Copyright Agrawal, 2007 ELEC6270 Fall 07, Lecture 3 1 ELEC 5270/6270 Fall 2007 Low-Power Design of...

Copyright Agrawal, 2007Copyright Agrawal, 2007 ELEC6270 Fall 07, Lecture 3ELEC6270 Fall 07, Lecture 3 11

ELEC 5270/6270 Fall 2007ELEC 5270/6270 Fall 2007Low-Power Design of Electronic CircuitsLow-Power Design of Electronic Circuits

Logic-Level Power EstimationLogic-Level Power Estimation

Vishwani D. AgrawalVishwani D. AgrawalJames J. Danaher ProfessorJames J. Danaher Professor

Dept. of Electrical and Computer EngineeringDept. of Electrical and Computer EngineeringAuburn University, Auburn, AL 36849Auburn University, Auburn, AL 36849

[email protected]://www.eng.auburn.edu/~vagrawal/COURSE/E6270_Fall07/course.html


Power AnalysisPower AnalysisMotivation:Motivation:

SpecificationSpecificationOptimizationOptimizationReliabilityReliability

ApplicationsApplicationsDesign analysis and optimizationDesign analysis and optimizationPhysical designPhysical designPackagingPackagingTestTest


Abstraction, Complexity, AccuracyAbstraction, Complexity, Accuracy

Abstraction levelAbstraction level Computing resourcesComputing resources Analysis accuracyAnalysis accuracy

AlgorithmAlgorithm LeastLeast WorstWorst

Software and systemSoftware and system

Hardware behaviorHardware behavior

Register transferRegister transfer

LogicLogic

CircuitCircuit

DeviceDevice MostMost BestBest


SpiceSpice Circuit/device level analysisCircuit/device level analysis

Circuit modeled as network of transistors, capacitors, resistors Circuit modeled as network of transistors, capacitors, resistors and voltage/current sources.and voltage/current sources.

Node current equations using Kirchhoff’s current law.Node current equations using Kirchhoff’s current law. Average and instantaneous power computed from supply Average and instantaneous power computed from supply

voltage and device current.voltage and device current.

Analysis is accurate but expensiveAnalysis is accurate but expensive Used to characterize parts of a larger circuit.Used to characterize parts of a larger circuit.

Original references:Original references: L. W. Nagel and D. O. Pederson, “SPICE – Simulation L. W. Nagel and D. O. Pederson, “SPICE – Simulation

Program With Integrated Circuit Emphasis,” Memo ERL-M382, Program With Integrated Circuit Emphasis,” Memo ERL-M382, EECS Dept., University of California, Berkeley, Apr. 1973.EECS Dept., University of California, Berkeley, Apr. 1973.

L. W. Nagel, L. W. Nagel, SPICE 2, A Computer program to Simulate SPICE 2, A Computer program to Simulate Semiconductor CircuitsSemiconductor Circuits, PhD Dissertation, University of , PhD Dissertation, University of California, Berkeley, May 1975.California, Berkeley, May 1975.


Ca

Logic Model of MOS CircuitLogic Model of MOS Circuit

Cc

Cb

VDD

a

b

c

pMOS FETs

nMOSFETs

Ca , Cb , Cc and Cd are

node capacitances

Dc

Da ca

b

Da and Db are

interconnect or propagation delays

Dc is inertial delay

of gate

Db

Cd


Spice Characterization of a 2-Input Spice Characterization of a 2-Input NAND GateNAND Gate

Input data patternInput data pattern Delay (ps)Delay (ps) Dynamic energy (pJ)Dynamic energy (pJ)

aa = = bb = 0 → 1 = 0 → 1 6969 1.551.55

aa = 1, = 1, bb = 0 → 1 = 0 → 1 6262 1.671.67

a = 0 → 1, b = 1a = 0 → 1, b = 1 5050 1.721.72

aa = = bb = 1 → 0 = 1 → 0 3535 1.821.82

aa = 1, = 1, bb = 1 → 0 = 1 → 0 7676 1.391.39

a = 1 → 0, b = 1a = 1 → 0, b = 1 5757 1.941.94


Spice Characterization (Cont.)Spice Characterization (Cont.)

