b.sc cs-ii-u-1.5 digital logic circuits, digital component
TRANSCRIPT
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Digital Logic Circuits, Digital Component and Data Representation
Course: B.Sc-CS-II Subject: Computer Organization And
Architecture Unit-1
1
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Basic Definitions
• Binary Operators– AND
z = x • y = x y z=1 if x=1 AND y=1
– ORz = x + y z=1 if x=1 OR y=1
– NOTz = x = x’ z=1 if x=0
• Boolean Algebra– Binary Variables: only ‘0’ and ‘1’ values– Algebraic Manipulation
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Boolean Algebra Postulates
• Commutative Lawx • y = y • x x + y = y + x
• Identity Elementx • 1 = x x + 0 = x
• Complementx • x’ = 0 x + x’ = 1
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Boolean Algebra Theorems• Duality– The dual of a Boolean algebraic expression is obtained
by interchanging the AND and the OR operators and replacing the 1’s by 0’s and the 0’s by 1’s.
– x • ( y + z ) = ( x • y ) + ( x • z )– x + ( y • z ) = ( x + y ) • ( x + z )
• Theorem 1– x • x = x x + x = x
• Theorem 2– x • 0 = 0 x + 1 = 1
Applied to a valid equation produces a valid equation
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Boolean Algebra Theorems
• Theorem 3: Involution– ( x’ )’ = x ( x ) = x
• Theorem 4: Associative & Distributive– ( x • y ) • z = x • ( y • z ) ( x + y ) + z = x + ( y + z )– x • ( y + z ) = ( x • y ) + ( x • z )
x + ( y • z ) = ( x + y ) • ( x + z )
• Theorem 5: De Morgan– ( x • y )’ = x’ + y’ ( x + y )’ = x’ • y’– ( x • y ) = x + y ( x + y ) = x • y
• Theorem 6: Absorption– x • ( x + y ) = x x + ( x • y ) = x
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Boolean Functions[1]
• Boolean ExpressionExample: F = x + y’ z
• Truth TableAll possible combinationsof input variables
• Logic Circuit
x y z F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1xyz
F
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Algebraic Manipulation
• Literal:A single variable within a term that may be complemented or not.
• Use Boolean Algebra to simplify Boolean functions to produce simpler circuits
Example: Simplify to a minimum number of literals
F = x + x’ y ( 3 Literals)
= x + ( x’ y )
= ( x + x’ ) ( x + y )
= ( 1 ) ( x + y ) = x + y ( 2 Literals)
Distributive law (+ over •)
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Complement of a Function
• DeMorgan’s Theorm
• Duality & Literal Complement
CBAF CBAF
CBAF
CBAF CBAF
CBAF
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Canonical Forms[1]
• Minterm– Product (AND function)– Contains all variables– Evaluates to ‘1’ for a
specific combination
Example
A = 0 A B C
B = 0 (0) • (0) • (0)
C = 01 • 1 • 1 = 1
A B C Minterm
0 0 0 0 m0
1 0 0 1 m1
2 0 1 0 m2
3 0 1 1 m3
4 1 0 0 m4
5 1 0 1 m5
6 1 1 0 m6
7 1 1 1 m7
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Canonical Forms[1]
• Maxterm– Sum (OR function)– Contains all variables– Evaluates to ‘0’ for a
specific combination
Example
A = 1 A B C
B = 1 (1) + (1) + (1)
C = 10 + 0 + 0 = 0
A B C Maxterm
0 0 0 0 M0
1 0 0 1 M1
2 0 1 0 M2
3 0 1 1 M3
4 1 0 0 M4
5 1 0 1 M5
6 1 1 0 M6
7 1 1 1 M7
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Canonical Forms[1]
• Truth Table to Boolean FunctionCBAF CBA CBA ABCA B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
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Canonical Forms• Sum of Minterms
• Product of Maxterms
ABCCBACBACBAF 7541 mmmmF
)7,5,4,1(F
CABBCACBACBAF
CABBCACBACBAF
CABBCACBACBAF
))()()(( CBACBACBACBAF 6320 MMMMF
(0,2,3,6)F
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Standard Forms• Sum of Products (SOP)
ABCCBACBACBAF BA
BA
CCBA
)1(
)(
AC
BBAC
)(
CB
AACB
)(
)()()( BBACCCBAAACBF
ACBACBF
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Standard Forms• Product of Sums (POS)
CABBCACBACBAF
)( CCBA
)( AACB
)( BBCA )()()( AACBCCBABBCAF
CBBACAF
))()(( CBBACAF
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Two - Level Implementations• Sum of Products (SOP)
• Product of Sums (POS)
B’C
FB’A
AC
AC
FB’A
B’C
ACBACBF
))()(( CBBACAF
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Logic Operators
• AND
• NAND (Not AND)
xy
x • y
xy
x • y
x y AND0 0 00 1 01 0 01 1 1
x y NAND0 0 10 1 11 0 11 1 0
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Logic Operators
• OR
• NOR (Not OR)
xy
x + y
xy
x + y
x y OR0 0 00 1 11 0 11 1 1
x y NOR0 0 10 1 01 0 01 1 0
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Logic Operators
• XOR (Exclusive-OR)
• XNOR (Exclusive-NOR) (Equivalence)
xy
x Å yx y + x y
xy
x Å yx � y
x y + x y
x y XOR0 0 00 1 11 0 11 1 0
x y XNOR0 0 10 1 01 0 01 1 1
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Logic Operators
• NOT (Inverter)
• Buffer
x x
x x
x NOT
0 1
1 0
x Buffer
0 0
1 1
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Multiple Input Gates
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De Morgan’s Theorem on Gates
• AND Gate– F = x • y F = (x • y) F =
x + y
• OR Gate– F = x + y F = (x + y) F =
x • y
Change the “Shape” and “bubble” all lines
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References
1. Computer Organization and Architecture, Designing for performance by William Stallings, Prentice Hall of India.
2. Modern Computer Architecture, by Morris Mano, Prentice Hall of India.
3. Computer Architecture and Organization by John P. Hayes, McGraw Hill Publishing Company.
4. Computer Organization by V. Carl Hamacher, Zvonko G. Vranesic, Safwat G. Zaky, McGraw Hill Publishing Company.