1. CS M51A/EE M16 Winter’05 Section 1 Logic Design of Digital Systems Lecture 3
January 19
W’05
Yutao He
4532B Boelter Hall
CSM51A/EEM16-Sec.1 W’05
Y. 12/9/2018

2. Outline Administrative Matters Recap: Data presentation
Chapter , App. A
Switching Function
Switching Expression
Switching (Boolean) Algebra
Canonical Forms
Summary

3. Administrative Matter
HW #2 is posted
HW #1 Solution is posted
Quiz #1 will be held on Friday
About Lecture note posting

4. Review - Data Representation
Encode with binary number 0s and 1s
Character
ASCII
Unicode
Number (positive integer)
Positional Number Systems
Context is very important!
What’s the output?
int a = 12; printf (“a is %d”, a);
What if
char a = ‘2’; printf (“a is %d”, a);
How about
int a = 12; printf (“a is %x”, a);

5. Review - Code for Decimal Numbers
Binary-Coded-Decimal (BCD) Code:
Encode each decimal digit individually using binary codes
Example:
Encode decimal number sixteen (16):
Binary Code: 10000
BCD Code:

6. Binary-Level Specification
High-Level Spec FH XH YH
Binary-Level
Spec FB n m xB yB
Encoding
Decoding

7. Truth Table Tabular representation of switching functionsx y f
x y f
0 0 0
0 1 1
1 0 0
1 1 1 i 1 2 3
There are 2^n rows for a n-input switching function

8. 2-D Truth Table

9. Truth Table of 1-Input S. F.x f
X f f f f i 1 2 3
0-Constant
Identity
(Buffer)
Complement
(NOT)
1-Constant

10. Truth Table of 2-Input S. F.x y f
x y possible functions (f0–f15)
AND OR
XOR
NOR
EQU
NAND
How many different switching functions can be obtained from a n-input switching function? 2 n

11. Incompletely Specified S. F.
Output value is not specified for a given input combination: Don’t care

12. One/Zero Set Notations
One/Zero/D.C. Set Notations:
f (x,y,z) = one-set(1,4,7), d.c.-set(3,6) or
f(x,y,z) = zero-set(0,2,5), d.c.-set(3,6)

13. Switching (Boolean) Algebra
Symbols: 0, 1
• is logic AND, + is logic OR, ‘ is logic NOT
It holds all algebra axioms:
1. set B contains at least two elements, a, b, such that a  b 2. closure: a + b is in B a • b is in B 3. commutativity: a + b = b + a a • b = b • a 4. associativity: a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c 5. identity: a + 0 = a a • 1 = a 6. distributivity: a + (b • c) = (a + b) • (a + c) a • (b + c) = (a • b) + (a • c) 7. complementarity: a + a' = 1 a • a' = 0

14. Switching Expressions
Any switching function that can be expressed as a truth table can be written as an expression in Boolean algebra using operators: ', +, and •, i.e. Switching Expressions.
X Y X • Y
1 1 1
X Y X' X' • Y
X Y X' Y' X • Y X' • Y' ( X • Y ) + ( X' • Y' )

15. S. F and Logic Gates NOT X' X ~X AND X • Y XY X  Y OR X + Y X  Y
X Y Z
1 1 1 X Y Z
X Y Z
1 1 1 X Z Y

16. X xor Y = X Y' + X' Y X or Y but not both ("inequality", "difference")
Logic Gates (Cont.)
X Y Z
1 1 0
NAND
NOR
XOR X Y
XNOR X Z Y
X Y Z
1 1 0 X Z Y
X Y Z
1 1 0
X xor Y = X Y' + X' Y X or Y but not both ("inequality", "difference") X Z Y
X Y Z
1 1 1
X xnor Y = X Y + X' Y' X and Y are the same ("equality", "coincidence") X Z Y

17. Universal sets
Can we implement all switching functions from NOT, NOR, and NAND?
Univerval sets:
{NOT, AND, OR}
{NOR}
{NAND}

18. Axioms of Boolean Algebra
Identity X + 0 = X 1D. X • 1 = X
Null X + 1 = 1 2D. X • 0 = 0
Idempotency: X + X = X 3D. X • X = X
Involution: (X')' = X
Complementarity: X + X' = 1 5D. X • X' = 0
Commutativity: X + Y = Y + X 6D. X • Y = Y • X
Associativity: (X + Y) + Z = X + (Y + Z) 7D. (X • Y) • Z = X • (Y • Z)

19. Axioms of Boolean Algebra (Cont.)
Distributivity: X • (Y + Z) = (X • Y) + (X • Z) 8D. X + (Y • Z) = (X + Y) • (X + Z)
Uniting: X • Y + X • Y' = X 9D. (X + Y) • (X + Y') = X
Absorption: 10. X + X • Y = X 10D. X • (X + Y) = X 11. (X + Y') • Y = X • Y 11D. (X • Y') + Y = X + Y
Factoring: 12. (X + Y) • (X' + Z) = 12D. X • Y + X' • Z = X • Z + X' • Y (X + Z) • (X' + Y)
Concensus: 13. (X • Y) + (Y • Z) + (X' • Z) = D. (X + Y) • (Y + Z) • (X' + Z) = X • Y + X' • Z (X + Y) • (X' + Z)

