2. D IGITAL C IRCUITS Book Title: Digital Design Edition: Fourth Author: M. Morris Mano 1

3. 2 CHAPTER 1 DIGITAL SYSTEMS AND BINARY NUMBERS

4. O UTLINE OF C HAPTER 1 1.1 Digital Systems 1.2 Binary Numbers 1.3 Number-base Conversions 1.4 Octal and Hexadecimal Numbers 1.5 Complements 1.6 Signed Numbers 1.7 Binary codes (BCD) 1.9 Binary Logic 33

5. DIGITAL SYSTEMS AND BINARY NUMBERS Digital computers General purposes Many scientific, industrial and commercial applications Digital systems Digital Telephone Digital camera Electronic calculators Digital TV o These devices have a special ‐ purpose digital computer embedded within them. 44

6. These devices have graphical user interfaces (GUIs), which enable them to execute commands that appear to the user to be simple. It can follow a sequence of instructions, called a program, that operates on given data. Discrete information-processing systems Manipulate discrete elements of information For example, {1, 2, 3, …} and {A, B, C, …}… Discrete elements of information are represented in a digital system by physical quantities called signals. Electrical signals such as voltages and currents are the most common. 5

7. A NALOG AND D IGITAL S IGNAL Analog system The physical quantities or signals may vary continuously over a specified range. Digital system The physical quantities or signals can assume only discrete values. 6 t X(t) kT X(kT) Analog signalDigital signal 6

8. B INARY D IGITAL S IGNAL For digital systems, the variable takes discrete values. Two level, or binary values. Binary values are represented abstractly by: Digits 0 and 1 (A binary digit is called a bit ) Words (symbols) False (F) and True (T) Words (symbols) Low (L) and High (H) And words Off and On 7

9. D ECIMAL N UMBER S YSTEM Base (also called radix) = 10 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } Digit Position Integer & fraction Digit Weight Weight = ( Base) Position Magnitude Sum of “ Digit x Weight ” Formal Notation 8 102-2 51274 1010.11000.01 5001020.70.04 d 2 *B 2 +d 1 *B 1 +d 0 *B 0 +d -1 *B -1 +d -2 *B -2 (512.74) 10 8

10. O CTAL N UMBER S YSTEM Base = 8 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 } Weights Weight = ( Base) Position Magnitude Sum of “ Digit x Weight ” Formal Notation 9 102-2 811/8641/64 51274 5 *8 2 +1 *8 1 +2 *8 0 +7 *8 -1 +4 *8 -2 =(330.9375) 10 (512.74) 8 9

11. B INARY N UMBER S YSTEM Base = 2 2 digits { 0, 1 }, called b inary dig its or “ bits ” Weights Weight = ( Base) Position Magnitude Sum of “ Bit x Weight ” Formal Notation Groups of bits 8 bits = Byte 10 102-2 211/241/4 10101 1 *2 2 +0 *2 1 +1 *2 0 +0 *2 -1 +1 *2 -2 =(5.25) 10 (101.01) 2 1 1 0 0 0 1 0 1 10

12. H EXADECIMAL N UMBER S YSTEM Base = 16 16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F } Weights Weight = ( Base) Position Magnitude Sum of “ Digit x Weight ” Formal Notation 11 102-2 1611/162561/256 1E57A 1 *16 2 +14 *16 1 +5 *16 0 +7 *16 -1 +10 *16 -2 =(485.4765625) 10 (1E5.7A) 16 11

13. T HE P OWER OF 2 12 n2n2n 02 0 =1 12 1 =2 22 2 =4 32 3 =8 42 4 =16 52 5 =32 62 6 =64 72 7 =128 n2n2n 82 8 =256 92 9 =512 102 10 =1024 112 11 =2048 122 12 =4096 202 20 =1M 302 30 =1G 402 40 =1T Mega Giga Tera Kilo 12

14. A DDITION Decimal Addition 13 55 55 + 011 = Ten ≥ Base  Subtract a Base 11Carry 13

15. B INARY A DDITION Column Addition 14 101111 11110 + 0000111 ≥ (2) 10 111111 = 61 = 23 = 84 14

16. B INARY S UBTRACTION Borrow a “Base” when needed 15 001110 11110 − 0101110 = (10) 2 2 2 2 2 1 000 1 = 77 = 23 = 54 15

17. B INARY M ULTIPLICATION Bit by bit 16 01111 0110 00000 01111 01111 0 0000 01101110 x

18. N UMBER B ASE C ONVERSIONS 17 Decimal (Base 10) Octal (Base 8) Binary (Base 2) Hexadecimal (Base 16) Evaluate Magnitude 17

19. D ECIMAL ( I NTEGER ) TO B INARY C ONVERSION Divide the number by the ‘Base’ (=2) Take the remainder (either 0 or 1) as a coefficient Take the quotient and repeat the division 18 Example: ( 13 ) 10 QuotientRemainder Coefficient Answer: (13) 10 = (a 3 a 2 a 1 a 0 ) 2 = (1101) 2 MSB LSB 13 / 2 = 61 a 0 = 1 6 / 2 = 30 a 1 = 0 3 / 2 = 11 a 2 = 1 1 / 2 = 01 a 3 = 1 18