Input data patternInput data pattern Static power (pW)Static power (pW)

aa = = bb = 0 = 0 5.055.05

a = 0, b = 1a = 0, b = 1 13.113.1

aa = 1, = 1, bb = 0 = 0 5.105.10

aa = = bb = 1 = 1 28.528.5


Switch-Level PartitioningSwitch-Level Partitioning Circuit partitioned into channel-connected components Circuit partitioned into channel-connected components

for Spice characterization.for Spice characterization. Reference: R. E. Bryant, “A Switch-Level Model and Reference: R. E. Bryant, “A Switch-Level Model and

Simulator for MOS Digital Systems,” Simulator for MOS Digital Systems,” IEEE Trans. IEEE Trans. ComputersComputers, vol. C-33, no. 2, pp. 160-177, Feb. 1984., vol. C-33, no. 2, pp. 160-177, Feb. 1984.

G1

G2

G3

Internal switching nodes not seen by logic simulator


Delay and Discrete-Event SimulationDelay and Discrete-Event Simulation (NAND gate)(NAND gate)

b

a

c (CMOS)

Time units 0 5

c (zero delay)

c (unit delay)

c (multiple delay)

c (minmax delay)

Inp

uts

Log

ic s

imul

atio

n

min =2, max =5

rise=5, fall=5

Transient region

Unknown (X)

X


Event-Driven Simulation ExampleEvent-Driven Simulation Example

2

2

4

2

a =1

b =1

c =1→0

d = 0

e =1

f =0

g =1

Time, t 0 4 8

g

t = 0 12345678

Scheduledevents

c = 0

d = 1, e = 0

g = 0

f = 1

g = 1

Activitylist

d, e

f, g

gTim

e s

tack


Time Wheel (Circular Stack)Time Wheel (Circular Stack)

t=0

1

2

3

4

5

6

7

maxCurrenttimepointer Event link-list


Gate-Level Power AnalysisGate-Level Power AnalysisPre-simulation analysis:Pre-simulation analysis:

Partition circuit into channel connected gate Partition circuit into channel connected gate components.components.

Determine node capacitances from layout Determine node capacitances from layout analysis (accurate) or from wire-load model* analysis (accurate) or from wire-load model* (approximate).(approximate).

Determine dynamic and static power from Spice Determine dynamic and static power from Spice for each gate.for each gate.

Determine gate delays using Spice or Elmore Determine gate delays using Spice or Elmore delay model.delay model.

* Wire-load model estimates capacitance of a net by its pin-count. See Yeap, p. 39.


Elmore Delay ModelElmore Delay Model W. Elmore, “The Transient Response of Damped Linear Networks W. Elmore, “The Transient Response of Damped Linear Networks

with Particular Regard to Wideband Amplifiers,” with Particular Regard to Wideband Amplifiers,” J. Appl. PhysJ. Appl. Phys., vol. ., vol. 19, no.1, pp. 55-63, Jan. 1948.19, no.1, pp. 55-63, Jan. 1948.

s 1

2

3

4

5

R1

R2

R3

R4

R5

C1

C2

C3

C5

C4

Shared resistance:

R45 = R1 + R3R15 = R1R34 = R1 + R3


Elmore Delay FormulaElmore Delay Formula

NDelay at node k = 0.69 Σ Cj × Rjk

j=1

where N = number of capacitive nodes in the network

Example:

Delay at node 5 = 0.69[R1 C1 + R1 C2 + (R1+R3)C3 + (R1+R3)C4 + (R1+R3+R5)C5]


Gate-Level Power Analysis (Cont.)Gate-Level Power Analysis (Cont.) Run discrete-event (event-driven) logic Run discrete-event (event-driven) logic

simulation with a set of input vectors.simulation with a set of input vectors. Monitor the toggle count of each net and obtain Monitor the toggle count of each net and obtain

capacitive power dissipation:capacitive power dissipation:

PPcapcap == ΣΣ CCk k V V 22 ff all nodes all nodes kk

Where:Where: CCkk is the total node capacitance being switched, as is the total node capacitance being switched, as

determined by the simulator.determined by the simulator. VV is the supply voltage. is the supply voltage. ff is the clock frequency, i.e., the number of vectors applied is the clock frequency, i.e., the number of vectors applied

per unit timeper unit time


Gate-Level Power Analysis (Cont.)Gate-Level Power Analysis (Cont.)

