APPLICATIONS OF BOOLEAN ALGEBRA MINTERM AND MAXTERM ...
APPLICATIONS OF BOOLEAN ALGEBRA MINTERM AND MAXTERM EXPANSIONS – Module 4 MULTIPLEXERS AND DECODERS – Module 9
A. B. C. D.
Combinational Logic Design Using a Truth Table Minterm and Maxterm Expansions General Minterm and Maxterm Expansions Incompletely Specified Functions
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Conversion of English Sentences to Boolean Equations The three main steps in designing a single-output combinational switching circuit are 1. Find a switching function that specifies the desired behavior of the circuit. 2. Find a simplified algebraic expression for the function.
3. Realize the simplified function using available logic elements.
Combinational Logic Design using a Truth Table Suppose we want the output of a circuit to be f = 1 if N ≥ 0112 and f = 0 if N < 0112. the truth table is:
Then
Next, we will derive an algebraic expression for f from the truth table by using the combinations of values of A, B, and C for which f = 1. For example, the term A′BC is 1 only if A = 0, B = 1, and C = 1. Finding all terms such that f = 1 and ORing them together yields:
f = A′BC + AB′C′ + AB′C + ABC′ + ABC
(4-1)
The equation can be simplified by first combining terms and then eliminating A′:
f = A′BC + AB′ + AB = A′BC + A = A + BC This equation leads directly to the following circuit:
(4-2)
Instead of writing f in terms of the 1’s of the function, we may also write f in terms of the 0’s of the function. Observe that the term A + B + C is 0 only if A = B = C = 0. ANDing all of these ‘0’ terms together yields:
f = (A + B + C)(A + B + C′)(A + B′ + C)
(4-3)
By combining terms and using the second distributive law, we can simplify the equation:
f = (A + B + C)(A + B + C′)(A + B′ + C) f = (A + B)(A + B′ + C) = A + B(B′ + C) = A + BC
(4-3) (4-4)
Minterm and Maxterm Expansions f = A′BC + AB′C′ + AB′C + ABC′ + ABC (4-1) Each of the terms in Equation (4-1) is referred to as a minterm. In general, a minterm of n variables is a product of n literals in which each variable appears exactly once in either true or complemented form, but not both. (A literal is a variable or its complement)
Section 4.3 (p. 93)
Table 4-1 Minterms and Maxterms for Three Variables
Minterm expansion for a function is unique. Equation (4-1) can be rewritten in terms of m-notation as:
f = A′BC + AB′C′ + AB′C + ABC′ + ABC (4-1) f (A, B, C) = m3 + m4 + m5 + m6 + m7
(4-5)
This can be further abbreviated by listing only the decimal subscripts in the form: f (A, B, C) = Ʃ m(3, 4, 5, 6, 7)
(4-5)
Minterm Expansion Example
Find the minterm expansion of f(a,b,c,d) = a'(b' + d) + acd'.
Section 4.3 (p. 95)
Design a comparator • Design a circuit returns 1 if number (a1 a0)2 >= (b1 b0)2. • We assume that the numbers are > 0 • Build the truth table of the comparator • Write the minterm expansion formula • Write the maxterm expansion formula
Table 4-2. General Truth Table for Three Variables Table 4-2 represents a truth table for a general function of three variables. Each ai is a constant with a value of 0 or 1.
General Minterm and Maxterm Expansions We can write the minterm expansion for a general function of three variables as follows:
The maxterm expansion for a general function of three variables is:
Section 4.4 (p. 97)
Conversion of Forms
QUIZ • Let’s assume that f( A, B, C) = ∑ m(0, 1, 2, 3); g( A, B, C) = ∏M(0, 1, 6, 7). What is the minimal form of h = f g A)A’B B)A’ + B’ C)AB + A’B’ D)B’C’ E)None of the above
QUIZ • Let’s assume that f( A, B, C) = ∑ m(0, 3, 4, 5); g( A, B, C) = ∏M(1, 2, 3, 6). What is the minimal (simplified) form of h = f XOR g A)A’BC B)ABC C)BC D)B’C’ E)None of the above
Incompletely Specified Functions A large digital system is usually divided into many subcircuits. Consider the following example in which the output of circuit N1 drives the input of circuit N2:
Truth Table with Don't Cares
Let us assume the output of N1 does not generate all possible combinations of values for A, B, and C. In particular, we will assume there are no combinations of values for w, x, y, and z which cause A, B, and C to assume values of 001 or 110.
