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PERI: Technique For Extending Delay/Slew Metrics For Ramps
Chandramouli V. KashyapCharles J. Alpert
Frank LiuAnirudh Devgan
IBM Corp.11400 Burnet RoadAustin, TX-78758
A Practical Problem
What are the sink delays and slews?
scope of this work
Prior Work
step input ramp input
delay
slew
Well studied
Somewhatstudied
Inefficient/inaccurate
Inefficient/inaccurate
Impulse Response and PDF
V(t)
t
PDF (Probability Density Function)Impulse response of circuit
median mean
median
meanFirst observed by ElmoreGeneralized by Pileggi et al.
circuit delay
Circuit Response To Input
V(t)
t
input signal derivative of output response(impulse response)
medianmean t
Circuit Response To Input
t
input signal derivative of output response(impulse response)
V(t)
mean tmedian
Circuit Response To Input
V(t)
t t
input signal derivative of output response
medianmean
Circuit Response To Input
V(t)
t t
input signal derivative of output response
medianmean
Circuit Response To Input
V(t)
t t
input signal derivative of output response
medianmean
Known Facts Time domain signal : like a CDF
Its time derivative is the PDF Output response is given by
convolution Probability moments add in
convolution
Proposed Delay Formula
stepelmore delaydelaydelay )()1(
delayelmore is the negative of first moment m1
delaystep is the delay for a step input, assumed known
alpha=1 delay =delaystep
alpha=0 delay =delayelmore
Computing Alpha
2/522
12
212 )
12/2
2(
Tmm
mm
T is 0-100 ramp slew
T=0 alpha=1 delay =delaystep
T=Inf. alpha=0 delay =delayelmore
Notes On Delay Derivation Distance between mean and
median is proportional to Pearson’s coefficient Easily computed from circuit moments
The constant of proportionality is independent of input slew This is a simplifying assumption we
make
Observations For Slew
step inputt
V(t)
t
V(t)
step response
timpulse response
Observations For Slew
ramp inputt
V(t)
t
V(t)
ramp response
t
derivative of response
Proposed Slew Formula
22stepinout slewslewslew
slewstep is the output slew for a step input: assumed known
slewin is the input slew
Notes On Slew Derivation Slew is proportional to standard
deviation of signal PDF Observed by Elmore, Pileggi Std.-Dev. Easily computed from moments
Constants of proportionality are identical for ramp, impulse and output This is a simplifying assumption we make
Experimental Setup 432 routed nets in 0.18um technology 4th order RICE used as golden Step delay/slew computed using RICE Measured delay(formula)/delay(rice)
Likewise, slew(formula)/slew(rice) Distribution of sinks:
#sinks
1-2 3-4 5-8 9-1314-19
Total
#nets
169 122 46 54 41 432
Results For Delay
Near-end Far-endMiddle-region
0
0.5
1
1.5
2
2.5
near middle far
AvgMaxMinDesired
Results For Slew
Near-end Far-endMiddle-region
0
0.25
0.5
0.75
1
1.25
near middle far
AvgMaxMin
Desired
Summary Provides a practical and useful
method for fast delay and slew computation
Renders prior research on step delay and slew metrics usable in a tool flow
Useful for physical design and optimization