Monitor dynamic energy events at the input Monitor dynamic energy events at the input of each gate and obtain internal switching of each gate and obtain internal switching power dissipation:power dissipation:

PPintint = = ΣΣ ΣΣ E(g,e) F(g,e)E(g,e) F(g,e) gates gates g g events events ee

WhereWhereE(g,e) = E(g,e) = energy of event energy of event ee of gate of gate gg, pre-computed , pre-computed

from Spice.from Spice.F(g,e) = F(g,e) = occurrence frequency of the eventoccurrence frequency of the event e e at gateat gate

g, g, observed by logic simulationobserved by logic simulation..


Gate-Level Power Analysis (Cont.)Gate-Level Power Analysis (Cont.)

Monitor the static power dissipation state of each Monitor the static power dissipation state of each gate and obtain the static power dissipation:gate and obtain the static power dissipation:

PPstatstat = = ΣΣ ΣΣ P(g,s) T(g,s)/ TP(g,s) T(g,s)/ T gates gates gg states states ss

WhereWhere P(g,s) = P(g,s) = static power dissipation of gatestatic power dissipation of gate g g for statefor state s, s,

obtained from Spiceobtained from Spice.. T(g,s) = T(g,s) = duration of stateduration of state s s at gateat gate g, g, obtained from logic obtained from logic

simulationsimulation.. T = T = vector periodvector period..


Gate-Level Power AnalysisGate-Level Power Analysis Sum up all three components of power:Sum up all three components of power:

P = PP = Pcap cap + P+ Pintint + P + Pstatstat

References:References: A. Deng, “Power Analysis for CMOS/BiCMOS Circuits,” A. Deng, “Power Analysis for CMOS/BiCMOS Circuits,” Proc. Proc.

International Workshop Low Power DesignInternational Workshop Low Power Design, 1994., 1994. J. Benkoski, A. C. Deng, C. X. Huang, S. Napper and J. Tuan, J. Benkoski, A. C. Deng, C. X. Huang, S. Napper and J. Tuan,

“Simulation Algorithms, Power Estimation and Diagnostics in “Simulation Algorithms, Power Estimation and Diagnostics in PowerMill,” PowerMill,” Proc. PATMOSProc. PATMOS, 1995., 1995.

C. X. Huang, B. Zhang, A. C. Deng and B. Swirski, “The Design C. X. Huang, B. Zhang, A. C. Deng and B. Swirski, “The Design and Implementation of PowerMill,” and Implementation of PowerMill,” Proc. International Symp. Low Proc. International Symp. Low Power DesignPower Design, 1995, pp. 105-109., 1995, pp. 105-109.


Probabilistic AnalysisProbabilistic Analysis

View signals as a random processesView signals as a random processes

Prob{s(t) = 1} = p1 p0 = 1 – p1

C

0→1 transition probability = (1 – p1) p1

Power, P = (1 – p1) p1 CV2fck


Source of InaccuracySource of Inaccuracy

1/fck

p1 = 0.5 P = 0.5CV 2 fck

p1 = 0.5 P = 0.33CV 2 fck

p1 = 0.5 P = 0.167CV 2 fck

Observe that the formula, Power, P = (1 – p1) p1 C V 2 fck , is notCorrect.


Switching FrequencySwitching Frequency

Number of transitions per unit time:

N(t)T = ───

t

For a continuous signal:

N(t)T = lim ───

t→∞ t

T is defined as transition density.


Static Signal ProbabilitiesStatic Signal Probabilities

Observe signal for interval Observe signal for interval t t 0 + 0 + t t 11Signal is 1 for duration Signal is 1 for duration t t 11Signal is 0 for duration Signal is 0 for duration t t 00Signal probabilities:Signal probabilities:

p p 1 = 1 = t t 1/(1/(t t 0 + 0 + t t 1)1) p p 0 = 0 = t t 0/(0/(t t 0 + 0 + t t 1) = 1 – 1) = 1 – p p 11


Static Transition ProbabilitiesStatic Transition Probabilities

Transition probabilities:Transition probabilities: T T 01 = 01 = p p 0 Prob{signal is 1 | signal was 0} = 0 Prob{signal is 1 | signal was 0} = p p 0 0 pp11 T T 10 = 10 = p p 1 Prob{signal is 0 | signal was 1} = 1 Prob{signal is 0 | signal was 1} = p p 1 1 p p 00 TT = = T T 01 + 01 + T T 10 = 2 10 = 2 p p 0 0 p p 1 = 2 1 = 2 p p 1 (1 – 1 (1 – p p 1)1)

Transition density: Transition density: TT = 2 = 2 p p 1 (1 – 1 (1 – p p 1)1)


Static Transition FrequencyStatic Transition Frequency

0 0.25 0.5 0.75 1.0

0.25

0.2

0.1

0.0

p1

f =

p1

(1 –

p1

)


Inaccuracy in Transition DensityInaccuracy in Transition Density

1/fck

p1 = 0.5 T = 1.0

p1 = 0.5 T = 4/6

p1 = 0.5 T = 1/6

Observe that the formula, T = 2 p1 (1 – p1), is not correct.


