Synthesizing a Representative Critical Path for Post-Silicon Delay - - PowerPoint PPT Presentation

Synthesizing a Representative Critical Path for Post-Silicon Delay Prediction

Qunzeng Liu and Sachin S. Sapatnekar Department of Electrical and Computer Engineering University of Minnesota

Variations in Digital Circuits

• Variations in nanoscale

technologies

– Process: across-die/within- die – Environment: T, Vdd

• These lead to circuit

performance deviations

• Timing PDF
• Power PDF

2 Poly Diffusion

• S. Tyagi

Normalized Leakage Normalized Frequency

1 2 3 4 5 0.9 1.0 1.1 1.2 1.3 1.4

30% 5X

Intel

Design Phase: Pre-Silicon

3

Pre-Silicon Optimization Deterministic/Statistical Timing/Power Analysis … Synthesis Gate Sizing …

Statistical Timing Analysis result Statistical Power Analysis result

Pre-Silicon Analysis

Statistical Static Timing Analysis (SSTA)

• Spatial correlation
• Canonical form

4

( )

I p p a , ~ N R d

c T c c

+ + = μ

1 2 3

Design Phase: Post-Silicon

5

Post-Silicon Optimization

• Delay Analysis,
• Design-Silicon Correlation,

…

• Adaptive Body Bias(ABB),
• Adaptive Voltage Scaling

(AVS), …

Post-Silicon Analysis

[amd.com]

Adaptive Body Bias (ABB)

6

[Tschanz et al., JSSC02]

Limitations of Critical Path Replica (CPR)

7

• Representative Critical Path (RCP)
• Always predicts the worst case delay
• Only the nominal critical path is replicated
• Numerous near-critical paths in modern VLSI circuits
• Nominal critical path not necessarily critical in the

manufactured die (process variations)

Outline

8

Problem Formulation

9

Circuit with Gaussian process parameter variations

Original Circuit RCP

Build the RCP to reveal most information about the

• riginal circuit.

Representative Critical Path (RCP) should be related to the original circuit

c

d

p

d

Mathematical Formulation

10

From SSTA (Pre-Silicon) From measurement data (Post-Silicon) Our goal:

ρ

Minimum Maximum

Conditional PDF

Full Correlated Case

11

Fully Correlated

k b a b a b a

n n =

= = = Λ

2 2 1 1

( )

p p c c

d k d μ μ − = −

Outline

12

Representative Critical Path (RCP) Synthesis (Method I)

13

Maximum improvement Maximum improvement

Comments on Method I

• Advantage

– Guaranteed to do no worse than CPR – Exact solution when there is clearly one dominating path

• Drawback

– Flexibility of the solution is limited

• Runtime: O(Ks)

– Saved by only updating the SSTA results of stages adjacent to the one sized up – K: number of iterations – s:number of stages

14

RCP Generation (Method II)

15

1

ρ ρ

2

ρ

1 − s

ρ

s

ρ

Max

Comments on Method II

• Advantage

– More flexibility, not tied to a specific path

• Drawbacks

– No exact solution when there is only one dominating path – Not guaranteed to be always better than CPR

• Runtime: O(kcs)

– k: number of starting locations – c: number of choices for each iteration – s: maximum number of stages

16

Outline

17

Comparison Metric

18

• Average error, maximum error w.s.t. Monte-Carlo

analysis

Guard band

• Guard band

Experimental Results (Method I)

19

Scatter Plots (Method I)

20

Experimental Results (Method II)

21

Number of Paths vs. Delay

22

500 1000 20 40 60 80 100 delay (ps) num ber of paths # Paths vs. delay for delay optimized s13207

200 400 600 800 5 10 15 20 25 30 delay (ps) num ber of paths # Paths vs. Delay for delay optimized s9234

Conclusion and Future Work

• Two novel methods for synthesizing a representative

critical path under process variations are presented

• Average prediction error: below 2.8%
• To ensure 99% of the predictions pessimistic,

requiring guard band 30% smaller than CPR

• Future work

– Test on real silicon

23

Thank you!

24

Extra Slides

25

Scatter Plots (Method II)

26

250 300 350 400 450 500 250 300 350 400 450 500 true delay (ps) predicted delay (ps) s35932 by Critical Path Replica

250 300 350 400 450 500 250 300 350 400 450 500 true delay (ps) predicted delay (ps) s35932 by Method II

An Example RCP (Method II)

590 590 x direction y direction RCP for s38417

27

ρ vs. iteration number (Method II)

28

10 20 30 40 50 0.7 0.75 0.8 0.85 0.9 0.95 1 Iteration Correlation coefficient Correlation coefficient trend for s38417

Equations for Editing

29

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡

2 2

, ~

p T T c p c p c

N d d σ σ μ μ b a b a c T c c

R d + + = p a μ

2 2

c

R T c

σ σ + = a a

p T p p

R d + + = p b μ

2 2

p

R T p

σ σ + = b b

c

d

pr p

d d = ( ) ( )

2

, ~ | σ μ N d d d

pr p c

=

( )

p pr p T c

d μ σ μ μ − + = b a ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

p c T c

σ σ σ σ b a 1

2 2

p T p p

R d + + = p b3 μ

2 1 2 1 1

p

R c T

σ σ ρ + = b b b a

Parameter Variations

30

Use the Grid-Based Correlation Model ([ Chang, ICCAD03] )

Parameter Variations

σ