in these slides you will learn about digital logic design ...

1. DIGITAL LOGIC DESIGN
LECTURE 03
BY-FARHAT ULLAH

2. SIGNED AND UNSIGNED
NUMBERS
Unsigned Binary Numbers
Signed Binary Numbers
Most significant bit represents sign
0 represents a positive number
1 represents a negative number

3. 2’S COMPLEMENT FORM
1’s complement form
2’s complement form
Binary number 01101 (13)
1’s complement 10010
+ 1
2’s complement 10011 (-13)

4. ADDITION AND SUBTRACTION
WITH 2’S COMPLEMENT
0101 +5 0101 +5
0010 +2 1110 -2
0111 +7 0011 +3
1011 -5 1011 -5
1110 -2 0010 +2
1001 -7 1101 -3

5. RANGE OF NUMBERS
Maximum Range
Number of digits
Decimal number example
Binary number example
Overflow

6. RANGE OF BINARY NUMBERS
Decimal
Number
Sign Magnitude
2n-1, 2n-1 -1
2’s Complement
2n-1, 2n-1 -1
Unsigned
2n, 2n - 1
0 0000 0000 000
1 0001 0001 001
2 0010 0010 010
3 0011 0011 011
4 0100 0100 100
5 0101 0101 101
6 0110 0110 110
7 0111 0111 111

7. RANGE OF BINARY NUMBERS
Decimal
Number
Sign Magnitude
2n-1-1, -(2n-1 -1)
2’s Complement
2n-1, -2n-1
Unsigned
-8 1000
-7 1111 1001
-6 1110 1010
-5 1101 1011
-4 1100 1100
-3 1011 1101
-2 1010 1110
-1 1001 1111

8. RANGE OF NUMBERS
Unsigned
 Positive Numbers Only (0 to 7)
 3-bit
Signed Magnitude
 Positive & Negative Numbers (-7 to 7)
 4-bit
2’s Complement
 Positive & Negative Numbers (-8 to 7)
 4-bit

9. RANGE & OVERFLOW
2 + 7 using 3-bit unsigned binary?
9 overflow?
2 – 7?
Can not represent -7 in unsigned binary

10. RANGE AND OVERFLOW
1011 -5
1101 -3
11000 -8
1011 -5 0101
+5
1100 -4 0100 +4
10111 -9 1001 +9

11. OCTAL NUMBER SYSTEM
Base 8
0, 1, 2, 3, 4, 5, 6, 7
Representing Binary in compact form
11011000001102 = 154068

12. COUNTING IN OCTAL
Decimal Binary Octal
0 000 0
1 001 1
2 010 2
3 011 3
4 100 4
5 101 5
6 110 6
7 111 7

13. COUNTING IN OCTAL
Decimal Octal Decimal Octal Decimal Octal
8 10 16 20 24 30
9 11 17 21 25 31
10 12 18 22 26 32
11 13 19 23 27 33
12 14 20 24 28 34
13 15 21 25 29 35
14 16 22 26 30 36
15 17 23 27 31 37

14. BINARY-OCTAL
CONVERSION
Binary to Octal Conversion
11010110101110010110
011 010 110 101 110 010 110
3 2 6 5 6 2 6
Octal to Binary Conversion
1726
001 111 010 110

15. DECIMAL-OCTAL CONVERSION
Decimal to Octal Conversion
Indirect Method
Decimal →Binary → Octal
Repeated Division by 8
Octal to Decimal Conversion
Indirect Method
Octal →Binary → Decimal
Sum-of-Weights

16. REPEATED DIVISION BY 8
Number Quotient Remainder
2075 259 3 (O0)
259 32 3 (O1)
32 4 0 (O2)
4 0 4 (O3)

17. SUM-OF-WEIGHTS
4033
(4 x 83) + (0 x 82) + (3 x 81) +
(3 x 80)
(4 x 512) + (0 x 64) + (3 x 8) +
(3 x 1)
2048 + 0 + 24 + 3
2075

18. OCTAL ADDITION
Carry 1
7602 2+1=3d 3O
+ 5771 0+7=7d 7O
15573 6+7=13d 15O
1+7+5=13d 15O

19. OCTAL SUBTRACTION
Borro 11
7602 2-1=1d
1O
- 5771 8-7=1d
1O
1611 13-
7=6d 6O
6-5=1d 1O

20. HEXADECIMAL NUMBER
SYSTEM
Base 16
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Representing Binary in compact form
11011000001102 = 1B06 H

