Transcript TAU Workshop

PERI: Technique For Extending
Delay/Slew Metrics For Ramps
Chandramouli V. Kashyap
Charles J. Alpert
Frank Liu
Anirudh Devgan
IBM Corp.
11400 Burnet Road
Austin, TX-78758
A Practical Problem
What are the sink delays and slews?
Prior Work
step input
delay
slew
Well studied
Somewhat
studied
ramp input
Inefficient/
inaccurate
Inefficient/
inaccurate
scope of
this work
Impulse Response and PDF
V(t)
PDF (Probability Density Function)
Impulse response of circuit
median
circuit delay
mean First observed by Elmore
Generalized by Pileggi et al.
median mean
t
Circuit Response To Input
V(t)
t
input signal
median mean
t
derivative of output response
(impulse response)
Circuit Response To Input
V(t)
t
input signal
median mean
t
derivative of output response
(impulse response)
Circuit Response To Input
V(t)
t
input signal
median mean
t
derivative of output response
Circuit Response To Input
V(t)
t
input signal
median
mean
t
derivative of output response
Circuit Response To Input
V(t)
t
input signal
median
mean
t
derivative of output response
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
delay  (1   )delayelmore  ( )delaystep
delayelmore is the negative of first moment m1
delaystep is the delay for a step input, assumed known
alpha=0
delay =delayelmore
alpha=1
delay =delaystep
Computing Alpha
 (
T=0
T=Inf.
2m2  m12
2m2  m12  T 2 / 12
alpha=1
alpha=0
)5 / 2
T is 0-100 ramp slew
delay =delaystep
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
V(t)
V(t)
step input
t
step response
impulse response
t
t
Observations For Slew
V(t)
V(t)
ramp input
t
ramp response
t
t
derivative of response
Proposed Slew Formula
2
slewout  slewin  slewstep
2
slewin is the input slew
slewstep is the output slew for a step input: assumed known
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
#nets
169
122
46
9-13 14-19 Total
54
41
432
Results For Delay
Far-end
Middle-region
Near-end
2.5
2
1.5
Desired
Avg
Max
Min
1
0.5
0
near
middle
far
Results For Slew
Far-end
Middle-region
Near-end
1.25
Desired
1
0.75
Avg
Max
Min
0.5
0.25
0
near
middle
far
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