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Adders, Multipliers, and Read-Only Memories in ABEL: Design Examples and Comparators

Learn about designing adders and multipliers, and explore the functionality of read-only memories in ABEL language. Understand comparator design examples, including 4-bit comparators and equality checking mechanisms.

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Adders, Multipliers, and Read-Only Memories in ABEL: Design Examples and Comparators

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  1. EE121 John Wakerly Lecture #6 Adders Multipliers Read-Only Memories Barrel-Shifter Design Example

  2. 4-bit comparator EQ_L Equality Comparators • 1-bit comparator

  3. 8-bit Magnitude Comparator

  4. Other conditions

  5. Comparators in ABEL • Equality checking • PEQQ = ([P7..P0] == [Q7..Q0]); • PEQQ_L = !([P7..P0] == [Q7..Q0]); • 16 product terms • Magnitude comparison • PGTQ = ([P7..P0] > [Q7..Q0]); • PGTQ_L = !([P7..P0] > [Q7..Q0]); • 255 product terms (try it in Foundation!)

  6. X Y Cin S Cout 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 Adders • Basic building block is “full adder” • 1-bit-wide adder, produces sum and carry outputs • Truth table:

  7. Full-adder circuit

  8. Ripple adder • Speed limited by carry chain • Faster adders eliminate or limit carry chain • 2-level AND-OR logic ==> 2n product terms • 3 or 4 levels of logic, carry lookahead

  9. 74x2834-bit adder • Uses carry lookahead internally

  10. “generate” “half sum” carry-in from previous stage “propagate”

  11. Ripple carry between groups

  12. Lookahead carry between groups

  13. Addition in ABEL • Easy? • No -- huge number of product terms! • On the order of 2n for bit n • @CARRY directive

  14. Subtraction • Subtraction is the same as addition of the two’s complement. • The two’s complement is the bit-by-bit complement plus 1. • Therefore, X – Y = X + Y + 1 . • Complement Y inputs to adder, set Cin to 1. • For a borrow, set Cin to 0.

  15. Full subtractor = full adder, almost

  16. Multipliers • 8x8 multiplier

  17. 4x4 multiplier in ABEL

  18. Full-adder array

  19. Faster carry chain

  20. Read-Only Memories

  21. Why “ROM”? • Program storage • Boot ROM for personal computers • Complete application storage for embedded systems. • Actually, a ROM is a combinational circuit, basically a truth-table lookup. • Can perform any combinational logic function • Address inputs = function inputs • Data outputs = function outputs

  22. Logic-in-ROM example

  23. 4x4 multiplier example

  24. Internal ROM structure PDP-11 boot ROM (64 words, 1024 diodes)

  25. Two-dimensional decoding ?

  26. Larger example, 32Kx8 ROM

  27. Today’s ROMs • 256K bytes, 1M byte, or larger • Use MOS transistors

  28. EEPROMs, Flash PROMs • Programmable and erasableusing floating-gate MOS transistors

  29. Typical commercial EEPROMs

  30. EEPROM programming • Apply a higher voltage to force bit change • E.g., VPP = 12 V • On-chip high-voltage “charge pump” in newer chips • Erase bits • Byte-byte • Entire chip (“flash”) • One block (typically 32K - 66K bytes) at a time • Programming and erasing are a lot slower than reading (milliseconds vs. 10’s of nanoseconds)

  31. Microprocessor EPROM application

  32. ROM control and I/O signals

  33. ROM timing

  34. Next time • Sequential circuits (Chapter 7)

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