Cause for Error and CorrectionCause for Error and CorrectionProbability of transition is not independent of Probability of transition is not independent of

the present state of the signal.the present state of the signal.Determine probability Determine probability p p 01 of a 0→1 01 of a 0→1

transition.transition.Recognize Recognize p p 01 ≠ 01 ≠ p p 0 × 0 × p p 11We obtain We obtain p p 1 = (1 – 1 = (1 – p p 1)1)p p 01 + 01 + p p 1 1 p p 1111

p p 0101p p 1 = ─────────1 = ─────────

1 – 1 – p p 11 + 11 + p p 0101


Correction (Cont.)Correction (Cont.) Since Since p p 11 + 11 + p p 10 = 1, i.e., given that the 10 = 1, i.e., given that the

signal was previously 1, its present value can signal was previously 1, its present value can be either 1 or 0.be either 1 or 0.

Therefore,Therefore, p p 0101

p p 1 = ──────1 = ────── p p 10 + 10 + p p 0101

This uniquely gives signal probability as a This uniquely gives signal probability as a function of transition probabilities.function of transition probabilities.


Transition and Signal ProbabilitiesTransition and Signal Probabilities

1/fck

p01 = p10 = 0.5 p1 = 0.5

p01 = p10 = 1/3 p1 = 0.5

p1 = 0.5p01 = p10 = 1/6


Probabilities: p0, p1, p00, p01, p10, p11Probabilities: p0, p1, p00, p01, p10, p11

p p 01 + 01 + p p 00 =100 =1 p p 11 + 11 + p p 10 = 110 = 1 p p 0 = 1 – 0 = 1 – p p 11

p p 0101

p p 1 = ───────1 = ───────

p p 10 + 10 + p p 0101


Transition DensityTransition Density

TT = 2 = 2 p p 1 (1 – 1 (1 – p p 1) = 1) = p p 0 0 p p 01 + 01 + p p 1 1 p p 1010

= 2 = 2 p p 10 10 p p 01 / (01 / (p p 10 + 10 + p p 01)01)

= 2 = 2 p p 1 1 p p 10 = 2 10 = 2 p p 0 0 p p 0101


Power CalculationPower Calculation

Power can be estimated if transition Power can be estimated if transition density is known for all signals.density is known for all signals.

Calculation of transition density requiresCalculation of transition density requiresSignal probabilitiesSignal probabilitiesTransition densities for primary inputs; Transition densities for primary inputs;

computed from vector statisticscomputed from vector statistics


Signal ProbabilitiesSignal Probabilities

x1

x2

x1 x2

x1

x2

x1 + x2 – x1x2

x1 1 - x1


Signal ProbabilitiesSignal Probabilities x1

x2 x3

x1 x2

y = 1 - (1 - x1x2) x3 = 1 - x3 + x1x2x3 = 0.625

X1 X2 X3 Y0 0 0 10 0 1 00 1 0 10 1 1 01 0 0 11 0 1 01 1 0 11 1 1 1

0.5

0.5

0.5

0.25 0.625

Ref: K. P. Parker and E. J. McCluskey,“Probabilistic Treatment of General Combinational Networks,” IEEE Trans. on Computers, vol. C-24, no. 6, pp. 668-670, June 1975.


Correlated Signal ProbabilitiesCorrelated Signal Probabilities

x1

x2

x1 x2

y = 1 - (1 - x1x2) x2 = 1 – x2 + x1x2x2 = 1 – x2 + x1x2 = 0.75 (correct value)

X1 X2 Y0 0 10 1 01 0 11 1 1

0.5

0.5 0.25 0.625?



x1

x2

x1 + x2 – x1x2

y = (x1 + x2 – x1x2) x2 = x1x2 + x2x2 – x1x2x2 = x1x2 + x2 – x1x2 = x2 = 0.5 (correct value)

X1 X2 Y0 0 00 1 11 0 01 1 1

0.5

0.5 0.75 0.375?


