1. Digital Logic Review: Part II
ECE511: Digital System & Microprocessor

2. What we will learn in this session:
Negative number representation:
2’s Complement method.
Minimizing Boolean expressions:
K-map
Boolean algebra.
Comparison between Active High, and Active Low signals.

3. 2’s Complement Representation

4. 2’s Complement Used by M68k to represent negative numbers. Advantages:
Simple representation, conversion method.
Can perform arithmetic operations directly.
Can use existing circuits.
MSB regarded as sign bit.
If 0, then positive number.
If 1, then negative number.
Max value for 8-bits  -128 to 127

5. 2’s Complement +34 (decimal) 00100010 -54 (decimal) 11001010 Value
Sign bit
Value
Sign bit

6. Converting to 2’s Complement
1. Start with positive number
10 (decimal) = (binary)
2. Invert all the bits
(invert)
3. Add 1 to inverted result
2’s Complement (-10)

7. Converting 2’s Complement Back
Converting -5 to 5:
1. The 2’s complement representation:
-5 (decimal) = (binary)
2. Invert all the bits
(invert)
3. Add 1 to inverted result
Positive value (+5)

8. Max Value  +127 1 1 1 1 1 1 1
Sign bit
Value
-127 =

9. Min Value  -128 1
Sign bit
Value
+128 = ???

10. What about zero? Sign bit Value invert 1 1 1 1 1 1 1 1 +1
Sign bit
Value
invert 1 1 1 1 1 1 1 1 +1
1 carried out,
not counted.

11. Calculating the Maximum Range
Calculated using the following formula:
-(2n-1)< x < +(2n-1-1)
Where n is number of bits.

12. Example: Calculating 2’s Complement Range
What is the range for 32-bit in 2’s complement representation?
n = 32
-( ) < x < 232-1
-2,147,483,649 < x < 2,147,483,648

13. Active High vs. Active Low

14. Active High and Active Low
Some signals are active lows.
They are active when they are low.
Marked with bar:

15. Active High and Active Low
5 V is ACTIVE
0 V is INACTIVE
Active Low
5 V is INACTIVE
0 V is ACTIVE
TRUE
FALSE
TRUE
FALSE

16. Why are active low signals preferred in control?
More noise immunity:
All electronic circuits affected by noise.
Less likely to false trigger.
Not affected by voltage surge.

17. Noise Immunity Example: Resets system when active (pulled high). + =
False Triggering,
system reset accidentally. 5V 5V 5V + = 0V 0V 0V
Signal
Noise
New Signal

18. Noise Immunity Example: Resets system when active (pulled low). + =
No effect, since RESET is active low. 5V 5V 5V + = 0V 0V 0V
Signal
Noise
New Signal

19. Noise Immunity Example: Reads from memory when low,
writes to memory when high.
False Triggering, memory
contents will be lost. 5V 5V 5V + = 0V 0V 0V
Signal
Noise
New Signal

20. Noise Immunity Example: Reads from memory when high,
writes to memory when low.
No data lost even
when false triggering
occurred. 5V 5V 5V + = 0V 0V 0V
Signal
Noise
New Signal

21. Logic Minimization

22. Importance of Minimization
In electronics, we want:
Functionality – desired objective.
Minimal circuit area – wafers, board space.
Minimal cost – more IC, more $$$.
Maximum reliability – more components, more fail.
Logic minimization + good design achieves all this.
2 popular methods:
Boolan Algebra.
Karnaugh Map.

23. Boolean Algebra

24. Boolean Algebra Named after George Boole.
Based on set theory and algebra.
Application to electronics – C. Shannon.
Application to computers: J. V. Atanasoff.
Important in computer emergence.

25. Boolean Identities Set of fundamental rules:
Defines Boolean behaviors.
Mathematics and Set Theory.
Used for minimization.
No set guideline for minimization:
When to use what.
Depends on luck, experience.
Better to use K-Map.

26. Boolean Identities Identity Law Dominance Law Idempotent Law
Inverse Law
Commutative Law
Associative Law
Distributive Law
Absorption Law
De Morgan Law
Double Complement Law
*You don’t have to memorize all these, you just have to know about them. Just use K-Map for minimization.

