CPS3340 COMPUTER ARCHITECTURE Fall Semester, 2013 09/17/2013 Lecture 6: Binary Addition &...
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Transcript of CPS3340 COMPUTER ARCHITECTURE Fall Semester, 2013 09/17/2013 Lecture 6: Binary Addition &...
CPS3340 COMPUTER
ARCHITECTURE Fall Semester, 2013
CPS3340 COMPUTER
ARCHITECTURE Fall Semester, 2013
09/17/2013
Lecture 6: Binary Addition & Subtraction,
1-bit ALU
Instructor: Ashraf YaseenDEPARTMENT OF MATH & COMPUTER SCIENCECENTRAL STATE UNIVERSITY, WILBERFORCE, OH
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Review
Last Class Integrated Circuits Decoder, Multiplexor PLA, ROM Don’t Care, Bus
This Class Assignemnt2 Representation of Integer Addition & Subtraction 1-bit ALU
Next Class Quiz2 32-bit ALU
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Bit, Byte, and Word
1 Bit – 0 or 1 1 Byte – 8 bits 1 Word – N bytes (in general)
4 bytes in a word (in our book)
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Most Significant Bit and Least Significant Bit
Most Significant Bit (High-Order Bit) The bit position having the greatest value Usually the left-most bit
Least Significant Bit (Low-Order Bit) The bit position having the smallest value Usually the right-most bit
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Binary Representation of Decimal Number
Binary 1 0 0 1 0 1 0 1 1 0 1
Decimal
1×210 + 0×29 + 0×28 + 1×27 + 0×26 + 1×25 + 0×24 + 1×23 + 1×22 + 0×21 + 1×20 = 1197
Using a binary number to represent a decimal number
Example
What is the maximum number a byte can represent?
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Binary Representation of Integers Unsigned Integers
0 and positive integers only Signed Integers
0, negative, and positive integers Three ways
Sign-Magnitude 1’s Complement 2’s Complement
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Unsigned Integers
Unsigned Integers Consider a word = 4 bytes Can represent numbers from 0 to 4294967295Decimal:
0 to 232-1Binary:
0 to 11111111111111111111111111111111 Example671210 = 00000000 00000000 00011010
001110002
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Signed Integer – Sign Magnitude
Sign Magnitude Use the most significant bit of the word to represent
the sign 0 – Positive 1 – Negative
Rest of the number is encoded in magnitude part Example 671210 = 00000000 00000000 00011010 001110002
-671210 = 10000000 00000000 00011010 001110002
Two representations of 0 0 = 00000000 00000000 00000000 00000000
-0 = 10000000 00000000 00000000 00000000 Cumbersome in Arithmetic
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1’s Complement
1’s Complement Negative number is stored as bit-wise complement of
corresponding positive number Use the most significant bit of the word to represent the
sign 0 – Positive 1 – Negative
Example 671210 = 00000000 00000000 00011010 001110002
-671210 = 11111111 11111111 11100101 110001112
Still two representations of zero 0 = 00000000 00000000 00000000 00000000
-0 = 11111111 11111111 11111111 11111111
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2’s Complement
2’s Complement Positive number represented in the same way as
sign-magnitude and 1’s complement Negative number obtained by taking 1’s
complement of positive number and adding 1
671210 = 00000000 00000000 00011010 001110002
1’s comp: -671210 = 11111111 11111111 11100101 110001112
2’s comp: -671210 = 11111111 11111111 11100101 110010002
One version of 0 Convenient in arithmetic
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Integer Addition
Example: 7 + 600000000 00000000 00000000 00000111
+ 00000000 00000000 00000000 0000011000000000 00000000 00000000 00001101
§3.2 Addition and S
ubtraction
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Integer Subtraction
Subtraction is actually an addition Example: 7 – 6 = 7 + (-6) 2’s complement
00000000 00000000 00000000 00000111- 11111111 11111111 11111111 11111010
00000000 00000000 00000000 00000001
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Overflow
Overflow if result out of range Adding +value and –value operands, no
overflow Adding two +value operands
Overflow if result sign is 1 Adding two –value operands
Overflow if result sign is 0
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Arithmetic Logic Unit
Arithmetic Logic Unit (ALU) Heart of a CPU Operations
Arithmetic operations Addition Subtraction
Logical operations NOT AND OR
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1-bit Logical Unit for AND and OR
1-bit logical unit for AND and OR
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1-bit adder16
1-bit adder truth table
can express the output functions Carry Out and Sum as logical equations, and these equations can in turn be implemented with logic gates
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Simplifying 1-bit adder
If a and b and CarryIn are true, then the three other terms are true as well
can be simplified as
Values when CarryOut is true
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Logic of CarryOut Bit19
Logic of Sum Bit20
Overall 1-bit ALU21
Summary
Bit, Byte, Word Binary Representation of Integer Addition Subtraction Overflow 1-bit ALU
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What I want you to do
Review Appendix C and Class Slides Work on Assignment 2 Prepare for Quiz 2
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