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Synthesizing a Representative Critical Path for Post-Silicon Delay Prediction
Qunzeng Liu and Sachin S. SapatnekarDepartment 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
DiffusionS. Tyagi
Normalized LeakageNor
mal
ized
Fr
eque
ncy
1 2 3 4 50.9
1.0
1.1
1.2
1.3
1.4
30%
5X
Inte
l
Design Phase: Pre-Silicon
3
Pre-Silicon Optimization
Deterministic/StatisticalTiming/Power Analysis
…
SynthesisGate Sizing
…
Statistical Timing Analysis result
Statistical Power Analysis result
Pre-Silicon Analysis
Statistical Static Timing Analysis (SSTA)
• Spatial correlation
• Canonical form
4
( )I0ppa ,~ NRd cT
cc ++= 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 original circuit.
Representative Critical Path (RCP) should be related to the original circuit
cd
pd
Mathematical Formulation
10
From SSTA (Pre-Silicon)
From measurement data(Post-Silicon)
Our goal:
ρ MinimumMaximum
Conditional PDF
Full Correlated Case
11
Fully Correlated
kba
ba
ba
n
n ==== Λ2
2
1
1( )ppcc dkd μμ −=−
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
1ρ 0ρ2ρ1−sρsρMax
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
0 500 10000
20
40
60
80
100
delay (ps)
num
ber o
f pat
hs
# Paths vs. delay for delay optimized s13207
0 200 400 600 8000
5
10
15
20
25
30
delay (ps)
num
ber o
f pat
hs
# 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)
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250 300 350 400 450 500250
300
350
400
450
500
true delay (ps)
pred
icte
d de
lay
(ps)
s35932 by Critical Path Replica
250 300 350 400 450 500250
300
350
400
450
500
true delay (ps)
pred
icte
d de
lay
(ps)
s35932 by Method II
An Example RCP (Method II)
0 5900
590
x direction
y di
rect
ion
RCP for s38417
27
ρ vs. iteration number (Method II)
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0 10 20 30 40 500.7
0.75
0.8
0.85
0.9
0.95
1
Iteration
Cor
rela
tion
coef
ficie
ntCorrelation coefficient trend for s38417
Equations for Editing
29
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡2
2
,~p
T
Tc
p
c
p
c Ndd
σσ
μμ
baba
cT
cc Rd ++= paμ22
cRT
c σσ += aa pT
pp Rd ++= pbμ
22pR
Tp σσ += bb cd prp dd = ( ) ( )2,~| σμNddd prpc =
( )pprp
T
c d μσ
μμ −+=ba
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
pc
T
c σσσσ ba122 p
Tpp Rd ++= pb3μ
21
21
1
pRc
T
σσρ
+=
bbba
Parameter Variations
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Use the Grid-Based Correlation Model ([Chang, ICCAD03])
Parameter Variations
σ