The minterm expansion for Table 4-5 is:
The maxterm expansion for Table 4-5 is:
Table 4-5
Example of Incomplete Specified Function • Let’s assume that we want do display the decimal equivalent of a 4 bit number. • For this we have available – two LED display modules. – two BCD drivers for the LED modules – a module that converts the 4 bit number into 2 decimal digits represented using 8421 – BCD code
Example of Incomplete Specified Function LED g Digit 1 d3 d2 d1 d0
a b
LED b
…
a0
b31 b21 b11 b01
Display Driver
a2 a1
g
LED a d3 d2 d1 d0
a
b b
…
b30 b20 b10 b00
Display Driver
4 bit – 2 BCD digits converter
4 bit binary number
a3
g a
0,
Derive the truth tables for b3 …, b0 and a. 0,
Digit 0
Multiplexers A multiplexer has a group of data inputs and a group of control inputs used to select one of the data inputs and connect it to the output terminal. Z = A′I0 + AI1
2-to-1 Multiplexer and Switch Analog
Figure 9-1:
8-to-1 MUX equation: Z = A′B′C′I0 + A′B′CI1 + A′BC′I2 + A′BCI3 AB′C′I4 + AB′CI5 + ABC′I6 + ABCI7 (9-2)
+
8-to-1 MUX equation: Z = A′B′C′I0 + A′B′CI1 + A′BC′I2 + A′BCI3 AB′C′I4 + AB′CI5 + ABC′I6 + ABCI7 (9-2)
+
Implement f(A, B, C) = ∑ m(0, 1, 2, 7) using an 8-to-1 MUX
Decoders The decoder is another commonly used type of integrated circuit. The decoder generates all of the minterms of the input variables. Exactly one of the output lines will be 1 for each combination of the values of the input variables.
Figure 9-13:
3-to-8 Line Decoder
Decoders Implement f(a, b, c) = ∑ m(0, 1, 2, 7) using a 3-to-8 lines decoder and one additional gate.
A
B
C
D
7442
(b) Block diagram
A 4-to-10 Line Decoder
A 4-to-10 Line Decoder
(c) Truth Table
Decoders Implement f(a, b, c) = ∑ m(0, 1, 2, 7) using a 7442 module and one single additional gate with 4 inputs.
A
B
C
7442
D
Conclusions • Functional specification in terms of Minterm and Maxterm expansion • Incompletely specified functions • Logic implementation using Multiplexers and Decoders
A. B. C. D.
Combinational Logic Design Using a Truth Table Minterm and Maxterm Expansions General Minterm and Maxterm Expansions Incompletely Specified Functions
Click the mouse to move to the next page. Use the ESC key to exit this chapter.
Conversion of English Sentences to Boolean Equations The three main steps in designing a single-output combinational switching circuit are 1. Find a switching function that specifies the desired behavior of the circuit. 2. Find a simplified algebraic expression for the function.
3. Realize the simplified function using available logic elements.
Combinational Logic Design using a Truth Table Suppose we want the output of a circuit to be f = 1 if N ≥ 0112 and f = 0 if N < 0112. the truth table is:
Then
Next, we will derive an algebraic expression for f from the truth table by using the combinations of values of A, B, and C for which f = 1. For example, the term A′BC is 1 only if A = 0, B = 1, and C = 1. Finding all terms such that f = 1 and ORing them together yields:
f = A′BC + AB′C′ + AB′C + ABC′ + ABC
(4-1)
The equation can be simplified by first combining terms and then eliminating A′:
f = A′BC + AB′ + AB = A′BC + A = A + BC This equation leads directly to the following circuit:
(4-2)
Instead of writing f in terms of the 1’s of the function, we may also write f in terms of the 0’s of the function. Observe that the term A + B + C is 0 only if A = B = C = 0. ANDing all of these ‘0’ terms together yields:
f = (A + B + C)(A + B + C′)(A + B′ + C)
(4-3)
By combining terms and using the second distributive law, we can simplify the equation:
f = (A + B + C)(A + B + C′)(A + B′ + C) f = (A + B)(A + B′ + C) = A + B(B′ + C) = A + BC
(4-3) (4-4)
Minterm and Maxterm Expansions f = A′BC + AB′C′ + AB′C + ABC′ + ABC (4-1) Each of the terms in Equation (4-1) is referred to as a minterm. In general, a minterm of n variables is a product of n literals in which each variable appears exactly once in either true or complemented form, but not both. (A literal is a variable or its complement)
Section 4.3 (p. 93)
Table 4-1 Minterms and Maxterms for Three Variables
Minterm expansion for a function is unique. Equation (4-1) can be rewritten in terms of m-notation as:
f = A′BC + AB′C′ + AB′C + ABC′ + ABC (4-1) f (A, B, C) = m3 + m4 + m5 + m6 + m7
(4-5)
This can be further abbreviated by listing only the decimal subscripts in the form: f (A, B, C) = Ʃ m(3, 4, 5, 6, 7)
(4-5)
Minterm Expansion Example
Find the minterm expansion of f(a,b,c,d) = a'(b' + d) + acd'.
