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Adders and Magnitude Comparators

ENGI 251ELEC 241

Signed Binary Numbers

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You Must be able to do the Following:

• Form the two's complement of a hexadecimal integer• Convert signed binary to decimal• Convert signed decimal to binary• Convert signed decimal to hexadecimal• Convert signed hexadecimal to decimal

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Signed Integers• The highest bit indicates the sign. 1 = negative,

0 = positive

1 1 1 1 0 1 1 0

0 0 0 0 1 0 1 0

sign bit

Negative

Positive

If the highest digit of a hexadecmal integer is > 7, the value is negative. Examples: 8A, C5, A2, 9D

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Forming the Two's Complement• Negative numbers are stored in two's complement notation• Additive Inverse of any binary integer• Steps:

– Complement (reverse) each bit– Add 1

Note that 00000001 + 11111111 = 00000000

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Ranges of Signed Integers

The most significant bit (MSB) is reserved for the sign. This limits the range:

What is the largest positive value that may be stored in 20 bits?

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Adder• Signed Binary Numbers

– Conversion between signed binary and decimal• Performs the function of Binary Addition

– MSI Chip • Uses XOR for SUM• Two types

– Half Adder– Full Adder

• Performs Subtraction using 2’s Compliment Addition

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1-Bit Half Adder

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1-Bit Full Adder from 2 Half Adders

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Logic Diagram for 1-Bit Full Adder

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2-Bit Full Adder

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Truth Table for 2-Bit Full Adder

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Logic Diagram for 2-Bit Full Adder

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4-Bit Full Adder

P3P2P1P0Q3Q2Q1Q0

C0Σ3Σ2Σ1Σ0

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4-Bit Full Adder

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8-Bit Parallel Adder

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Full Adder with Look Ahead Carry

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Carry Propagate and Generate Illustration

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Carry Propagate and Generate Logic

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4-Bit Full Adder

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4 Stage Look Ahead Adder

Magnitude Comparators

ENGI 251ELEC 241

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Magnitude Comparator• Logic behavior similar to Adder

– Does not return Sum or Carry, but whether one value is larger, smaller, or equal

– Uses XOR• Returns result of comparison of two quantities stating

– Px < Qx– Px = Qx– Px > Qx

• Can return either one or all• Usually MSI

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Magnitude Comparator

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2-Bit Magnitude Comparator

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Example

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4-Bit Magnitude Comparator

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Logic Symbol for 4-Bit Magnitude Comparator

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Logic Diagram of 4-Bit Magnitude Comparator

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8-Bit Magnitude Comparator

Q = Q7 Q6 Q5 Q4 Q3 Q2 Q1 Q0

P = P7 P6 P5 P4 P3 P2 P1 P0