chapter 7 arithmetic operations and circuits 1. 7-4 hexadecimal arithmetic 4 binary bits represent a...
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Chapter 7
Arithmetic Operations and Circuits
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7-4 Hexadecimal Arithmetic
• 4 binary bits represent a single hexadecimal digit
• Addition– Add the digits in decimal– If sum is less than 16, convert to hexadecimal– Is sum is more than 16, subtract 16, convert to
hexadecimal and carry 1 to the next-more-significant column
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Example 7-12
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Hexadecimal Arithmetic
• Subtraction– When you borrow, the borrower increases by 16– See example 7-15
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Example 7-15
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7-5 BCD Arithmetic
• Group 4 binary digits to get combinations for 10 decimal digits
• Range of valid numbers 0000 to 1001• Addition
– Add as regular binary numbers– If sum is greater than 9 or if carry out
generated:• Add 6 (0110) saving any carry out
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7-6 Arithmetic Circuits• Only two inputs are of concern in the LSB
column.• More significant columns must include the
carry-in from the previous column as a third input.
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Arithmetic Circuits• The addition of the third input (Cin) is shown in
the truth table below.
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Arithmetic Circuits• Half-Adder
– No carry in (LSB column)
– The 0 output is HIGH when A or B, but not both, is high.
• Exclusive-OR function
– Cout is high when A and B are high.• AND function
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Arithmetic Circuits• The half-adder can also be implemented
using NOR gates and one AND gate.– The NOR output is Ex-OR.– The AND output is the carry.
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Arithmetic Circuits
• Full-Adder– Provides for a carry input
– The 1 output is high when the 3-bit input is odd.
• Even parity generator
– Cout is high when any two inputs are high.• 3 AND gates and an OR
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Arithmetic Circuits
• Full-adder sum from an even-parity generator
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Arithmetic Circuits
• Full-adder carry out function
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Arithmetic Circuits
• Logic diagram of a complete full-adder
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Arithmetic Circuits
• Block diagrams of a half-adder (HA) and a full adder (FA).
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Arithmetic Circuits
• Block diagram of a 4-bit binary adder
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7-7 Four-Bit Full-Adder ICs
• Four full-adders in a single package• Will add two 4-bit binary words plus one
carry input bit.
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Four-Bit Full-Adder ICs
• Functional diagram of the 7483
• Note that some manufacturers label inputs A0B0 to A1B3
• The carry-out is internally connected to the carry-in of the next full-adder.
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Four-Bit Full-Adder ICs
• Logic diagram for the 7483.
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Four-Bit Full-Adder ICs
• Logic symbol for the 7483
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Four-Bit Full-Adder ICs
• Fast-look-ahead carry– Evaluates 4 low-order inputs– High-order bits added at same time– Eliminates waiting for propagation ripple
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7-9 System Design Application
• Two’s-Complement Adder/Subtractor Circuit
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System Design Application
• BCD Adder Circuit
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7-10 Arithmetic/Logic Units• The ALU is a multipurpose device• Available in LSI
package• 74181 (TTL)• 74HC181 (CMOS)• Mode Control input
– Arithmetic (M = L)– Logic (M = H)
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Arithmetic/Logic Units
• Function Select - selects specific function to be performed
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Summary
• The binary arithmetic functions of addition, subtraction, multiplication, and division can be performed bit-by-bit using several of the same rules of regular base 10 arithmetic.
• The two’s-complement representation of binary numbers is commonly used by computer systems for representing positive and negative numbers.
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Summary
• Two’s-complement arithmetic simplifies the process of subtraction of binary numbers.
• Hexadecimal addition and subtraction is often required for determining computer memory space and locations.
• When performing BCD addition a correction must be made for sums greater than 9 or when a carry to the next more significant digit occurs.
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Summary
• Binary adders can be built using simple combinational logic circuits.
• A half-adder is required for addition of the least significant bits
• A full-adder is required for addition of the more significant bits.
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Summary
• Multibit full-adder ICs are commonly used for binary addition and two’s-complement arithmetic.
• Arithmetic/logic units are multipurpose ICs capable of providing several different arithmetic and logic functions.
• The logic circuits for adders can be described in VHDL using integer arithmetic.
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Summary
• The Quartus II software provides 7400-series macrofunctions and a Library of Parameterized Modules (LPMs) to ease in the design of complex digital systems.
• Conditional assignments can be made using the IF-THEN-ELSE VHDL statements.
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