Prediction of Stochastic Eye Diagrams via IC Equivalents and Lagrange Polynomials Paolo Manfredi, Igor S. Stievano, Flavio G. Canavero Department of Electronics and Telecommunications (DET) Politecnico di Torino, Italy paolo.manfredi@polito.it
Motivation The increasing miniaturization is stressing the impact of process variations on system performance K.J. Kuhn et al., "Process technology variation," IEEE Trans. Electron Devices, vol. 58, no. 8, Aug. 2011 Circuit waveforms no longer have a deterministic description 2
Impact of process variations on data links Process variations + uncontrolled parameters affect the properties of both ICs and interconnections (connectors + lines) Different IC samples have different driving strengths Etching process impact on line characteristic impedance (Z0) Substrate quality affects propagation delay (TD) etc. A statistical assessment of system performance is necessary 3
Stochastic eye pattern Interconnect performance usually assessed via eye pattern Due to variability, different fabricated devices will have different eye properties Width, height? Eye properties more suitably described as random variables 4
Monte Carlo simulation The traditional approach uses bruteforce sampling-based methods, such as Monte Carlo analysis 1. Identify random parameters and their distribution 2. Generate a set of random samples according to their distribution 3. Perform a deterministic simulation for the scenario associated to 5 each sample 4. Aggregate and analyze results to obtain statistical information Inefficient for eye patterns because of 1. Large number of runs 2. Length of bit streams 3. Complexity of models 5
Monte Carlo analysis e.g., single run with transistorlevel simulation requires mins 1000-bit stream with Gaussian jitter 10 000 simulations days IC equivalents [1] make the Monte Carlo analysis possible 1% accuracy i( t) w ( t) i ( v( t), t) w ( t) i ( v( t), t) H H L L 10 000 simulations 16 h Further improvements are still desirable [1] I.S. Stievano, I.A. Maio, F.G. Canavero, C. Siviero, "Reliable eye-diagram analysis of data links via device macromodels," IEEE Transactions on Advanced Packaging, vol. 29, no. 1, pp. 31 38, Feb. 2006 6
Lagrange interpolation Clever solution via the combination of IC models + Lagrange interpolation random variable ξ 01011 e.g., eye opening y t, ξ φ k : Lagrange polymomial y k = y(t, ξ k ) y y 0 φ 0 + y 1 φ 1 + 7
Choice of the interpolation points Several rules can be used for the generation of an optimal (low) number of interpolation points E.g., zeros of classical polynomals orthogonal to the distribution function (see Gaussian quadrature rules ) Uniform variability Legendre polynomials Gaussian variability Hermite polynomials Multidimensional interpolation: tensor product rule For large grid sizes, sparse grids 8
Application example Point-to-point interconnect: driver + interconnections + receiver Driver: IC equivalent i( t) w H ( t) i w ( t) i L H L ( v( t), t) ( v( t), t) LC model Microstrip traces Receiver: RC circuit + clamp diodes 9
Process variability Process variations affect for instance Driver strength out Trace width (etching process) time Substrate relative permittivity (material quality) Modeled as independent Gaussian random variables 10
Eye pattern Prediction of statistical fluctuation of eye pattern parameters:?? How does the opening vary among different devices? 11
Eye vertical opening (height) Probability distribution of eye height Monte Carlo Lagrange interpolation 99% probability opening: h: p H > h = 99% method h % error Monte Carlo 2.0666 V - Lagrange interp. 2.0652 V 0.07% 12
Eye horizontal opening (width) Probability distribution of eye width Monte Carlo Lagrange interpolation 99% probability opening: w: p W > w = 99% method w % error Monte Carlo 2.4337 ps - Lagrange interp. 2.4321 ps 0.07% 13
Eye opening profile (mask) 99%-confidence level opening mask Monte Carlo Lagrange interpolatation There is only a 1% chance the response will intersect the red area!! 14
Efficiency assessment Monte Carlo convergence rate is 1 N N ~ 10k For a clever choice of the interpolation points, Lagrange interpolation requires a small subset of responses, i.e. Proposed example: N LAGRANGE N MONTE CARLO Method Samples Simulation Speed-up Monte Carlo 10 000 16 hours - Lagrange interp. 64 5 min 47 s 160x 15
Conclusions Effective solution for the statistical assessment of eye patterns in high-speed links affected by process-induced variability Fast analysis via combination of IC equivalents and stochastic Lagrange interpolation Inclusion of multiple random variables Quantitative prediction (e.g., PDF) of the influence of process variability on the circuit performance Improved efficiency, good accuracy Thank you for your attention!! 16
Q&A 17
Choice of the interpolation points interpolation H, W Postprocessing Postprocessing H, W interpolation 18