Section 4.3 (p. 95)
Design a comparator • Design a circuit returns 1 if number (a1 a0)2 >= (b1 b0)2. • We assume that the numbers are > 0 • Build the truth table of the comparator • Write the minterm expansion formula • Write the maxterm expansion formula
Table 4-2. General Truth Table for Three Variables Table 4-2 represents a truth table for a general function of three variables. Each ai is a constant with a value of 0 or 1.
General Minterm and Maxterm Expansions We can write the minterm expansion for a general function of three variables as follows:
The maxterm expansion for a general function of three variables is:
Section 4.4 (p. 97)
Conversion of Forms
QUIZ • Let’s assume that f( A, B, C) = ∑ m(0, 1, 2, 3); g( A, B, C) = ∏M(0, 1, 6, 7). What is the minimal form of h = f g A)A’B B)A’ + B’ C)AB + A’B’ D)B’C’ E)None of the above
QUIZ • Let’s assume that f( A, B, C) = ∑ m(0, 3, 4, 5); g( A, B, C) = ∏M(1, 2, 3, 6). What is the minimal (simplified) form of h = f XOR g A)A’BC B)ABC C)BC D)B’C’ E)None of the above
Incompletely Specified Functions A large digital system is usually divided into many subcircuits. Consider the following example in which the output of circuit N1 drives the input of circuit N2:
Truth Table with Don't Cares
Let us assume the output of N1 does not generate all possible combinations of values for A, B, and C. In particular, we will assume there are no combinations of values for w, x, y, and z which cause A, B, and C to assume values of 001 or 110.
The minterm expansion for Table 4-5 is:
The maxterm expansion for Table 4-5 is:
Table 4-5
Example of Incomplete Specified Function • Let’s assume that we want do display the decimal equivalent of a 4 bit number. • For this we have available – two LED display modules. – two BCD drivers for the LED modules – a module that converts the 4 bit number into 2 decimal digits represented using 8421 – BCD code
Example of Incomplete Specified Function LED g Digit 1 d3 d2 d1 d0
a b
LED b
…
a0
b31 b21 b11 b01
Display Driver
a2 a1
g
LED a d3 d2 d1 d0
a
b b
…
b30 b20 b10 b00
Display Driver
4 bit – 2 BCD digits converter
4 bit binary number
a3
g a
0,
Derive the truth tables for b3 …, b0 and a. 0,
Digit 0
Multiplexers A multiplexer has a group of data inputs and a group of control inputs used to select one of the data inputs and connect it to the output terminal. Z = A′I0 + AI1
2-to-1 Multiplexer and Switch Analog
Figure 9-1:
8-to-1 MUX equation: Z = A′B′C′I0 + A′B′CI1 + A′BC′I2 + A′BCI3 AB′C′I4 + AB′CI5 + ABC′I6 + ABCI7 (9-2)
+
8-to-1 MUX equation: Z = A′B′C′I0 + A′B′CI1 + A′BC′I2 + A′BCI3 AB′C′I4 + AB′CI5 + ABC′I6 + ABCI7 (9-2)
+
Implement f(A, B, C) = ∑ m(0, 1, 2, 7) using an 8-to-1 MUX
Decoders The decoder is another commonly used type of integrated circuit. The decoder generates all of the minterms of the input variables. Exactly one of the output lines will be 1 for each combination of the values of the input variables.
Figure 9-13:
3-to-8 Line Decoder
Decoders Implement f(a, b, c) = ∑ m(0, 1, 2, 7) using a 3-to-8 lines decoder and one additional gate.
A
B
C
D
7442
(b) Block diagram
A 4-to-10 Line Decoder
A 4-to-10 Line Decoder
(c) Truth Table
Decoders Implement f(a, b, c) = ∑ m(0, 1, 2, 7) using a 7442 module and one single additional gate with 4 inputs.
A
B
C
7442
D
Conclusions • Functional specification in terms of Minterm and Maxterm expansion • Incompletely specified functions • Logic implementation using Multiplexers and Decoders