21. COUNTING IN HEXADECIMAL
Decimal Binary Hexadecimal Decimal Binary Hexadecimal
0 0000 0 8 1000 8
1 0001 1 9 1001 9
2 0010 2 10 1010 A
3 0011 3 11 1011 B
4 0100 4 12 1100 C
5 0101 5 13 1101 D
6 0110 6 14 1110 E
7 0111 7 15 1111 F

22. COUNTING IN HEXADECIMAL
Decimal Hexa-
Decimal
Decimal Hexa-
Decimal
Decimal Hexa-
Decimal
16 10 24 18 32 20
17 11 25 19 33 21
18 12 26 1A 34 22
19 13 27 1B 35 23
20 14 28 1C 36 24
21 15 29 1D 37 25
22 16 30 1E 38 26
23 17 31 1F 39 27

23. BINARY-HEXADECIMAL
CONVERSION
Binary to Hexadecimal Conversion
11010110101110010110
1101 0110 1011 1001 0110
D 6 B 9 6
Hexadecimal to Binary
Conversion
FD13
1111 1101 0001 0011

24. DECIMAL-HEXADECIMAL
CONVERSION
Decimal to Hexadecimal Conversion
Indirect Method
Decimal →Binary → Hexadecimal
Repeated Division by 16
Hexadecimal to Decimal Conversion
Indirect Method
Hexadecimal →Binary → Decimal
Sum-of-Weights

25. REPEATED DIVISION BY 16
Number Quotient Remainder
2096 131 0
131 8 3
8 0 8

26. SUM-OF-WEIGHTS
CA02
(C x 163) + (A x 162) + (0 x 161) + (2 x
160)
(12 x 163) + (10 x 162) + (0 x 161) + (2 x
160)
(12 x 4096) + (10 x 256) + (0 x 16) + (2
x 1)
49152 + 2560 + 0 + 2
51714

27. HEXADECIMAL ADDITION
Carry 1
2AC6 6+5=11d Bh
+ 92B5 C+B=23d 17h
BD7B A+2+1=13d Dh
2+9=11d Bh

28. HEXADECIMAL
SUBTRACTION
Borrow 111
92B5 21-6=15d Fh
- 2AC6 26-C=14d Eh
67EF 17-A=7d 7h
8-2=6d 6h

29. ALTERNATE REPRESENTATIONS
BCD (Binary Coded Decimal) Code
Decimal BCD Decimal BCD
0 0000 5 0101
1 0001 6 0110
2 0010 7 0111
3 0011 8 1000
4 0100 9 1001

30. BCD ADDITION
Multi-digit BCD numbers can be added
together
23 0010 0011
45 0100 0101
68 0110 1000
23 0010 0011
48 0100 1000
71 0110 1011
1011 is illegal BCD number

31. BCD ADDITION
Add a 0110 (6) to an invalid BCD number
Carry added to the most significant BCD
digit
23 0010 0011
48 0100 1000
71 0110 1011
0110
0111 0001

32. ALPHANUMERIC CODE
Numbers, Characters, Symbols
ASCII 7-bit Code
American Standard Code for Information
Interchange
10 Numbers (0-9)
26 Lower Case Characters (a-z)
26 Upper Case Characters (A-Z)
Punctuation and Symbols
32 Control Characters

33. ASCII CODE
Numbers 0 to 9
ASCII 0110000 (30h) to 0111001 (39h)
Alphabets a to z
ASCII 1100001 (61h) to 1111010 (7Ah)
Alphabets A to Z
ASCII 1000001 (41h) to 1011010 (5Ah)
Control Characters
ASCII 0000000 (0h) to 0011111 (1Fh)

34. ERROR DETECTION
Digital Systems are very Reliable
Errors during storage or transmission
Parity Bit
Even Parity
Odd Parity

35. ODD PARITY ERROR
DETECTION
Original data 10011010
With Odd Parity 110011010
1-bit error 110111010
Number of 1s even indicates 1-bit error
2-bit error 110110010
Number of 1s odd no error indicated
3-bit error 100110010
Number of 1s even indicates error