20. D ECIMAL ( F RACTION ) TO B INARY C ONVERSION Multiply the number by the ‘Base’ (=2) Take the integer (either 0 or 1) as a coefficient Take the resultant fraction and repeat the multiplication 19 Example: ( 0.625 ) 10 IntegerFraction Coefficient Answer: (0.625) 10 = (0.a -1 a -2 a -3 ) 2 = (0.101) 2 MSB LSB 0.625 * 2 = 1. 25 0.25 * 2 = 0. 5 a -2 = 0 0.5 * 2 = 1. 0 a -3 = 1 a -1 = 1 19

21. D ECIMAL TO O CTAL C ONVERSION 20 Example: ( 175 ) 10 QuotientRemainder Coefficient Answer: (175) 10 = (a 2 a 1 a 0 ) 8 = (257) 8 175 / 8 = 217 a 0 = 7 21 / 8 = 25 a 1 = 5 2 / 8 = 02 a 2 = 2 Example: ( 0.3125 ) 10 IntegerFraction Coefficient Answer: (0.3125) 10 = (0.a -1 a -2 a -3 ) 8 = (0.24) 8 0.3125 * 8 = 2. 5 0.5 * 8 = 4. 0 a -2 = 4 a -1 = 2 20

22. B INARY − O CTAL C ONVERSION 8 = 2 3 Each group of 3 bits represents an octal digit 21 OctalBinary 00 0 0 10 0 1 20 1 0 30 1 1 41 0 0 51 0 1 61 1 0 71 1 1 Example: ( 1 0 1 1 0. 0 1 ) 2 ( 2 6. 2 ) 8 Assume Zeros Works both ways (Binary to Octal & Octal to Binary) 21

23. B INARY − H EXADECIMAL C ONVERSION 16 = 2 4 Each group of 4 bits represents a hexadecimal digit 22 HexBinary 00 0 10 0 0 1 20 0 1 0 30 0 1 1 40 1 0 0 50 1 60 1 1 0 70 1 1 1 81 0 0 0 91 0 0 1 A1 0 B1 0 1 1 C1 1 0 0 D1 1 0 1 E1 1 1 0 F1 1 Example: ( 1 0 1 1 0. 0 1 ) 2 ( 1 6. 4 ) 16 Assume Zeros Works both ways (Binary to Hex & Hex to Binary) 22

24. O CTAL − H EXADECIMAL C ONVERSION Convert to Binary as an intermediate step 23 Example: ( 0 1 0 1 1 0. 0 1 0 ) 2 ( 1 6. 4 ) 16 Assume Zeros Works both ways (Octal to Hex & Hex to Octal) ( 2 6. 2 ) 8 Assume Zeros 23

25. D ECIMAL, B INARY, O CTAL AND H EXADECIMAL 24 DecimalBinaryOctalHex 000000000 010001011 020010022 030011033 040100044 050101055 060110066 070111077 081000108 091001119 10101012A 11101113B 12110014C 13110115D 14111016E 15111117F 24

26. C OMPLEMENTS Complements are used in digital computers to simplify the subtraction operation. There are two types of complements for each base- r system: the radix complement and diminished radix complement. Diminished (Radix = base) Complement (r-1)’s Complement Given a number N in base r having n digits, the ( r–1 )’s complement of N is defined as: (r n –1) – N Example for 4-digit binary numbers: 1’s complement is ( 2 n – 1) – N = (2 4 –1)– N = 1111– N 1’s complement of 1100 is 1111–1100 = 0011 Observation: Subtraction from (2 n – 1) will never require a borrow For binary: 1 – 0 = 1 and 1 – 1 = 0 25

27. C OMPLEMENTS 1’s Complement ( Diminished Radix Complement) All ‘0’s become ‘1’s All ‘1’s become ‘0’s Example (10110000) 2  (01001111) 2 If you add a number and its 1’s complement … 26 1 0 1 1 0 0 0 0 + 0 1 0 0 1 1 1 1 1 1 1 1 26

28. C OMPLEMENTS Radix Complement Example: Base-2 27 The r's complement of an n-digit number N in base r is defined as r n – N for N ≠ 0. Comparing with the (r  1) 's complement, we note that the r's complement is obtained by adding 1 to the (r  1) 's complement, since r n – N = [(r n  1) – N] + 1. The 2's complement of 1101100 is 0010100 The 2's complement of 0110111 is 1001001 27

29. C OMPLEMENTS 2’s Complement ( Radix Complement) Take 1’s complement then add 1 Toggle all bits to the left of the first ‘1’ from the right Example : Number: 1’s Comp.: 28 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 + 1 OR 1 0 1 1 0 0 0 0 00001010 28

30. C OMPLEMENTS Subtraction with Complements The subtraction of two n -digit unsigned numbers M – N in base r can be done as follows: 29

31. C OMPLEMENTS Example 1.7 Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X – Y ; and (b) Y  X, by using 2's complement. 30 There is no end carry. Therefore, the answer is Y – X =  (2's complement of 1101111) =  0010001. 30

