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EE121 John Wakerly Lecture #6 Adders Multipliers Read-Only Memories Barrel-Shifter Design Example 1 Equality Comparators • 1-bit comparator • 4-bit comparator EQ_L 2 8-bit Magnitude Comparator 3 Other conditions 4 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!) 5 Adders • Basic building block is “full adder” – 1-bit-wide adder, produces sum and carry outputs • Truth table: X Y Cin S Cout 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 1 6 Full-adder circuit 7 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 8 74x283 4-bit adder • Uses carry lookahead internally 9 “generate” “half sum” “propagate” carry-in from previous stage 10 Ripple carry between groups 11 Lookahead carry between groups 12 Addition in ABEL • Easy? – No -- huge number of product terms! – On the order of 2n for bit n – @CARRY directive 13 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. 14 Full subtractor = full adder, almost 15 Multipliers • 8x8 multiplier 16 4x4 multiplier in ABEL 17 Full-adder array 18 Faster carry chain 19 Read-Only Memories 20 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 21 Logic-in-ROM example 22 4x4 multiplier example 23 Internal ROM structure PDP-11 boot ROM (64 words, 1024 diodes) 24 Two-dimensional decoding ? 25 Larger example, 32Kx8 ROM 26 Today’s ROMs • 256K bytes, 1M byte, or larger • Use MOS transistors 27 EEPROMs, Flash PROMs • Programmable and erasable using floating-gate MOS transistors 28 Typical commercial EEPROMs 29 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) 30 Microprocessor EPROM application 31 ROM control and I/O signals 32 ROM timing 33 Next time • Sequential circuits (Chapter 7) 34