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EE121 John Wakerly Lecture #6

Adders

Multipliers

Read-Only Memories

Barrel-Shifter Design Example

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Equality Comparators

• 1-bit comparator

• 4-bit comparator

EQ_L

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8-bit Magnitude Comparator

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Other conditions

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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!)

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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 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

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Full-adder circuit

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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

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74x2834-bit adder

• Uses carry lookahead internally

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“generate”

“propagate”

“half sum”

carry-in from previous stage

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Ripple carry

between groups

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Lookahead carry between groups

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Addition in ABEL

• Easy? – No -- huge number of

product terms!

– On the order of 2n for bit n– @CARRY directive

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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.

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Full subtractor = full adder, almost

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Multipliers

• 8x8 multiplier

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4x4 multiplier in ABEL

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Full-adder array

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Faster carry chain

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Read-Only Memories

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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

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Logic-in-ROM example

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4x4 multiplier example

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Internal ROM

structure

PDP-11 boot ROM(64 words, 1024 diodes)

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Two-dimensional decoding

?

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Larger example, 32Kx8 ROM

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Today’s ROMs

• 256K bytes, 1M byte, or larger• Use MOS transistors

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EEPROMs, Flash PROMs

• Programmable and erasable using floating-gate MOS transistors

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Typical commercial EEPROMs

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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)

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Microprocessor EPROM application

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ROM control and I/O signals

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ROM timing

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Next time

• Sequential circuits (Chapter 7)