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ELEC 7770 Advanced VLSI Design Spring 2008 Retiming

ELEC 7770 Advanced VLSI Design Spring 2008 Retiming. Vishwani D. Agrawal James J. Danaher Professor ECE Department, Auburn University Auburn, AL 36849 vagrawal@eng.auburn.edu http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr08/course.html. Retiming.

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ELEC 7770 Advanced VLSI Design Spring 2008 Retiming

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  1. ELEC 7770Advanced VLSI DesignSpring 2008Retiming Vishwani D. Agrawal James J. Danaher Professor ECE Department, Auburn University Auburn, AL 36849 vagrawal@eng.auburn.edu http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr08/course.html ELEC 7770: Advanced VLSI Design (Agrawal)

  2. Retiming • Retiming is a function-preserving transformation of a synchronous sequential circuit. • Flip-flops are moved according to specific rules. • Original references: • C. E. Leiserson, F. Rose and J. B. Saxe, “Optimizing Synchronous Circuits by Retiming,” Proc. 3rd Caltech Conf. on VLSI, 1983, pp. 87-116. • C. E. Leiserson and J. B. Saxe, “Retiming Synchronous Circuitry,” Algorithmica, vol. 6, pp. 5-35, 1991. ELEC 7770: Advanced VLSI Design (Agrawal)

  3. A Trivial Example: Reduced Hardware FF FF FF ELEC 7770: Advanced VLSI Design (Agrawal)

  4. Example 2: Faster Clock FF FF ELEC 7770: Advanced VLSI Design (Agrawal)

  5. Example 3: Reduced Flip-Flops FF FF FF ELEC 7770: Advanced VLSI Design (Agrawal)

  6. Applications of Retiming • Performance optimization • Area optimization • Power optimization • Testability enhancement • FPGA optimization ELEC 7770: Advanced VLSI Design (Agrawal)

  7. Fundamental Operation of Retiming • A retiming move in a circuit is caused by moving all of the memory elements at the input of a combinational block to all of its outputs, or vice-versa. FF Combinational logic Combinational logic FF ≡ FF ELEC 7770: Advanced VLSI Design (Agrawal)

  8. A Correlator Circuit Adder delay = 7 + + + PO host PI = = = = a1 a2 a3 a4 Comparator delay = 3 Flip-flop ELEC 7770: Advanced VLSI Design (Agrawal)

  9. Graph Model f e g 0 0 7 7 7 0 0 0 0 0 0 h 1 3 3 3 3 1 1 1 a b c d Vertex, vi, combinational, delay = d(vi), assumed unchanged by retiming d(host) = 0 Edge, e(vi,vj) or eij, weight wij = number of flip-flops between vi and vj ELEC 7770: Advanced VLSI Design (Agrawal)

  10. Path Delay and Path Weight • A set of connected nodes specify a path. A path does not traverse through the host node. • Path delay = ∑ d(vi) = combinational delay of path • Path weight = ∑ wij = clock delay of path • Retiming of a node i is denoted by an integer ri • It represents the number of registers moved across, initially ri = 0 • Register moved from output to input, ri → ri + 1 • Register moved from input to output, ri → ri – 1 • After retiming, edge weight wij’ = wij + rj – ri ELEC 7770: Advanced VLSI Design (Agrawal)

  11. Example of Node Retiming r1 = 0 r2 = 0 r3 = 0 r4 = 0 r5 = 0 r6 =0 3 3 3 3 3 3 ∑ d(vi) = 12, ∑ wij = 0 r1 = 0 r2 = -1 r3 = 0 r4 = 0 r5 = 1 r6 =0 3 3 3 3 3 3 ∑ d(vi) = 12, ∑ wij = 2 ELEC 7770: Advanced VLSI Design (Agrawal)

  12. Legal Retiming • Retiming is legal if the retimed circuit has no negative weights. • A legally retimed circuit is functionally equivalent to the original circuit – proof by Leiserson and Saxe (1991) • Retiming is the most general method for changing the register count and position without knowing the functions of vertices. ELEC 7770: Advanced VLSI Design (Agrawal)

  13. Example FF a c b x d c 1 0 x host 0 0 ELEC 7770: Advanced VLSI Design (Agrawal)

  14. Example: Illegal Retiming 0 0 c 1 c 1 → 0 0 0 x x host host 0 0 → –1 0 0 → –1 0 0 0 0 →1 Retiming vector = {0, 0, 0} Retiming vector = {0, 0, –1} a c FF x b d ELEC 7770: Advanced VLSI Design (Agrawal)

  15. Example: Legal Retiming 0 →1 0 1 → 0 c 1 c 0 0 →1 x x host host 0 0 0 0 0 0 0 0 Retiming vector = {0, 1, 0} Retiming vector = {0, 0, 0} FF a c FF b x d ELEC 7770: Advanced VLSI Design (Agrawal)

  16. Correlator Circuit Critical path delay = 24 f g e 0 0 7 7 7 re=0 0 rf=0 rg=0 0 0 0 0 0 h rh=0 1 3 3 3 3 1 1 1 rd=0 rb=0 rc=0 ra=0 a c b d Initial retiming vector = {0,0,0,0,0,0,0,0} ELEC 7770: Advanced VLSI Design (Agrawal)

  17. Retimed Correlator Circuit Critical path delay = 13 f g e 0→1 0→1 7 7 7 re= -2 0 rf= -1 rg=0 0 0 0 0→1 0 h rh=0 1→0 1→0 1 3 3 3 3 1 rd= -2 rb= -1 rc= -2 ra= -1 a c b d retiming vector = {-1,-1,-2,-2,-2,-1,0,0} ELEC 7770: Advanced VLSI Design (Agrawal)

  18. Retiming Theorem • Given a network G(V, E, W) and a cycle time T, (r1, . . . ) is a feasible retiming if and only if: • ri – rj ≤ wij for all edges (vi,vj) ε E • ri – rj ≤ W(vi,vj) – 1 for all node-pairs vi, vj such that D(vi,vj) > T Where, W(vi,vj): is the minimum weight for all paths between vi and vj D(vi,vj): is the maximum delay among all minimum weight paths between vi and vj ELEC 7770: Advanced VLSI Design (Agrawal)

  19. Proof of Condition 1 • We assume that the original network is legal, i.e., all edge weights are positive. • For an arbitrary edge (vi,vj) ε E: ri – rj ≤ wij or wij + rj – ri ≥ 0, means that after retiming the new weight wij’ = wij + rj – ri will be positive. Thus, condition 1 ensures the legality of retiming. ELEC 7770: Advanced VLSI Design (Agrawal)

  20. Proof of Condition 2 • Given: d(vi) < T, for all i. • Any retimed path whose combinational delay exceeds clock period, will have at least one flip-flop. • The above is the requirement for correct operation. rj flip-flops ri flip-flops Wij flip-flops i j Path (i,j), D(i,j) > T Original weight, Wij Retimed weight, Wij’ = Wij + rj – ri ≥ 1 ELEC 7770: Advanced VLSI Design (Agrawal)

  21. References • Two papers by Leiserson et al. (see slide 2). • G. De Micheli, Synthesis and Optimization of Digital Circuits, New York: McGraw-Hill, 1994. • N. Maheshwari and S. S. Sapatnekar, Timing Analysis and Optimization of Sequential Circuits, Boston: Springer, 1999. ELEC 7770: Advanced VLSI Design (Agrawal)

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