1. Boolean Logic 2 ©Paul Godin Created September 2007 Last edit Sept 2009
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2. Pushing a Signal
Signal Pushing is a technique of applying an input value and following its progression through a circuit.
This method is used extensively when analysing and troubleshooting circuits.

3. Truth Tables and Signal Pushing
A truth table can be derived from a circuit by signal pushing.
All possible input combinations are applied to determine the output of the circuit.

4. Example 1: Signal Pushing1 1 1 1 1 1 1 1 1 1
INPUT
OUTPUT A B W 1 1 1 1 1 1
Animated
Apply all input combinations, follow the logic through the circuit and complete the truth table.

5. Example 2: Signal Pushing
INPUT
OUTPUT A B C Y 1

6. Exercise 1: Signal Pushing
INPUT
OUTPUT A B C W 1
Apply all input combinations and complete the truth table.

7. Exercise 2: Signal Pushing
INPUT
OUTPUT A B C D Z 1

8. Sum-of-Products (SOP)

9. Boolean from the Truth Table
A Boolean equation derived from the truth table takes on a “Sum-of-Products” form.
Sum-of-Products are ANDed statements (products) that are ORed together (sum). Example: (A●B)+(C●D)
From the Truth Table, for all output values that equal “1”, the ANDed input values are written.
If the input value is “0”, the complimented input is indicated.

10. SOP from the Truth Table
This example demonstrates how the S.O.P. equation is determined.
INPUT
OUTPUT A B W 1
(A●B)+(A●B)=W

11. S.O.P. Simplification (A●B)+(A●B)=W B(A+A)=W B(1)=W B=W
Once the Boolean equation in S.O.P. form is determined, standard simplification rules are applied.
Example:
(A●B)+(A●B)=W
B(A+A)=W
B(1)=W
B=W

12. Word Problem
Often the designer of a digital logic circuit is given input and output parameters of the circuit’s logic in the form of a word problem.
Often the easiest way to begin the design process is to express the word problem in the form of a truth table.
The SOP is derived from the truth table, simplified and then the logic circuit can be designed.

13. Example: Word Problem to Truth Table
INPUT
OUTPUT A B C W 1
Problem: A control circuit will produce a logic high whenever two of its 3 inputs are high, or if all inputs are high.
Build a truth table from the word problem.

14. 2- Truth Table to a Boolean Expression
INPUT
OUTPUT A B C W 1
(ABC)+(ABC)+(ABC)+(ABC) = W

15. 3-Boolean Expression Simplified
(ABC)+(ABC)+(ABC)+(ABC) = W
(ABC)+(ABC)+(ABC)+(ABC)+(ABC)+(ABC) = W
BC(A+A)+AB(C+C)+AC(B+B) = W
BC+AB+AC = W

16. 4-Boolean to Circuit
BC+AB+AC = W

17. In-Class Exercise: Word problem to circuit implementation
Design a circuit that will produce a logic high when input A is high, or when B and C are high at the same time.

18. Product-of-Sums
A Boolean equation derived from the truth table can also take on a “Products-of-Sums” form if the “0” output is selected and DeMorgan applied.
Products-of-Sums are ORed statements that are ANDed together. Ex: (A+B)●(C+D)
POS are used in some applications.
An example is when a truth table has many logic 1 outcomes but few logic 0 outcomes.

19. POS from the Truth Table
This example demonstrates how the POS equation is determined.
INPUT
OUTPUT A B W 1
(A●B)+(A●B)=W (SOP form)
(A●B)+(A●B)=W
(A●B)●(A●B)=W (DeMorgan)
(A+B)●(A+B)=W
(POS form)

20. POS Simplification (A+B)●(A+B)=W AA+AB+AB+BB=W 0+AB+AB+B=W AB+(AB+B)=W
Once the Boolean equation in POS form is determined, standard simplification rules are applied.
Example:
(A+B)●(A+B)=W
AA+AB+AB+BB=W
0+AB+AB+B=W
AB+(AB+B)=W
AB+B=W
B=W

21. END
©Paul R. Godin
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