ObservationObservation

Numerical computation of signal Numerical computation of signal probabilities is accurate for fanout-free probabilities is accurate for fanout-free circuits.circuits.


RemediesRemedies

Use Shannon’s expansion theorem to Use Shannon’s expansion theorem to compute signal probabilities.compute signal probabilities.

Use Boolean difference formula to Use Boolean difference formula to compute transition densities.compute transition densities.


Shannon’s Expansion TheoremShannon’s Expansion Theorem

C. E. Shannon, “A Symbolic Analysis of Relay C. E. Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” and Switching Circuits,” Trans. AIEETrans. AIEE, vol. 57, , vol. 57, pp. 713-723, 1938.pp. 713-723, 1938.

Consider:Consider: Boolean variables, X1, X2, . . . , XnBoolean variables, X1, X2, . . . , Xn Boolean function, F(X1, X2, . . . , Xn)Boolean function, F(X1, X2, . . . , Xn)

Then F = Xi F(Xi=1) + Xi’ F(Xi=0)Then F = Xi F(Xi=1) + Xi’ F(Xi=0) WhereWhere

Xi’ is complement of X1Xi’ is complement of X1 Cofactors, F(Xi=j) = F(X1, X2, . . , Xi=j, . . , Xn), j = 0 or 1Cofactors, F(Xi=j) = F(X1, X2, . . , Xi=j, . . , Xn), j = 0 or 1


Expansion About Two InputsExpansion About Two Inputs

F = XiXj F(Xi=1, Xj=1) + XiXj’ F(Xi=1, Xj=0)F = XiXj F(Xi=1, Xj=1) + XiXj’ F(Xi=1, Xj=0)

+ Xi’Xj F(Xi=0, Xj=1)+ Xi’Xj F(Xi=0, Xj=1)

+ Xi’Xj’ F(Xi=0, Xj=0)+ Xi’Xj’ F(Xi=0, Xj=0) In general, a Boolean function can be In general, a Boolean function can be

expanded about any number of input expanded about any number of input variables.variables.

Expansion about k variables will have 2Expansion about k variables will have 2kk terms.terms.



X1

X2

X1 X2

X1 X2 Y0 0 10 1 01 0 11 1 1

Y = X1 X2 + X2’

Shannon expansion about the reconverging input, X2:

Y = X2 Y(X2 = 1) + X2’ Y(X2 = 0) = X2 (X1) + X2’ (1)


Correlated SignalsCorrelated Signals

When the output function is expanded about all When the output function is expanded about all reconverging input variables,reconverging input variables,

All cofactors correspond to fanout-free circuits.All cofactors correspond to fanout-free circuits. Signal probabilities for cofactor outputs can be calculated Signal probabilities for cofactor outputs can be calculated

without error.without error. A weighted sum of cofactor probabilities gives the correct A weighted sum of cofactor probabilities gives the correct

probability of the output.probability of the output.

For two reconverging inputs:For two reconverging inputs:

f = xixj f(Xi=1, Xj=1) + xi(1-xj) f(Xi=1, Xj=0)f = xixj f(Xi=1, Xj=1) + xi(1-xj) f(Xi=1, Xj=0)

+ (1-xi)xj f(Xi=0, Xj=1) + (1-xi)(1-xj) f(Xi=0, Xj=0)+ (1-xi)xj f(Xi=0, Xj=1) + (1-xi)(1-xj) f(Xi=0, Xj=0)


Correlated Signal ProbabilitiesCorrelated Signal Probabilities X1

X2

X1 X2

X1 X2 Y0 0 10 1 01 0 11 1 1

Y = X1 X2 + X2’

Shannon expansion about the reconverging input, X2:

Y = X2 Y(X2=1) + X2’ Y(X2=0) = X2 (X1) + X2’ (1)

y = x2 (0.5) + (1-x2) (1) = 0.5 (0.5) + (1-0.5) (1) = 0.75


ExampleExample

Point of reconv.

Supergate0.5

0.5

0.5

0.5

0.25

10

0.50.0

0.01.0

0.51.0

Signal probability for supergate output = 0.5 Prob{rec. signal = 1} + 1.0 Prob{rec. signal = 0} = 0.5 × 0.5 + 1.0 × 0.5 = 0.75

0.375

Reconv. signal

S. C. Seth and V. D. Agrawal, “A New Model for Computation ofProbabilistic Testability in Combinational Circuits,” Integration, the VLSI Journal, vol. 7, no. 1, pp. 49-75, April 1989.


