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Arithmetic Circuits

Arithmetic Circuits. A Generic Digital Processor. 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.).

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Arithmetic Circuits

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  1. Arithmetic Circuits

  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

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