1 / 33

Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering

ELEC 5970-001/6970-001(Fall 2005) Special Topics in Electrical Engineering Low-Power Design of Electronic Circuits Power Analysis: Probabilistic Methods. Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University

elsu
Télécharger la présentation

Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ELEC 5970-001/6970-001(Fall 2005)Special Topics in Electrical EngineeringLow-Power Design of Electronic CircuitsPower Analysis: Probabilistic Methods Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University http://www.eng.auburn.edu/~vagrawal vagrawal@eng.auburn.edu ELEC 5970-001/6970-001 Lecture 11

  2. Basic Idea • View 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 ELEC 5970-001/6970-001 Lecture 11

  3. Source of Inaccuracy p1 = 0.5 P = 0.5CV2fck 1/fck p1 = 0.5 P = 0.33CV2fck p1 = 0.5 P = 0.167CV2fck Observe that the formula, Power, P = (1 – p1) p1 CV2fck, is not Correct. ELEC 5970-001/6970-001 Lecture 11

  4. Switching 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 the transition density. ELEC 5970-001/6970-001 Lecture 11

  5. Static Signal Probabilities • Observe signal for interval t0 + t1 • Signal is 1 for duration t1 • Signal is 0 for duration t0 • Signal probabilities: • p1 = t1/(t0 + t1) • p0 = t0/(t0 + t1) = 1 – p1 ELEC 5970-001/6970-001 Lecture 11

  6. Static Transition Probabilities • Transition probabilities: • T01 = p0 Prob{signal is 1 | signal was 0} = p0 p1 • T10 = p1 Prob{signal is 0 | signal was 1} = p1 p0 • T = T01 + T10 = 2 p0 p1 = 2 p1 (1 – p1) • Transition density: T = 2 p1 (1 – p1) • Transition frequency: f = T/ 2 • Power = CV2T/ 2 (correct formula) ELEC 5970-001/6970-001 Lecture 11

  7. Static Transition Frequency 0.25 0.2 0.1 0.0 f = p1(1 – p1) 0 0.25 0.5 0.75 1.0 p1 ELEC 5970-001/6970-001 Lecture 11

  8. Inaccuracy in Transition Density p1 = 0.5 T = 1.0 1/fck p1 = 0.5 T = 4/6 p1 = 0.5 T = 1/6 Observe that the formula, T = 2 p1 (1 – p1), is not correct. ELEC 5970-001/6970-001 Lecture 11

  9. Cause for Error and Correction • Probability of transition is not independent of the present state of the signal. • Consider probability p01of a 0→1 transition, • Then p01 ≠ p0 p1 • We can write p1 = (1 – p1)p01 + p1 p11 p01 p1 = ───────── 1 – p11 + p01 ELEC 5970-001/6970-001 Lecture 11

  10. Correction (Cont.) • Since p11 + p10 = 1, i.e., given that the signal was previously 1, its present value can be either 1 or 0. • Therefore, p01 p1 = ────── p10 + p01 This uniquely gives signal probability as a function of transition probabilities. ELEC 5970-001/6970-001 Lecture 11

  11. Transition and Signal Probabilities p01 = p10 = 0.5 p1= 0.5 1/fck p01 = p10 = 1/3 p1= 0.5 p01 = p10 = 1/6 p1= 0.5 ELEC 5970-001/6970-001 Lecture 11

  12. Probabilities: p0, p1, p00, p01, p10, p11 • p01 + p00 =1 • p11 + p10 = 1 • p0 = 1 – p1 • p01 p1 = ────── p10 + p01 ELEC 5970-001/6970-001 Lecture 11

  13. Transition Density • T = 2 p1(1 – p1) = p0 p01 + p1 p10 = 2 p10 p01/(p10 + p01) = 2 p1 p10 = 2 p0 p01 ELEC 5970-001/6970-001 Lecture 11

  14. Power Calculation • Power can be estimated if transition density is known for all signals. • Calculation of transition density requires • Signal probabilities • Transition densities for primary inputs; computed from vector statistics ELEC 5970-001/6970-001 Lecture 11

  15. Signal Probabilities x1 x2 x1 x2 x1 x2 x1 + x2 – x1x2 1 - x1 x1 ELEC 5970-001/6970-001 Lecture 11

  16. Signal Probabilities 0.5 x1 x2 x3 x1 x2 0.25 0.5 0.625 0.5 y = 1 - (1 - x1x2) x3 = 1 - x3 + x1x2x3 = 0.625 X1 X2 X3 Y 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1 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. ELEC 5970-001/6970-001 Lecture 11

  17. Correlated Signal Probabilities 0.5 x1 x2 x1 x2 0.5 0.25 0.625? y = 1 - (1 - x1x2) x2 = 1 – x2 + x1x2x2 = 1 – x2 + x1x2 = 0.75 X1 X2 Y 0 0 1 0 1 0 1 0 1 1 1 1 ELEC 5970-001/6970-001 Lecture 11

