CS231 (Spring 06) Review Session

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CS231 (Spring 06)Review Session

• Sangkyum Kim
• Feb 17, 2006

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Outline

• Addition

• Multiplication

• Subtraction

• MP Example

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A Complete ALU Circuit
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Binary addition by hand
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Adding two bits (half adder)
0 0 0 0 1 1 1 0 1 1 1 10
C XY S X Y X Y X ? Y
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Adding three bits
0 0 0 00 0 0 0 01 0 1 0 01 0
1 1 10 1 0 0 01 1 0 1 10 1 1
0 10 1 1 1 11
(These are the same functions from the decoder
and mux examples.)
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Full adder equations
S ?m(1,2,4,7) X Y Cin X Y Cin X
Y Cin X Y Cin X (Y Cin Y Cin) X
(Y Cin Y Cin) X (Y ? Cin) X (Y ?
Cin) X ? Y ? Cin Cout ?m(3,5,6,7) X Y
Cin X Y Cin X Y Cin X Y Cin (X Y X
Y) Cin XY(Cin Cin) (X ? Y) Cin XY
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Full adder circuit
S X ? Y ? Cin Cout (X ? Y) Cin XY
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A 4-bit adder
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An example of 4-bit addition
A1011 (eleven), B1110 (fourteen)
Woohoo! The final answer is 11001 (twenty-five).
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Hierarchical adder design
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Some other Issues

• Gate Delays
• Ripple Carry Adder
• Carry Lookahead Adder

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Quiz 8.4
For unsigned addition
Based on the 4-bit ripple adder figure. Addition
would be overflow if and only if _____

• CO 1

• (A3 B3 1) Or (A3 ? B3 1 C3 1)

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Outline

• Addition

• Multiplication

• Subtraction

• MP Example

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A 2x2 binary multiplier
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Outline

• Addition

• Multiplication

• Subtraction

• MP Example

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Converting signed numbers to decimal

• Convert 110101 to decimal, assuming this is a
  number in
• (a) signed magnitude format
• (b) 1s complement
• (c) 2s complement

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Comparing the signed number systems
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A twos complement subtraction circuit

• A B A B 1

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An adder-subtractor circuit
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Some other Issues

• Signed Overflow
• Sign Extension

• Going from 4-bit to 8-bit numbers
• 0101 (5) ? 0000 0101 (5)
• 1100 (-4) ? 1111 1100 (-4)

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Examples from the past Exams
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Examples from the past Exams
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Examples from the past Exams
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Examples from the past Exams
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Examples from the past Exams
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Examples from the past Exams
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Outline

• Addition

• Multiplication

• Subtraction

• MP Example

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MP Example - Problem

• Input

The input to your circuit will be the low 4
bits of the ASCII value for the character being
displayed. These values are shown as N3-N0 in the
truth table below. Since you are only
implementing letters A-J, you may assume that the
remaining six 4-bit patterns will never be used,
and so are don't care situations.
Character ASCII N3 N2 N1 N0
A 65 0 0 0 1
B 66 0 0 1 0

J 74 1 0 1 0
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MP Example - Problem

• Output

Your circuit will produce seven outputs, one for
each of the seven line segments. We'll name the
outputs T, U, V, W, X, Y, Z, as illustrated right.
A B C D E F G H
I J
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MP Example Truth Table
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MP Example K-Maps
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MP Example K-Map Simplification
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MP Example K-Map Simplification
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MP Example Two Level Implementation
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MP Example NAND Implementation
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MP Example Decoder Implementation
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MP Example Multiplexer Implementation
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THANK YOU!