philadelphia university faculty of information technology ... simplification using...
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Philadelphia University
Faculty of Information Technology
Department of Computer Science
Computer Logic Design
By
Dareen Hamoudeh
Dareen Hamoudeh 1
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Simplification Using Map Method
Dareen Hamoudeh 2
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Why map method?
• Complex algebraic expression Complex Logic gates.
• Several algebraic expressions for same function.
• Function minimization using algebraic expression is awkward no specific rules to predict each step in the manipulative process.
• Map Method:
– Provides simple, straightforward procedure in minimizing functions.
Dareen Hamoudeh 3
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Map method (K-map)
• Also known as: – Veitch diagram.
– Karnaugh map.
• The Diagram made up of squares , each square represents one minterm.
• Represents visual diagram of all possible ways a function may expressed in standard form.
• We will assume: the simplest algebraic expression is any one in(SOP) or (POS) that has minimum numbers of literals.
Dareen Hamoudeh 4
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Two Variables Map
Dareen Hamoudeh 5
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Two Variables Map
• There are 4 minterms for two variables, so the map consists of 4 squares one for each minterm.
• We mark 0 and 1 for each row & column designate x and y:
X: primed in row 0.
Unprimed in row 1.
y: primed in col. 0.
Unprimed in col. 1.
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Two Variables Map
• We only mark the squares whose minterm belong to the given function.
• If we have F=x.y, it is equal to m3 ,because it is = 1 when x=1 and y=1. so, we place 1 inside the square that belong to m3:
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Two Variables Map
• If we have F=x+y, then its minterms are:
X+y=X’.y+x.y’+x.y=m1+m2+m3
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Three Variables Map
Dareen Hamoudeh 9
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Three Variables Map • There are 8 minterms.
• Map consists of 8 squares.
• Minterms are arranged in a sequence similar to reflected code.
• Only one bit changes from 1 to 0 or from 0 to 1 in the sequence.
• There are 4 squares where each variable =1, and 4 squares where each variable =0.
• We write the variable with its letter symbol under the four squares where it is unprimed.
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Map in simplification
• Basic property for adjacent squares in the map:
– Any two adjacent squares differ by only one variable: primed in square & unprimed in the other.
– EX:
In m5 & m7 : y is primed in m5 and unprimed in m7, from postulates m5+m7= xy’z+ xyz = xy(y’+y) = xy.
Sum of minterms in adjacent squares can simplified to a single AND term with 2 literals.
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Example 1
• Simplify the Boolean function using K-map
F=x’yz + x’yz’+ xy’z’+ xy’z Solution:
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Represents x’.y
Represents x.y’
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(m0 + m2) and (m4 + m6)
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• Solution:
Dareen Hamoudeh 15
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Four adjacent squares
Dareen Hamoudeh 16
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Example 2
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• Solution:
Dareen Hamoudeh 18
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Example 3
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• Solution:
Dareen Hamoudeh 20
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Most minimization example
Dareen Hamoudeh 21
F= Z’
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Most minimization example
Dareen Hamoudeh 22
F= x’+y
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Four Variables Map
Dareen Hamoudeh 23
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Four Variables Map
Dareen Hamoudeh 24
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Four Variables Map
• Like three-variable map: we minimize function using Adjacent squares property.
• In addition the map is considered to lie on surface with the top and bottom edges as well as the right and the left, for Example:
– m0 and m2 form adjacent squares.
– m3 and m11 form adjacent squares.
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Four Variables Map
• Combination of adjacent squares is easily determined:
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Example 1
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Solution:
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Dareen Hamoudeh 28
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Example 2
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• Solution:
Dareen Hamoudeh 30
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Self Study & Practice
Five Variables Map
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Five Variables Map
• Number of squares = number of minterms: 25 =32 • Rows & columns are numbered in reflected code
sequence. • There are 16 squares where each variable =1, and 16
squares where each variable =0. • As it consists of 2 four-variable maps. • Each four-variable maps is recognized from the double
line in the center: – Each retains the previously defined adjacency, individually. – In addition, the center lines considered as the center of a
book, with each half of the map being a page
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Represented as 2 four-variable map
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Five Variables Map
• When the book is closed, two adjacent squares will fall one in each other, beside its four neighboring squares.
• Example: m31 is adjacent to m30,m15,m29,m23 and m27
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Five Variables Map
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Solution
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NAND and NOR Implementation
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NAND and NOR Implementation
• Digital circuits are frequently constructed with only NAND or NOR gates.
– because these gates are easier to fabricate with electronic components.
• Because of the importance of NAND and NOR in the design of digital circuits.
– rules and procedures have been developed for the conversion from Boolean functions in terms of AND, OR and NOT into equivalent NAND or NOR logic diagrams.
• NAND and NOR are called universal gates.
– because any digital system or Boolean function can be implemented with only these gates.
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NAND and NOR Implementation
• Two-level implementation is presented here.
• There are two other graphic symbols for these gates, to facilitate conversions.
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NAND and NOR Implementation
• NAND equivalent symbols:
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NAND equivalent symbols
• Consists of an AND symbol followed by small circle.
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NAND equivalent symbols:
• OR symbol preceded by small circles in all the inputs.
• It follows DeMorgan’s theorem where small circles denote complementation.
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NAND and NOR Implementation
• NOR equivalent symbols:
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NOR equivalent symbols
• Consists of an OR symbol followed by small circle.
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NOR equivalent symbols:
• AND symbol preceded by small circles in all the inputs.
• It follows DeMorgan’s theorem where small circles denote complementation.
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NAND and NOR Implementation
• One-input NAND or NOR gate:
– Inverter.
• Three different graphic symbols for inverter:
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NAND and NOR Implementation
• NAND Simple Examples:
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NAND and NOR Implementation
• NOR Simple Examples:
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