2 Course objectives This course provides you with a basic understanding of what digital devices are, how they operate, and how they can be designed to perform useful functions. The course is intended to give you an understanding of Binary systems, Boolean algebra, digital design techniques, logic gates, logic minimization, standard combinational circuits, sequential circuits, flip-flops, synthesis of synchronous sequential circuits, and arithmetic circuits.
3 Contents Week No. Topic Lecture Practical Total 1 Number Systems and Codes Number Systems and Codes Boolean Algebra and Logic Simplification Minimization Techniques- Karnaugh Map Minimization Techniques- Karnaugh Map Logic Gates Arithmetic Circuits-Adders Mid- Term Exam
5 Textbook Logic and Computer Design Fundamentals, 4th Edition by M. Morris Mano and Charles R. Kime, Prentice Hall, 2008
6 Chapter 1 Number Systems
7 1. Number Systems Location in course textbook Chapt. 1
8 Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, 7 No No Hexadecimal 16 0, 1, 9, A, B, F No No
9 Quantities/Counting (1 of 3) Decimal Binary Octal Hexadecimal p. 33
10 Quantities/Counting (2 of 3) Decimal Binary Octal Hexadecimal A B C D E F
11 Quantities/Counting (3 of 3) Decimal Binary Octal Hexadecimal Etc.
12 Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal pp
13 Quick Example = = 31 8 = Base
14 Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide
15 Weight => 5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = Base
16 Binary to Decimal Decimal Octal Binary Hexadecimal
17 Binary to Decimal Technique Multiply each bit by 2 n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results
18 Example Bit => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 =
19 Octal to Decimal Decimal Octal Binary Hexadecimal
20 Octal to Decimal Technique Multiply each bit by 8 n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results
21 Example => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 =
22 Hexadecimal to Decimal Decimal Octal Binary Hexadecimal
23 Hexadecimal to Decimal Technique Multiply each bit by 16 n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results
24 Example ABC 16 => C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 =
25 Decimal to Binary Decimal Octal Binary Hexadecimal
26 Decimal to Binary Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.
27 Example =? =
28 Decimal to Octal Decimal Octal Binary Hexadecimal
29 Decimal to Octal Technique Divide by 8 Keep track of the remainder
30 Example =? =
31 Decimal to Hexadecimal Decimal Octal Binary Hexadecimal
32 Decimal to Hexadecimal Technique Divide by 16 Keep track of the remainder
33 Example =? = D = 4D2 16
34 Octal to Binary Decimal Octal Binary Hexadecimal
35 Octal to Binary Technique Convert each octal digit to a 3-bit equivalent binary representation
36 Example =? =
37 Hexadecimal to Binary Decimal Octal Binary Hexadecimal
38 Hexadecimal to Binary Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation
39 Example 10AF 16 =? A F AF 16 =
40 Binary to Octal Decimal Octal Binary Hexadecimal
41 Binary to Octal Technique Group bits in threes, starting on right Convert to octal digits
42 Example =? =
43 Binary to Hexadecimal Decimal Octal Binary Hexadecimal
44 Binary to Hexadecimal Technique Group bits in fours, starting on right Convert to hexadecimal digits
45 Example =? B B = 2BB 16
46 Octal to Hexadecimal Decimal Octal Binary Hexadecimal
47 Octal to Hexadecimal Technique Use binary as an intermediary
48 Example =? E = 23E 16
49 Hexadecimal to Octal Decimal Octal Binary Hexadecimal
50 Hexadecimal to Octal Technique Use binary as an intermediary
51 Example 1F0C 16 =? 8 1 F 0 C F0C 16 =
52 Exercise Convert... Decimal Binary Octal Hexadecimal 1AF Don t use a calculator! Skip answer Answer
53 Exercise Convert Answer Decimal Binary Octal Hexadecimal C AF
54 Common Powers (1 of 2) Base 10 Power Preface Symbol pico p 10-9 nano n 10-6 micro 10-3 milli m 10 3 kilo k 10 6 mega M 10 9 giga G tera T Value
55 Common Powers (2 of 2) Base 2 Power Preface Symbol 2 10 kilo k 2 20 mega M 2 30 Giga G Value What is the value of k, M, and G? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies
56 Example In the lab 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 2 30 =
57 Exercise Free Space Determine the free space on all drives on a machine in the lab Drive Bytes Free space GB A: C: D: E: etc.
