1. Lecture 3 Combinational Circuits
CSCE 211 Digital Design
Lecture 3 Combinational Circuits
Topics
Basic Gates (3.1)
Adders: half-adder, full-adder
Gray Code (2.11)
N-cube (2.14)
Error Correcting codes (2.15)
Boolean Algebra (4.1)
Combinational-Circuit Analysis (4.2)
August 28, 2003

2. Overview Last Time On last Time’s Slides(what we didn’t get to) New
Conversion of Fractions decimal  base-r
Representations of Integers
Unsigned, signed magnitude, one’s complement, two’s complement
Arithmetic
BCD, representations of characters BCD, unicode
On last Time’s Slides(what we didn’t get to)
Basic gates (well almost on transparencies)
Adder circuits
New
Boolean algebra
Combinational circuits
VHDL
ButNot Table 4-36 page 277
History: George Boole

4. Basic Gates
AND OR
NOT

5. Basic Gates
NAND
NOR
XOR

6. Half Adder Circuit

7. Full Adder

8. Ripple Carry Adder

9. Gray Code
In some mechanical devices you want to encode positions as binary strings in such a way that positions close to each other are represented by strings that are close together.
Gray code – adjacent position differ in only one bit
000
001
011
010
110
111
101
100
Figure 2-6 encoding disk

10. Construction of Gray Codes
Gray Codes are reflective
Algorithm for construction of Gray Codes on n-bits
A 1-bit Gray code has two words 0 and 1.
The first 2n words of an (n+1)-bit gray code are the words of the n-bit gray code with a leading 0 added.
The next 2n words of an (n+1)-bit gray code are the words of the n-bit gray code but written in reverse order with a leading 1 added.

11. Construction of Gray Codes
1-bit gray code 1
2-bit gray code 00 01 11 10
3-bit gray code
0 00
0 01
0 11
0 10
1 10
1 11
1 01
1 00
4-bit gray code
0 000

12. Gray Code Generation Algorithm 2
The bits of an n-bit binary and n-bit Gray-code code word are numbered from right to left from 0 to n-1.
Bit i of a Gray-code word is 0 if bits i and i+1 of the corresponding binary code word are the same, else bit i is 1. (When i = n-1, bit n is considered to be 0.)
Example
Given
Add fake 0 on front bit n is 0 

13. Codes representing characters
ASCII (American Standard Code for Information Interchange) – 8 bits = 7 + parity
Unicode 16 bits

14. N-cubes
1-cube
2-cube
3-cube
Traversing in gray-code order

15. Parity Bits
Even parity bit – parity bit is set so that the number of ones is even
Odd parity bit

16. Error Correcting codes
For an n-bit code, consider the hypercube of dimension n
Choose some subset of the nodes as code words.
Suppose the distance between any two two code words is at least 3.
Now consider transmission errors.
Then if there is an error in transmitting just one bit then the distance from the received word to one code word is one, distances to other code words are at least two.
Single error correcting, double error detecting.
Such codes are called Hamming codes after their inventor Richard Hamming.

17. Boolean Algebra George Boole (1854) invented a two valued algebra
To “give expression … to the fundamental laws of reasoning in the symbolic language of a Calculus.”
1938 Claude Shannon at Bell Labs noted that this Boolean logic could be used to describe switching circuits. (Switching Algebra)
In Shannon’s view a relay had two positions open and closed and collections of relays satisfied the properties of Boolean algebra.

18. Boolean Algebra Axioms
The axioms of a mathematical system are a minimal set of properties that are assumed to be true.
Axioms of Boolean Algebra
X = 0 if X != X=1 if X!=0
If X = 0, then X’ = 1 if X=1 then X’=0
0 . 0 = = 1
1 . 1 = = 0
0 . 1 = = = = 1
Axioms 1-5 and 1-5’ completely define Boolean algebra.

19. Boolean Algebra Theorems
Proofs by perfect induction

20. More Theorems
N.B. T8¢, T10, T11

21. Duality
Swap 0 & 1, AND & OR
Result: Theorems still true
Why?
Each axiom (A1-A5) has a dual (A1¢-A5¢)
Counterexample: X + X × Y = X (T9) X × X + Y = X (dual) X + Y = X (T3¢) ????????????
X + (X × Y) = X (T9) X × (X + Y) = X (dual) (X × X) + (X × Y) = X (T8) X + (X × Y) = X (T3¢)
parentheses, operator precedence!

22. N-variable Theorems Prove using finite induction
Most important: DeMorgan theorems

23. Combinational Circuit Analysis
A combinational circuit is one whose outputs are a function of its inputs and only its inputs.
These circuits can be analyzed using:
Truth tables
Algebraic equations
Logic diagrams

24. Truth Tables