1. Chapter 3 Combinational Logic Design

2. Combinational Circuits
(logic gates)
n binary
inputs
m binary
outputs
n input bits  2n possible binary input combinations
For each possible input combination, there is one possible output value
truth table
Boolean functions (with n input variables)
Examples: adders, subtractors, comparators, decoders, encoders, and multiplexers.

3. Possible Design Steps Find out the number of inputs and outputs
Derive the truth table that defines the required relationship between inputs and outputs
Obtain a simplified Boolean function for each output
Draw the logic diagram
Verify the correctness of the design

4. Example: Design Process
BCD-to-2421 Converter
Verbal specification:
Given a BCD digit (i.e. {0, 1, …, 9}), the circuit computes 2421 code equivalent of the decimal number
Step 1: how many inputs and how many outputs?
four inputs and four outputs
Step 2:
Obtain the truth table
0000  0000
1001  1111
etc.

5. BCD-to-2421 Converter
Truth Table
Inputs
Outputs A B C D x y z t 1

6. BCD-to-2421 Converter
Step 3: Obtain simplified Boolean expression for each output
Output x: A B C D x 1
The rest X 10 11 01 00 CD AB 1 x x x x x x
x = BD + BC + A

7. Boolean Expressions for Outputs
Output y: CD AB 00 01 11 10 1 X A B C D y z 1
The rest X
Output z: CD AB 00 01 11 10 1 X

8. Boolean Expressions for Outputs
Output t: CD AB 00 01 11 10 1 X A B C D T 1
The rest X
t = D
Step 4: Draw the logic diagram
x = BC + BD + A
y = A + BD’ + BC
z = A + B’C + BC’D

9. Example: Logic DiagramB
x = BC + BD + A C D
y = A + BD’ + BC
z = A + B’C + BC’D
t = D

10. Example: Verification
Step 5: Check the functional correctness of the logic circuit
Apply all possible input combinations
And check if the circuit generates the correct outputs for each input combinations
For large circuits with many input combinations, this may not be feasible.
Statistical techniques may be used to verify the correctness of large circuits with many input combinations

11. Design Example Specification BCD to Excess-3 code converter
Transforms BCD code for the decimal digits to Excess-3 code for the decimal digits
BCD code words for digits 0 through 9: 4-bit patterns 0000 to 1001, respectively
Excess-3 code words for digits 0 through 9: 4-bit patterns consisting of 3 (binary 0011) added to each BCD code word

12. Design Example (continued)
Formulation (Find truth table)
Conversion of 4-bit codes can be most easily formulated by a truth table
Variables - BCD: A,B,C,D
Variables - Excess-3 W,X,Y,Z
Don’t Cares - BCD to 1111

13. Design Example (continued)
W = A + BC + BD
X = C + D + B
Y = CD +
Z = B C D A 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 X w z y x B B C D C D D

14. Design Example (continued)
Technology Mapping A B C D W X Y Z