58 Review multiplying powers For common bases, add powers a b a c = a b+c = 2 16 = 65,536 or = = 64k
59 Fractions Decimal to decimal (just for fun) 3.14 => 4 x 10-2 = x 10-1 = x 10 0 = pp
60 Fractions Binary to decimal => 1 x 2-4 = x 2-3 = x 2-2 = x 2-1 = x 2 0 = x 2 1 = pp
61 Fractions Decimal to binary x x x x x x etc. p. 50
62 Fractions Octal to decimal => 2 x 8-2 = x 8-1 = x 8 0 = x 8 1 = pp
63 Fractions Decimal to octal x x x x x x etc. p. 50
64 Fractions Hexadecimal to decimal 2B.84 => 4 x 16-2 = x 16-1 = 0.5 B x 16 0 = x 16 1 = pp
65 Fractions Decimal to Hexadecima x x x x x x etc. p. 50
66 Exercise Convert... Decimal Binary Octal Hexadecimal C.82 Don t use a calculator! Skip answer Answer
67 Exercise Convert Answer Decimal Binary Octal Hexadecimal D.CC D C C.82
68 Binary Addition (1 of 2) Two 1-bit values A B A + B two pp
69 Binary Addition (2 of 2) Two n-bit values Add individual bits Propagate carries E.g.,
70 Multiplication (2 of 3) Binary, two 1-bit values A B A B
71 Binary Subtraction (1 of 2) Two 1-bit values Borrow 1 A B A - B pp
72 Binary Subtraction (2 of 2) Two n-bit values Subtract individual bits Propagate borrows E.g.,
73 Binary Subtraction (2 of 2) Two n-bit values Subtract individual bits Propagate borrows E.g.,
74 Subtraction with Complements Complements are used for simplifying the subtraction operations. There are two types of complements for each base-r system: the r's complement and the (r l)'s complement. 2's complement and 1's complement for binary numbers, and the 10's complement and 9's com plement for decimal numbers.
75 9's complement The 9's complement of a decimal number is obtained by subtracting each digit from 9. The 9's complement of is: = The 9's complement of is: =
76 Binary numbers, the 1's complement The 1's complement of a binary number is formed by changing 1's to 0's and 0's to 1's. Examples: The 1's complement of is The 1's complement of is
77 10's complement The 10's complement can be formed by leaving all least significant 0's un changed, subtracting the first nonzero least significant digit from 10, and subtracting all higher significant digits from 9. The 10's complement of is The 10's complement of is
78 2's complement The 2's complement can be formed by leaving all least significant 0's and the first 1 unchanged, and replacing 1's with 0's and 0's with 1's in all other higher significant digits. The 2's complement of is The 2's complement of is
79 Subtraction with Complements The subtraction of two n-digit unsigned numbers M N in base r can be done as follows: Add M to the r's complement of N. If M N, the sum will produce an end carry, r n, which is discarded; what is left is the result M - N. If M < N, the sum does not produce an end carry. To obtain the answer in a familiar form, take the r's complement of the sum and place a negative sign in front.
80 Examples to illustrate the procedure Given the two binary numbers; X = and Y = , perform the subtraction: (a) X Y (b) Y X using 2's complements.
81 X Y= X = 's complement of Y = Sum = Discard end carry = Answer: X Y =
82 Y X= Y = 's complement of X = Sum = There is no end carry. Answer: Y - X = -(2's complement of ) =
83 Thank you Next topic
84 Binary Codes Binary codes are codes which are represented in binary system with modification from the original ones. Binary codes are classified as: Weighted Binary Systems Non Weighted Codes
85 Weighted Binary Systems Weighted binary codes are those which obey the positional weighting principles, Each position of the number represents a specific weight. The codes 8421, 2421, 5421, and 5211 are weighted binary codes.
