSFETs.  Resistors only have thermal noise. Accuracy and efficiency validity  SoP expansion method can achieve up to 488X speedup with 0.5% error in time domain, when compared with MC. CMOS comparator InverterOPAMComparatorOscillator MCTime(s)91.954266.642226.71146851.2 SoP Method Error0.43%0.93%1.78%1.62% Time(s)1.8752.3512.72300.91 Speedup49X81X175X488X Runtime Comparison on Different Circuits

8. Conclusion A fast non-Monte-Carlo transient noise analysis using Itô- Integral based SDAE and stochastic orthogonal polynomials (SoPs) The first solution of SDAE by SoPs  Expand all random variables with SoPs  Apply inner-product with SoPs to expansions (orthogonal property)  Obtain the SoP expansion of transient noise at each time-step To learn more come to poster session! To learn more come to poster session!