27. Identity Law + Dominance Law
1x = x
0 + x = x
0x = 0
1 + x = 1

28. Idempotent Law + Inverse Law
xx = x
x + x = x
xx = 0
x + x = x

29. Commutative Law + Associative Law
xy = yx
x + y = y + x
(xy)z = x(yz)
(x + y) + z = x + (y + z)

30. Distributive Law
x + yz = (x + y) (x + z)
x(y + z) = xy + xz

31. Absorption Law
x (x + y) = x
x + xy = x

32. Double Complement Law
x = x

33. De Morgan’s Law A . B = A + B A + B = A . B A . B A + B . = + A + B
+ = .

34. De Morgan’s Law A . B = A + B A + B = A . B A . B A + B . = + A + B
+ = .

35. Example 1 Z = ABC + ABC + ABC Z = ABC + B(AC + AC) Z = ABC + BC(A + A)
Minimize: Z = ABC + ABC + ABC
Z = ABC + ABC + ABC
Z = ABC + B(AC + AC)
Z = ABC + BC(A + A)
Z = ABC + BC
Distributive Law
Distributive Law
Inverse Law

36. What if… Z = ABC + ABC + ABC Z = A(BC + BC) + ABC
Minimize: Z = ABC + ABC + ABC
Z = ABC + ABC + ABC
Distributive Law
Z = A(BC + BC) + ABC
Stuck here…
* Can never get the answer!

37. Karnaugh Map

38. Karnaugh Maps To simplify Boolean expressions.
Invented by Maurice Karnaugh.
Simpler than Boolean Algebra.
Principles:
Group together common factors.
Delete unwanted variables.
Works best for two to four variables.

39. Table Layout – 2 Variables1 B A

40. Table Layout – 3 Variables11 01 00 10 C AB 1

41. Table Layout – 4 Variables11 01 00 10 CD AB

42. How to Construct the K-Map
Analyze function, create Truth Table (TT).
Draw K-Map based on no. of variables.
Fill the K-Map with values from TT.
Group 1’s together.
Extract simplified expression.

43. × Rules √ 1. You must not miss any 1’s.11 01 00 10 CD AB 1 × √
1. You must not miss any 1’s.
2. You can go right-left or up-down, but you cannot go diagonal.

44. × Rules √ 3. You can only have 1, 2, 4, 8, …, 2n elements in a group.11 01 00 10 CD AB 1 √
3. You can only have
1, 2, 4, 8, …, 2n elements
in a group. 11 01 00 10 CD AB 1 ×

45. × Rules √ 4. Try to cover all 1’s using the minimum number of groups.11 01 00 10 CD AB 1
4. Try to cover all 1’s using the
minimum number of groups. . √ 11 01 00 10 CD AB 1 ×
2 groups
6 groups

46. Rules √ √ √ 5. Overlapping groups are allowed. AB AB CD CD 11 01 00 101
5. Overlapping groups are allowed. √ 11 01 00 10 CD AB 1 √ √

47. Rules √ √ 6. Wrap-around is allowed. CD AB CD AB Corner wrap Side wrap11 01 00 10 CD AB 1 11 01 00 10 CD AB 1 √ √
Corner wrap
Side wrap

48. Rules 11 01 00 10 CD AB 1 √
Top-down wrap

49. × Rules √ 6. Don’t cares (X) can be grouped with 1’s if they help. CD11 01 00 10 CD AB X 1 11 01 00 10 CD AB X 1 × √
Don’t cares can
help make the group larger
(1 group).
Not selecting don’t
cares (2 groups)

50. Extracting the Results
*All AB cancel out, only
C and D are left. AB 00 01 11 10 AB 00 CD 00 01 11 10 00 1 1 1 1 01 CD 11 1 1 10 1 1 AB
*A and D cancel out, only
B and C are left.
Answer: 01 11 11 CD 10

51. Example
Minimize this logic equation:

52. Solution A B C Z 1

53. Solution AB 00 01 11 10 1 1 1 C 1 1 1 1
Values of Z (from TT)

54. Solution AB 00 01 11 10 1 1 1 C 1 1 1 1

55. Solution 11 01 00 10 C AB 1

56. Example 1 Z = ABC + BC C Minimize: Z = ABC + ABC + ABC AB 11 01 00 10C A B C Z 1
Truth Table:
Z = ABC + BC

57. Example 2 Minimize: Z = ABCD + ABCD + ABCD + ABCD B C D Z 1 A1 A
Truth Table:

58. Example 2 11 01 00 10 CD AB 1
Z = ACD + ABCD + ABCD

59. Conclusion

60. Conclusion Active low signals are active when they are low.
2’s Complement represents negative numbers in µP.
Boolean Logic and K-Map minimize equations.
K-Map simpler, less errors.
Both should have same answers.

61. The End
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