86 Weighted Binary Systems
87 8421 Code/BCD Code The BCD (Binary Coded Decimal) is a straight assignment of the binary equivalent. It is possible to assign weights to the binary bits according to their positions. The weights in the BCD code are 8,4,2,1. Example: The bit assignment 1001, can be seen by its weights to represent the decimal 9 because: 1x8+0x4+0x2+1x1 = 9 Ex. number 12 is represented in BCD as [ ]
88 2421 Code 2421 Code This is a weighted code, its weights are 2, 4, 2 and 1. A decimal number is represented in 4-bit form and the total four bits weight is = 9. Hence the 2421 code represents the decimal numbers from 0 to 9.
89 5211 Code 5211 Code This is a weighted code, its weights are 5, 2, 1 and 1. A decimal number is represented in 4-bit form and the total four bits weight is = 9. Hence the 5211 code represents the decimal numbers from 0 to 9.
90 Reflective Code Reflective Code A code is said to be reflective when code for 9 is complement for the code for 0, and so is for 8 and 1 codes, 7 and 2, 6 and 3, 5 and 4. Codes 2421, 5211, and excess-3 are reflective, whereas the 8421 code is not.
91 Sequential Codes Sequential Codes A code is said to be sequential when two subsequent codes, seen as numbers in binary representation, differ by one. This greatly aids mathematical manipulation of data. The 8421 and Excess-3 codes are sequential, whereas the 2421 and 5211 codes are not.
92 Excess-3 Code Excess-3 Code Excess-3 is a non weighted code used to express decimal numbers. The code derives its name from the fact that each binary code is the corresponding 8421 code plus 0011(3). Example: 1000 of 8421 = 1011 in Excess-3
93 Error Detecting and Correction Codes For reliable transmission and storage of digital data, error detection and correction is required.
94 Error Detecting Codes When data is transmitted from one point to another there are chances that data may get corrupted. To detect these data errors, we use special codes, which are error detection codes.
95 Parity check In parity codes, every binary message is checked if they have even number of ones or even number of zeros. Based on this information an additional bit is appended to the original data. At the receiver side, once again parity is calculated and matched with the received parity, and if they match, data is ok, otherwise data is corrupt. There are two types of parity: Even parity and Odd Parity
96 Parity There are two types of parity: Even parity: Checks if there is an even number of ones; if so, parity bit is zero. When the number of ones is odd then parity bit is set to 1. Message xyz Even parity code xyz p
97 Parity Odd Parity: Checks if there is an odd number of ones; if so, parity bit is zero. When number of ones is even then parity bit is set to 1. Message xyz Odd parity code xyz p
98 Alphanumeric Codes The binary codes that can be used to represent all the letters of the alphabet, numbers and mathematical symbols, punctuation marks, are known as alphanumeric codes or character codes. These codes enable us to interface the input-output devices like the keyboard, printers, video displays with the computer.
99 ASCII Code ASCII Code ASCII stands for American Standard Code for Information Interchange. It has become a world standard alphanumeric code for microcomputers and computers. It is a 7-bit code representing 2 7 = 128 different characters. These characters represent 26 upper case letters (A to Z), 26 lowercase letters (a to z), 10 numbers (0 to 9), 33 special characters and symbols and 33 control characters.
100 ASCII Code The 7-bit code is divided into two portions, The leftmost 3 bits portion is called zone bits and the 4-bit portion on the right is called numeric bits. Character A B 3 7-bit ASCII
101 ASCII Code An 8-bit version of ASCII code is known as ASCII-8. The 8-bit version can represent a maximum of 256 characters.
102 EBCDIC Code EBCDIC Code EBCDIC stands for Extended Binary Coded Decimal Interchange. It is mainly used with large computer systems like mainframes. EBCDIC is an 8-bit code and thus accommodates up to 256 characters. An EBCDIC code is divided into two portions: 4 zone bits (on the left) and 4 numeric bits (on the right).
Number Systems Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, 7 No No Hexadecimal 16 0, 1, 9, A, B, F No No Quantities/Counting
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