  18. Correlated Signal Probabilities 0.5 x1 + x2 – x1x2 x1 x2 0.75 0.5 0.375? y = (x1 + x2 – x1x2) x2 = x1x2 + x2x2 – x1x2x2 = x1x2 + x2 – x1x2 = x2 = 0.5 X1 X2 Y 0 0 0 0 1 1 1 0 0 1 1 1 ELEC 5970-001/6970-001 Lecture 11

  19. Observation • Numerical computation of signal probabilities is accurate for fanout-free circuits. ELEC 5970-001/6970-001 Lecture 11

  20. Remedies • Use Shannon’s expansion theorem to compute signal probabilities. • Use Boolean difference formula to compute transition densities. ELEC 5970-001/6970-001 Lecture 11

  21. Shannon’s Expansion Theorem • C. E. Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” Trans. AIEE, vol. 57, pp. 713-723, 1938. • Consider: • Boolean variables, X1, X2, . . . , Xn • Boolean function, F(X1, X2, . . . , Xn) • Then F = Xi F(Xi=1) + Xi’ F(Xi=0) • Where • Xi’ is complement of X1 • Cofactors, F(Xi=j) = F(X1, X2, . . , Xi=j, . . , Xn), j = 0 or 1 ELEC 5970-001/6970-001 Lecture 11

  22. Expansion About Two Inputs • 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=0) • In general, a Boolean function can be expanded about any number of input variables. • Expansion about k variables will have 2k terms. ELEC 5970-001/6970-001 Lecture 11

  23. Correlated Signal Probabilities X1 X2 X1 X2 Y = X1 X2 + X2’ X1 X2 Y 0 0 1 0 1 0 1 0 1 1 1 1 Shannon expansion about the reconverging input: Y = X2 Y(X2=1) + X2’ Y(X2=0) = X2 (X1) + X2’ (1) ELEC 5970-001/6970-001 Lecture 11

  24. Correlated Signals • When the output function is expanded about all reconverging input variables, • All cofactors correspond to fanout-free circuits. • Signal probabilities for cofactor outputs can be calculated without error. • A weighted sum of cofactor probabilities gives the correct probability of the output. • For two reconverging inputs: 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) ELEC 5970-001/6970-001 Lecture 11

  25. Correlated Signal Probabilities X1 X2 X1 X2 Y = X1 X2 + X2’ X1 X2 Y 0 0 1 0 1 0 1 0 1 1 1 1 Shannon expansion about the reconverging input: 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 ELEC 5970-001/6970-001 Lecture 11

  26. Example 0.5 Supergate 0.25 Point of reconv. 0.5 0.0 0.5 1.0 0.5 1 0 0.0 1.0 0.5 0.375 0.5 Reconv. signal 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 S. C. Seth and V. D. Agrawal, “A New Model for Computation of Probabilistic Testability in Combinational Circuits,” Integration, the VLSI Journal, vol. 7, no. 1, pp. 49-75, April 1989. ELEC 5970-001/6970-001 Lecture 11

  27. Probability Calculation Algorithm • Partition circuit into supergates. • Definition: A supergate is a circuit partition with a single output such that all fanouts that reconverge at the output are contained within the supergate. • Identify reconverging and non-reconverging inputs of each supergate. • Compute signal probabilities from PI to PO: • For a supergate whose input probabilities are known • Enumerate reconverging input states • For each input state do gate by gate probability computation • Sum up corresponding signal probabilities, weighted by state probabilities ELEC 5970-001/6970-001 Lecture 11

  28. Calculating Transition Density 1 Boolean function x1, T1 . . . . . xn, Tn y, T(Y) = ? n ELEC 5970-001/6970-001 Lecture 11

  29. Boolean Difference ∂Y Boolean diff(Y, Xi) = ── = Y(Xi=1) ⊕ Y(Xi=0) ∂Xi • Boolean diff(Y, Xi) = 1 means that a path is sensitized from input Xi to output Y. • Prob(Boolean diff(Y, Xi) = 1) is the probability of transmitting a toggle from Xi to Y. • Probability of Boolean difference is determined from the probabilities of cofactors of Y with respect to 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. ELEC 5970-001/6970-001 Lecture 11

  30. Transition Density n T(y) = Σ T(Xi) Prob(Boolean diff(Y, Xi) = 1) i=1 F. Najm, “Transition Density: A New Measure of Activity in Digital Circuits,” IEEE Trans. CAD, vol. 12, pp. 310-323, Feb. 1993. ELEC 5970-001/6970-001 Lecture 11

  31. Power Computation • For each primary input, determine signal probability and transition density for given vectors. • For each internal node and primary output Y, find the transition density T(Y), using supergate partitioning and the Boolean difference formula. • Compute power, P = Σ 0.5CY V2 T(Y) all Y where CY is the capacitance of node Y and V is supply voltage. ELEC 5970-001/6970-001 Lecture 11

  32. Transition Density and Power 0.2, 1 X1 X2 X3 0.06, 0.7 0.3, 2 0.436, 3.24 Ci Y CY 0.4, 3 Transition density Signal probability Power = 0.5 V2 (0.7Ci + 3.24CY) ELEC 5970-001/6970-001 Lecture 11

  33. Prob. Method vs. Logic Sim. * CONVEX c240 ELEC 5970-001/6970-001 Lecture 11

More Related