32. S IGNED B INARY N UMBERS To represent negative integers, we need a notation for negative values. Signed-magnitude represents the sign with a bit placed in the leftmost position of the number and the rest of the bits represent the number. The convention is to make the sign bit 0 for positive and 1 for negative. Example: +9 is represented only by 00001001 three different ways to represent -9 31

33. 32  Table 1.3 lists all possible four-bit signed binary numbers in the three representations. 2 n-1 -1 -(2 n-1 -1)

34. S IGNED B INARY N UMBERS Arithmetic addition The addition of two numbers in the signed-magnitude system follows the rules of ordinary arithmetic. If the signs are the same, we add the two magnitudes and give the sum the common sign. If the signs are different, we subtract the smaller magnitude from the larger and give the difference the sign of the larger magnitude. This is a process that requires a comparison of the signs and magnitudes and then performing either addition or subtraction. The addition of two signed binary numbers with negative numbers represented in signed-2's-complement form is obtained from the addition of the two numbers, including their sign bits. A carry out of the sign-bit position is discarded. 33

35. 34 Example: The main problem with signed-magnitude system is that it doesn’t support binary arithmetic (which is what the computer would naturally do). That is, if you add 10 and -10 binary you won’t get 0 as a result. 00001010 (decimal 10) (signed-magnitude) + 10001010 (decimal -10) -------------------------------- 10010100 (decimal -20) (wrong answer)

36. S IGNED B INARY N UMBERS Arithmetic Subtraction In 2’s-complement form: Example: 35 1.Take the 2’s complement of the subtrahend (including the sign bit) and add it to the minuend (including sign bit). 2.A carry out of sign-bit position is discarded. (  6)  (  13)(11111010  11110011) (11111010 + 00001101) 00000111 (+ 7) 35

37. 36 BCD numbers are decimal numbers and not binary numbers, although they use bits in their representation. The only difference between a decimal number and BCD is that decimals are written with the symbols 0, 1, 2,..., 9 and BCD numbers use the binary code 0000, 0001, 0010..... 1001. A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9. The binary combinations 1010 through 1111 are not used and have no meaning in BCD. B INARY C ODES

38. 37 BCD Code (Decimal computers) A number with k decimal digits will require 4k bits in BCD. Decimal 396 is represented in BCD with 12bits as 0011 1001 0110, with each group of 4 bits representing one decimal digit. BCD is very common in electronic systems where a numeric value is to be displayed, especially in systems consisting only of digital logic, and not containing a microprocessor. 37

39. 38 ASCII C HARACTER C ODES American Standard Code for Information Interchange (Refer to Table 1.7) A popular code used to represent information sent as character-based data. It uses 7-bits to represent: 94 Graphic printing characters. 34 Non-printing characters (Control Characters). Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return). (Format effectors) Other non-printing characters are Communication CC (e.g. STX and ETX start and end text areas). Information separators are used to separate the data into divisions such as paragraphs and pages. 38

40. 39 ASCII C HARACTER C ODE American Standard Code for Information Interchange (ASCII) Character Code 39

41. 40 ASCII P ROPERTIES The seven bits of the code are designated by b1 through b7. with b7 the most significant bit. The letter A. for example is represented in ASCII as 1000001 (column 100, row 0001). ASCII has some interesting properties: Digits 0 to 9 span Hexadecimal values 30 16 to 39 16 Upper case A-Z span 41 16 to 5A 16 Lower case a-z span 61 16 to 7A 16 Lower to upper case translation (and vice versa) occurs by flipping bit 6. 40

42. 41  ASCII is a seven-bit code, but most computers manipulate an eight-bit quantity as a single unit called a byte. Therefore, ASCII characters often are stored one per byte.  Extended ASCII (8 bits) adds the Greek alphabets.

43. 42 B INARY L OGIC (B OOLEAN ALGEBRA ) Definition of Binary Logic Binary logic consists of binary variables and a set of logical operations. The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each variable having two and only two distinct possible values: 1 and 0, Three basic logical operations: AND, OR, and NOT. 42

44. 43 BINARY LOGIC Truth Tables, Boolean Expressions, and Logic Gates xyz 000 010 100 111 xyz 000 011 101 111 xz 01 10 ANDORNOT z = x y = x yz = x + yz = x = x’

45. 44 S WITCHING C IRCUITS ANDOR 44

46. 45 L OGIC GATES Logic gates are electronic circuits that operate on one or more input signals to produce an output signal. 0 1 2 3 Logic 1 Logic 0 Un-define Figure 1.3 Example of binary signals 45

47. 46 L OGIC GATES Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.4 Symbols for digital logic circuits Fig. 1.5 Input-Output signals for gates 46

48. 47 Logic gates Graphic Symbols and Input-Output Signals for Logic gates with multiple inputs: 47

49. The problems of chapter one: 1.2, 1.3, 1.4, 1.7, 1.8, 1.9, 1.13, 1.14(a, c), 1.18(a, c),1.35 48

50. 49