1. CEC 220 Digital Circuit Design
CEC 220 Digital Circuit Design Truth Tables, Incomplete Functions , & Full Adders
Wed, January 28
CEC 220 Digital Circuit Design

2. CEC 220 Digital Circuit Design
Lecture Outline
Examples of Truth Table Construction
Incompletely Specified Functions
Design of Adders and Subtractors
Wed, January 28
CEC 220 Digital Circuit Design

3. Examples of Truth Table Construction
Design a binary adder that adds two 1-bit binary numbers (A and B) to give a 2-bit sum (X Y) A B
X Y
0 0 1 A B
X Y 1 A B
X Y
0 0 1
0 1
1 0 A B
X Y
0 0 1
0 1 A B
X Y
0 0 1
0 1 + A X A
+ B
X Y B Y
This device is sometimes called a half-adder
Wed, January 28
CEC 220 Digital Circuit Design

4. Examples of Truth Table Construction
Design a binary adder that adds two UNSIGNED 2-bit binary numbers to form a 3-bit sum.
A B
+ C D
X Y Z
X= S m (7,10,11,13,14,15)
Y= S m (2,3,5,6,8,9,12,15)
Z= S m (1,3,4,6,9,11,12,14)
Wed, January 28
CEC 220 Digital Circuit Design

5. Incompletely Specified Functions
The four inputs to a circuit (A,B,C,D) represent an BCD digit. The output should be 1 iff the decimal number represented by the inputs is exactly divisible by three (i.e. a remainder of “0”).
Assume that only valid BCD digits occur as inputs.
These inputs will not occur.
Do I care what the associated outputs are?
Wed, January 28
CEC 220 Digital Circuit Design

6. Incompletely Specified Functions
Consider the truth table
If we make both don’t cares “0”
If we make 𝑑 1 a “1” and 𝑑 6 a “0”
If we make 𝑑 1 a “0” and 𝑑 6 a “1”
If we make 𝑑 1 a “1” and 𝑑 6 a “1” A B C 1 F 1 F 1 X F 1 F 1 F 1
We can select the “don’t cares” to make our expression simpler!!
Wed, January 28
CEC 220 Digital Circuit Design

7. Design of Adders and Subtracters A Binary Adder
Design a 1-bit Full-Adder A B
Cin
Cout S 1 A
+ B
Cout S
+ Cin
= Sum
Wed, January 28
CEC 220 Digital Circuit Design

8. Design of Adders and Subtracters A 4-bit Binary Adder
Design a 4-bit adder (S=A+B):
Considering 2’s complement arithmetic:
Ignore Cout3
Overflow
Set Cin0 = 0
1-bit Full Adder A3 B3 S3
Cin3
Cout3
1-bit Full Adder A2 B2 S2
Cin2
Cout2
1-bit Full Adder A1 B1 S1
Cin1
Cout1
1-bit Full Adder A0 B0 S0
Cin0
Cout0
Wed, January 28
CEC 220 Digital Circuit Design

9. Design of Adders and Subtracters
Our adder is “slow” as it propagates the carry
Can develop carry look-ahead circuitry
How can I do subtraction
Add A+(-B)
-B = Flip all of the bits and add “1”
1-bit Full Adder A3 B3 S3
Cin3
Cout3
1-bit Full Adder A2 B2 S2
Cin2
Cout2
1-bit Full Adder A1 B1 S1
Cin1
Cout1
1-bit Full Adder A0 B0 S0
Cin0
Cout0 =1
Wed, January 28
CEC 220 Digital Circuit Design

10. CEC 220 Digital Circuit Design
Next Lecture
Karnaugh Maps
Wed, January 28
CEC 220 Digital Circuit Design