Chapter 1 Digital Systems and Binary Numbers

1. Chapter 1 Digital Systems and Binary Numbers 授課教師: 張傳育 博士 (Chuan-Yu Chang Ph.D.) E-mail: chuanyu@yuntech.edu.tw Tel: (05)5342601 ext. 4337 Office: EB212 http://MIPL.yuntech.edu.tw

2. Text Book M. M. Mano and M. D. Ciletti, “Digital Design," 4th Ed., Pearson Prentice Hall, 2007. Reference class notes Grade Quizzes: Mid-term: Final: Appearance: Digital Logic Design

3. Digital age and information age Digital computers general purposes many scientific, industrial and commercial applications Digital systems telephone switching exchanges digital camera electronic calculators, PDA's digital TV Discrete information-processing systems manipulate discrete elements of information Chapter 1: Digital Systems and Binary Numbers

4. An information variable represented by physical quantity For digital systems, the variable takes on discrete values Two level, or binary values are the most prevalent values Binary values are represented abstractly by: digits 0 and 1 words (symbols) False (F) and True (T) words (symbols) Low (L) and High (H) and words On and Off. Binary values are represented by values or ranges of values of physical quantities Signal

5. Signal Example – Physical Quantity: Voltage Threshold Region

6. Signal Examples Over Time Time Continuous in value & time Analog Digital Discrete in value & continuous in time Asynchronous Discrete in value & time Synchronous

7. A Digital Computer Example Inputs: Keyboard, mouse, modem, microphone Outputs: CRT, LCD, modem, speakers Synchronous or Asynchronous?

8. Information Types Numeric Must represent range of data needed Represent data such that simple, straightforward computation for common arithmetic operations Tight relation to binary numbers Non-numeric Greater flexibility since arithmetic operations not applied. Not tied to binary numbers Binary Numbers and Binary Coding

9. Given n digits in radix r, there are rn distinct elements that can be represented. But, you can represent m elements, m < rn Examples: You can represent 4 elements in radix r = 2 with n = 2 digits: (00, 01, 10, 11). You can represent 4 elements in radix r = 2 with n = 4 digits: (0001, 0010, 0100, 1000). This second code is called a "one hot" code. Number of Elements Represented

10. Given n binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2n binary numbers. Example: Abinary codefor the sevencolors of therainbow Code 100 is not used Binary Number 000 001 010 011 101 110 111 Non-numeric Binary Codes Color Red Orange Yellow Green Blue Indigo Violet

11. Base or radix Power Binary Numbers … a5a4a3a2a1.a1a2a3… • Decimal number Decimal point Example: • General form of base-r system Coefficient: aj= 0 to r 1

12. Binary Numbers Example: Base-2 number Example: Base-5 number Example: Base-8 number Example: Base-16 number

13. To convert to decimal, use decimal arithmetic to form S (digit × respective power of 2). Example:Convert 110102to N10: Converting Binary to Decimal

14. Binary Numbers Example: Base-2 number Special Powers of 2 • 210 (1024) is Kilo, denoted "K" • 220 (1,048,576) is Mega, denoted "M" • 230 (1,073, 741,824)is Giga, denoted "G" Powers of two Table 1.1

15. Arithmetic operation Arithmetic operations with numbers in base r follow the same rules as decimal numbers.

16. Single Bit Addition with Carry Multiple Bit Addition Single Bit Subtraction with Borrow Multiple Bit Subtraction Multiplication BCD Addition Binary Arithmetic

17. Single Bit Binary Addition with Carry

18. Extending this to two multiple bit examples: Carries 00 Augend 01100 10110 Addend +10001+10111 Sum Note: The 0 is the default Carry-In to the least significant bit. Multiple Bit Binary Addition

19. Subtraction Minuend: 101101 Subtrahend: 100111 Augend: 101101 Addend: +100111 Difference: 000110 Sum: 1010100 Binary Arithmetic • Addition • Multiplication

20. Number-Base Conversions Name Radix Digits Binary 2 0,1 Octal 8 0,1,2,3,4,5,6,7 Decimal 10 0,1,2,3,4,5,6,7,8,9 Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F • The six letters (in addition to the 10 integers) in hexadecimal represent: 10, 11, 12, 13, 14, and 15, respectively.

