Digital Integrated Circuits A Design Perspective

1. Digital Integrated CircuitsA Design Perspective Jan M. Rabaey Anantha Chandrakasan Borivoje Nikolic Arithmetic Circuits January, 2003

2. A Generic Digital Processor

3. Building Blocks for Digital Architectures Arithmetic unit Bit-sliced datapath (adder, multiplier, shifter, comparator, etc.) - Memory - RAM, ROM, Buffers, Shift registers Control - Finite state machine (PLA, random logic.) - Counters Interconnect - Switches - Arbiters - Bus

4. An Intel Microprocessor Itanium has 6 integer execution units like this

5. Bit-Sliced Design

6. Bit-Sliced Datapath

7. Itanium Integer Datapath Fetzer, Orton, ISSCC’02

8. Adders

9. Full-Adder

10. The Binary Adder

11. Express Sum and Carry as a function of P, G, D Define 3 new variable which ONLY depend on A, B Generate (G) = AB Propagate (P) = A B Å Delete = A B S C D and P Can also derive expressions for and based on o Note that we will be sometimes using an alternate definition for + Propagate (P) = A B

12. The Ripple-Carry Adder Worst case delay linear with the number of bits td = O(N) tadder = (N-1)tcarry + tsum Goal: Make the fastest possible carry path circuit

13. Complimentary Static CMOS Full Adder 28 Transistors

14. Inversion Property

15. Minimize Critical Path by Reducing Inverting Stages Exploit Inversion Property

16. A Better Structure: The Mirror Adder

17. Mirror Adder Stick Diagram

18. The Mirror Adder • The NMOS and PMOS chains are completely symmetrical. A maximum of two series transistors can be observed in the carry-generation circuitry. • When laying out the cell, the most critical issue is the minimization of the capacitance at node Co. The reduction of the diffusion capacitances is particularly important. • The capacitance at node Co is composed of four diffusion capacitances, two internal gate capacitances, and six gate capacitances in the connecting adder cell . • The transistors connected to Ci are placed closest to the output. • Only the transistors in the carry stage have to be optimized for optimal speed. All transistors in the sum stage can be minimal size.

19. Transmission Gate Full Adder

20. Manchester Carry Chain

21. Manchester Carry Chain

22. Manchester Carry Chain Stick Diagram

23. Carry-Bypass Adder Also called Carry-Skip

24. Carry-Bypass Adder (cont.) tadder = tsetup + Mtcarry + (N/M-1)tbypass + (M-1)tcarry + tsum

25. Carry Ripple versus Carry Bypass

26. Carry-Select Adder

27. Carry Select Adder: Critical Path

28. Linear Carry Select

29. Square Root Carry Select

30. Adder Delays - Comparison

31. LookAhead - Basic Idea

32. Look-Ahead: Topology Expanding Lookahead equations: All the way:

33. Logarithmic Look-Ahead Adder

34. Carry Lookahead Trees Can continue building the tree hierarchically.

35. Tree Adders 16-bit radix-2 Kogge-Stone tree

36. Tree Adders 16-bit radix-4 Kogge-Stone Tree

37. Sparse Trees 16-bit radix-2 sparse tree with sparseness of 2

38. Tree Adders Brent-Kung Tree

39. Example: Domino Adder Propagate Generate

40. Example: Domino Adder Propagate Generate

41. Example: Domino Sum

42. Multipliers

43. The Binary Multiplication

44. The Binary Multiplication

45. The Array Multiplier

46. The MxN Array Multiplier— Critical Path Critical Path 1 & 2

47. Carry-Save Multiplier

48. Multiplier Floorplan

49. Wallace-Tree Multiplier

50. Wallace-Tree Multiplier