chapter3: gate-level minimization part 2 originally reham s. al-majed imam muhammad bin saud...
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Chapter3: Gate-Level MinimizationPart 2
Originally Reham S. Al-Majed
Imam Muhammad Bin Saud University
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Outline
Don’t-Care Conditions
NAND – NOR implementation.
Exclusive-OR Function
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Don’t Care Conditions
Minterms associated with a function specifies the conditions under
which the function is equal to 1.
The function is equal to 0 for the rest.
In some applications, the function is not specified for certain
combinations of the variables (e.g. BCD)
Incompletely specified functions: are functions that have
unspecified outputs for some input combinations.
Unspecified minterms of a function are called don’t-care conditions
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Don’t Care Conditions (cont.)
Don’t-care conditions can be used for further simplification.
Marked with an X in the map.
In choosing adjacent squares to simplify the function in a map,
the don’t-care minterms may be assumed to be either 0 or 1.
Example: Simplify the Boolean function:
F(w,x,y,z) = ∑(1,3,7,11,15)
d(w,x,y,z)= ∑(0,2,5)
F=yz+w’x’ F=yz+w’z
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NAND-NOR Implementation
Digital circuits uses NAND or NOR rather than AND and OR gates.
Easier to fabricate with electronic components and are the basic
gates used in all IC digital logic families.
They are called universal gates Any Boolean function can be implemented with them.
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NAND Circuit The NAND gate represents the complement of the AND operation
Abbreviation of Not AND
Two equivalent graphic symbols for the NAND gate:
Can implement any digital systems.
NOT Gate
AND Gate OR Gate
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NAND Circuit (Cont.)
The implementation of Boolean functions with NAND gate
requires that the function be in SoP form
Procedure : 1. Simplify and express the function in SoP.
2. Draw a NAND gate for each product term containing at least 2 literal. ( first-level gates)
3. Draw single NAND gate in the second level with outputs of first-level gates.
4. A term with a single literal requires inverter.
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Example
Implement the following functions with NAND gates
F= AB+CD
( , , ) (1,2,3,4,5,7)F x y z ( , , )F x y z xy x y z
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Multilevel NAND Circuits
If we choose to implement the function as a mixed notation
(without reducing the expression into a SoP form)
The procedure is1. to change every AND gate to an AND-invert graphic symbol
2. and every OR gate to an invert-OR symbol .
3. Make sure that there are two bubbles along the same line
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Example
Implement the following functions with NAND gates.
F = A(CD + B)+ B C’
F = (AB+ AB )(C+D)
(page 109-110)
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NOR Implementation
The NOR gate represents the complement of the OR operation
Abbreviation of Not OR
Two graphic symbols
Can implement any digital systems
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NOR Circuit (Cont.)
A two-level implementation with NOR gate requires that the
function be simplified into PoS form
Procedure : 1. Simplify and express the function in SoP.
2. Draw a NOR gate (OR-invert symbol) for each sum term containing at least 2 literal. ( first-level gates)
3. Draw single NOR gate (invert-AND symbol)in the second level with outputs of first-level gates.
4. A term with a single literal requires inverter.
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Example
Implement the following function with NOR gates.
F = (A + B)(C + D)E
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Multilevel NOR Circuits
The procedure for converting a multilevel AND-OR diagram to
an all-NOR diagram is1. to change every AND gate to an invert-AND graphic symbol
2. and every OR gate to an OR-invert symbol .
3. Make sure that there are two bubbles along the same line
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Example
Implement the following functions with NOR gates.
F = (AB+ AB )(C+D)
(page 111)
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(XOR) Exclusive-OR function
x 0 = x
x 1 = x
x x = 0
x x = 1
Commutative and associative
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XOR implementation
A two-input XOR function is constructed with conventional gates
Or with NAND gates
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Odd Function The multiple- variable XOR is an odd function: it is
equal to 1 if the inputs variables have an odd number of 1's.
Example : Three variables F = (1,2,4,7)
Figure 2.8 3-input XOR gate
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(XOR) Exclusive-OR function
Only a limited number of Boolean functions can be expressed in
terms of XOR operations. Nevertheless, this function is
particularly useful in arithmetic operations , error detection and
correction circuits.
Parity Generation and Checking (page num 120)
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Reading 3.1 3.2 3.3 3.5 3.6 3.7 3.9