Probability Calculation AlgorithmProbability Calculation Algorithm

Partition circuit into supergates.Partition circuit into supergates. Definition: A supergate is a circuit partition with a single output Definition: A supergate is a circuit partition with a single output

such that all fanouts that reconverge at the output are contained such that all fanouts that reconverge at the output are contained within the supergate. within the supergate.

Identify reconverging and non-reconverging inputs Identify reconverging and non-reconverging inputs of each supergate.of each supergate.

Compute signal probabilities from PI to PO:Compute signal probabilities from PI to PO: For a supergate whose input probabilities are knownFor a supergate whose input probabilities are known

Enumerate reconverging input statesEnumerate reconverging input states For each input state do gate by gate probability computationFor each input state do gate by gate probability computation Sum up corresponding signal probabilities, weighted by state Sum up corresponding signal probabilities, weighted by state

probabilitiesprobabilities


Calculating Transition DensityCalculating Transition Density

Boolean function

1

n

x1, T1..... xn, Tn

y, T(Y) = ?


Boolean DifferenceBoolean Difference

Boolean diff(Y, Xi) = 1 means that a path is sensitized from input Boolean diff(Y, Xi) = 1 means that a path is sensitized from input Xi to output Y.Xi to output Y.

Prob(Boolean diff(Y, Xi) = 1) is the probability of transmitting a Prob(Boolean diff(Y, Xi) = 1) is the probability of transmitting a toggle from Xi to Y.toggle from Xi to Y.

Probability of Boolean difference is determined from the Probability of Boolean difference is determined from the probabilities of cofactors of Y with respect to Xi. probabilities of cofactors of Y with respect to Xi.

∂YBoolean diff(Y, Xi) = ── = Y(Xi=1) ⊕ Y(Xi=0)

∂Xi

F. F. Sellers, M. Y. Hsiao and L. W. Bearnson, “Analyzing Errors with the Boolean Difference,” IEEE Trans. on Computers, vol. C-17, no. 7, pp. 676-683, July 1968.


Transition DensityTransition Density

nT(y) = Σ T(Xi) Prob(Boolean diff(Y, Xi) = 1)

i=1

F. Najm, “Transition Density: A New Measure of Activity in DigitalCircuits,” IEEE Trans. CAD, vol. 12, pp. 310-323, Feb. 1993.


Power ComputationPower Computation For each primary input, determine signal probability and For each primary input, determine signal probability and

transition density for given vectors.transition density for given vectors. For each internal node and primary output Y, find the For each internal node and primary output Y, find the

transition density T(Y), using supergate partitioning and transition density T(Y), using supergate partitioning and the Boolean difference formula.the Boolean difference formula.

Compute power,Compute power,

P =P = ΣΣ 0.5C0.5CYY V V22 T(Y) T(Y)

all Yall Y

where Cwhere CYY is the capacitance of node Y and V is supply is the capacitance of node Y and V is supply

voltage.voltage.


Transition Density and PowerTransition Density and Power

X1

X2 X3

0.2, 1

0.3, 2

0.4, 3

0.06, 0.7

0.436, 3.24

Transition densitySignal probability

YCi

CY

Power = 0.5 V 2 (0.7Ci + 3.24CY)


Prob. Method vs. Logic Sim. Prob. Method vs. Logic Sim.

CircuitCircuit No. of No. of gatesgates

Probability methodProbability method Logic SimulationLogic SimulationErrorError

%%Av. densityAv. density CPU s*CPU s* Av. densityAv. density CPU s*CPU s*

C432C432 160160 3.463.46 0.520.52 3.393.39 6363 +2.1+2.1

C499C499 202202 11.3611.36 0.580.58 8.578.57 241241 +29.8+29.8

C880C880 383383 2.782.78 1.061.06 3.253.25 132132 -14.5-14.5

C1355C1355 346346 4.194.19 1.391.39 6.186.18 408408 -32.2-32.2

C1908C1908 880880 2.972.97 2.002.00 5.015.01 464464 -40.7-40.7

C2670C2670 11931193 3.503.50 3.453.45 4.004.00 619619 -12.5-12.5

C3540C3540 16691669 4.474.47 3.773.77 4.494.49 10821082 -0.4-0.4

C5315C5315 23072307 3.523.52 6.416.41 4.794.79 16161616 -26.5-26.5

C6288C6288 24062406 25.1025.10 5.675.67 34.1734.17 3105731057 -26.5-26.5

C7552C7552 35123512 3.833.83 9.859.85 5.085.08 27132713 -24.2-24.2* CONVEX c240


Probability Waveform MethodsProbability Waveform Methods F. Najm, R. Burch, P. Yang and I. Hajj, “CREST – A F. Najm, R. Burch, P. Yang and I. Hajj, “CREST – A

Current Estimator for CMOS Circuits,” Current Estimator for CMOS Circuits,” Proc. IEEE Int. Proc. IEEE Int. Conf. on CADConf. on CAD, Nov. 1988, pp. 204-207., Nov. 1988, pp. 204-207.

