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