1 Adders and Magnitude Comparators ENGI 251 ELEC 241 Signed Binary Numbers
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Adders and Magnitude Comparators
ENGI 251ELEC 241
Signed Binary Numbers
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ENGI 251/ELEC 241 Multibit Adder/Comparator Design 3
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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ENGI 251/ELEC 241 Multibit Adder/Comparator Design 7
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
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