C.-S. Ding, C.-S. Ding, et alet al., “Gate-Level Power Estimation using ., “Gate-Level Power Estimation using Tagged Probabilistic Simulation,” Tagged Probabilistic Simulation,” IEEE Trans. on CADIEEE Trans. on CAD, , vol. 17, no. 11, pp. 1099-1107, Nov. 1998.vol. 17, no. 11, pp. 1099-1107, Nov. 1998.

F. Hu and V. D. Agrawal, “Dual-Transition Glitch Filtering F. Hu and V. D. Agrawal, “Dual-Transition Glitch Filtering in Probabilistic Waveform Power Estimation,” in Probabilistic Waveform Power Estimation,” Proc. IEEE Proc. IEEE Great Lakes Symp. VLSIGreat Lakes Symp. VLSI, Apr. 2005, pp. 357-360., Apr. 2005, pp. 357-360.

F. Hu and V. D. Agrawal,F. Hu and V. D. Agrawal, “ “Enhanced Dual-Transition Enhanced Dual-Transition Probabilistic Power Estimation with Selective Supergate Probabilistic Power Estimation with Selective Supergate Analysis,” Analysis,” Proc. IEEE Int. Conf. Computer DesignProc. IEEE Int. Conf. Computer Design, Oct. , Oct. 2005. pp. 366-369.2005. pp. 366-369.


Problem 1Problem 1For equiprobable inputs analyze the 0→1 transition probabilities of all gates in the two implementations of a four-input AND gate shown below. Assuming that the gates have zero delays, which implementation will consume less average dynamic power?

Chain structure Tree structure

ABCD

EF

G

ABCD

E

F

G


Problem 1 SolutionProblem 1 SolutionGiven the primary input probabilities, P(A) = P(B) = P(C) = P(D) = 0.5, signal and transition (0→1) probabilities are as follows:

Signalname

Chain Tree

Prob(sig.= 1) Prob(0→1) Prob(sig.=1) Prob(0→1)

E 0.2500 0.1875 0.2500 0.1875

F 0.1250 0.1094 0.2500 0.1875

G 0.0625 0.0586 0.0625 0.0586

Total transitions/vector

0.3555 0.4336

The tree implementation consumes 100×(0.4336 – 0.3555)/0.3555 = 22% more average dynamic power. This advantage of the chain structure may be somewhat reduced because of glitches caused by unbalanced path delays.


Problem 2Problem 2Assume that the two-input AND gates in Problem 1 each has one unit of delay. Find input vector pairs for each implementation that will consume the peak dynamic power. Which implementation consumes less peak dynamic power?

Chain structure Tree structure

ABCD

EF

G

ABCD

E

F

G


Problem 2 SolutionProblem 2 SolutionFor the chain structure, a vector pair {A B C D} = {1110},{1011} will produce four gate transitions as shown below.

A

BCD

EF

G

A=11

B=10

E=10

C=11

F=10

D=01

G=00

Time units0 1 2 3


Problem 2 Solution (Cont.)Problem 2 Solution (Cont.)The tree structure has balanced delay paths. So it cannot make more than 3 gate transitions. A vector pair {ABCD} = {1111},{1010} will produce three transitions as shown below.

A

B

C

D

E

F

G

A=11

B=10

E=10

C=11

D=10

F=10

G=10Time units

0 1 2 3

Therefore, just counting the gate transitions, we find that the chain consumes 100(4 – 3)/3 = 33% higher peak power than the tree.

Copyright Agrawal, 2007 ELEC6270 Fall 07, Lecture 3 1 ELEC 5270/6270 Fall 2007 Low-Power Design of...

Documents

Transcript of Copyright Agrawal, 2007 ELEC6270 Fall 07, Lecture 3 1 ELEC 5270/6270 Fall 2007 Low-Power Design of...