20. De Morgan’s Law
de Morgan's Law: 14. (X + Y )' = X' • Y' 14D. (X • Y )' = X' + Y'
establishes relationship between • and +
also called “bubble logic” X y X y

21. Equivalent Switching Expressions
Two switching expressions are equivalent if they represent the same switching function:
Have the same number of inputs variables
Output is same for all combinations of input values
Example:

22. Equivalence Proof Show that E1 and E2 are equivalent:
E1 = bc+cb’+ac
E2 = c
Show that a+a’b = a+b
Show that xyz+x’y+xyz’ = y

23. Canonical Forms
Truth table is the unique signature of a switching function
Many alternative switching expressions may have the same truth table - equivalence
Canonical forms
Standard forms for a switching expression
Provides a unique algebraic signature

24. Sum Of Products (S.O.P.) Also known as sum of minterms
Corresponding to two-level AND-OR networks
F =
+ ABC
A'B'C
+ ABC'
+ A'BC
+ AB'C
F =
A B C F F'
F' = A'B'C' + A'BC' + AB'C'

25. Sum Of Products (Cont.) Minterm
ANDed product of literals – input combination for which output is true
Each variable appears exactly once, in true or inverted form (but not both)
F in canonical form: F(A, B, C) = m(1,3,5,6,7) = m1 + m3 + m5 + m6 + m7
= A'B'C + A'BC + AB'C + ABC' + ABC
canonical form  minimal form F(A, B, C) = A'B'C + A'BC + AB'C + ABC + ABC'
= (A'B' + A'B + AB' + AB)C + ABC'
= ((A' + A)(B' + B))C + ABC' = C + ABC' = ABC' + C
= AB + C
A B C minterms A'B'C' m0
0 0 1 A'B'C m1
0 1 0 A'BC' m2
0 1 1 A'BC m3
1 0 0 AB'C' m4
1 0 1 AB'C m5
1 1 0 ABC' m6
1 1 1 ABC m7
m-Notation
short-hand notation for minterms of 3 variables

26. Product of Sums Also known as product of maxterm
Corresponds to two-level OR-AND networks
F = F =
(A + B' + C)
(A' + B + C)
(A + B + C)
A B C F F'
F' = (A + B + C') (A + B' + C') (A' + B + C') (A' + B' + C) (A' + B' + C')

27. Product of Sums (Cont.) Maxterm
ORed sum of literals – input combination for which output is false
each variable appears exactly once, in true or inverted form (but not both)
A B C maxterms A+B+C M0
0 0 1 A+B+C' M1
0 1 0 A+B'+C M2
0 1 1 A+B'+C' M3
1 0 0 A'+B+C M4
1 0 1 A'+B+C' M5
1 1 0 A'+B'+C M6
1 1 1 A'+B'+C' M7
F in canonical form: F(A, B, C) = M(0,2,4) = M0 • M2 • M4
= (A + B + C) (A + B' + C) (A' + B + C)
canonical form  minimal form F(A, B, C) = (A + B + C) (A + B' + C) (A' + B + C)
= (A + B + C) (A + B' + C)
(A + B + C) (A' + B + C)
= (A + C) (B + C)
M-Notation
short-hand notation for maxterms of 3 variables

28. Canonical form conversions
Minterm to maxterm conversion
Use maxterms whose indices do not appear in minterm expansion
e.g., F(A,B,C) = m(1,3,5,6,7) = M(0,2,4)
Maxterm to minterm conversion
Use minterms whose indices do not appear in maxterm expansion
e.g., F(A,B,C) = M(0,2,4) = m(1,3,5,6,7)
Minterm expansion of F to minterm expansion of F'
Use minterms whose indices do not appear
e.g., F(A,B,C) = m(1,3,5,6,7) F'(A,B,C) = m(0,2,4)
Maxterm expansion of F to maxterm expansion of F'
Use maxterms whose indices do not appear
e.g., F(A,B,C) = M(0,2,4) F'(A,B,C) = M(1,3,5,6,7)

29. An Example: Binary coded decimal increment by 1
BCD digits encode decimal digits 0 – 9 in bit patterns 0000 – 1001
A B C D W X Y Z
X X X X
X X X X
X X X X
X X X X
X X X X
X X X X
0-set of W
1-set of W
don't care (DC) set of W
these inputs patterns should never be encountered in practice – "don't care" about associated output values, can be exploited in minimization

30. An Example - Cont.
Canonical representations of the BCD increment by 1 function:
Z = m0 + m2 + m4 + m6 + m8 + d10 + d11 + d12 + d13 + d14 + d15
Z =  [ m(0,2,4,6,8) + d.c.(10,11,12,13,14,15) ]
Z = M1 • M3 • M5 • M7 • M9 • D10 • D11 • D12 • D13 • D14 • D15
Z =  [ M(1,3,5,7,9) • D.C.(10,11,12,13,14,15) ]

31. Summary Truth tables and switching functions
NOT, AND, OR, NAND, NOR, XOR
Axioms and theorems of Boolean algebra
Proofs by re-writing
Canonical forms
SOP, POS, m/M-notation, 1/0/dc-sets
Expression equivalence
Incompletely specified Functions

32. Next Lecture
Chapter 3 – Combinational IC’s