21. Number-Base Conversions Example1.1 Convert decimal 41 to binary. The process is continued until the integer quotient becomes 0.

22. Number-Base Conversions  The arithmetic process can be manipulated more conveniently as follows:

23. Number-Base Conversions Example 1.2 Convert decimal 153 to octal. The required base r is 8. Example1.3 Convert (0.6875)10 to binary. The process is continued until the fraction becomes 0 or until the number of digits has sufficient accuracy.

24. Number-Base Conversions Example1.3 To convert a decimal fraction to a number expressed in base r, a similar procedure is used. However, multiplication is by r instead of 2, and the coefficients found from the integers may range in value from 0 to r 1 instead of 0 and 1.

25. Number-Base Conversions Example1.4 Convert (0.513)10 to octal. (41.6875)10 = (101001.1011)2  From Examples 1.1 and 1.3: (153.513)10 = (231.406517)8  From Examples 1.2 and 1.4:

26. Octal and Hexadecimal Numbers  Numbers with different bases: Table 1.2.

27. Octal and Hexadecimal Numbers  Conversion from binary to octal can be done by positioning the binary number into groups of three digits each, starting from the binary point and proceeding to the left and to the right.  Conversion from binary to hexadecimal is similar, except that the binary number is divided into groups of four digits:  Conversion from octal or hexadecimal to binary is done by reversing the preceding procedure.

28. Complements  There are two types of complements for each base-r system: the radix complement and diminished radix complement. the r's complement and the second as the (r 1)'s complement. ■Diminished Radix Complement Example:  For binary numbers, r = 2 and r – 1 = 1, so the 1's complement of N is (2n 1) – N. Example:

29. Complements (cont.) ■Radix Complement The r's complement of an n-digit number N in base r is defined as rn – N for N ≠ 0 and as 0 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 rn – N = [(rn 1) – N] + 1. Example: Base-10 The 10's complement of 012398 is 987602 The 10's complement of 246700 is 753300 Example: Base-10 The 2's complement of 1101100 is 0010100 The 2's complement of 0110111 is 1001001

30. Complements (cont.) ■Subtraction with Complements The subtraction of two n-digit unsigned numbers M–N in base r can be done as follows:

31. Complements (cont.) Example 1.5 Using 10's complement, subtract 72532 – 3250. Example 1.6 Using 10's complement, subtract 3250 – 72532 There is no end carry. Therefore, the answer is – (10's complement of 30718) =  69282.

32. Complements (cont.) 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. There is no end carry. Therefore, the answer is Y–X =  (2's complement of 1101111) =  0010001.

33. Complements (cont.) Subtraction of unsigned numbers can also be done by means of the (r 1)'s complement. Remember that the (r  1) 's complement is one less then the r's complement. Example 1.8 Repeat Example 1.7, but this time using 1's complement. There is no end carry, Therefore, the answer is Y–X =  (1's complement of 1101110) =  0010001.

34. Signed Binary Numbers •  To represent negative integers, we need a notation for negative values. • It is customary to represent the sign with a bit placed in the leftmost position of the number. • The convention is to make the sign bit 0 for positive and 1 for negative. Example:  Table 3 lists all possible four-bit signed binary numbers in the three representations.

35. Signed Binary Numbers (cont.)

36. Signed Binary Numbers (cont.) ■ 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 if the larger magnitude. • 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. Example:

37. Signed Binary Numbers (cont.) ■ Arithmetic Subtraction  In 2’s-complement form: • Take the 2’s complement of the subtrahend (including the sign bit) and add it to the minuend (including sign bit). • A carry out of sign-bit position is discarded. Example: ( 6)  ( 13) (11111010  11110011) (11111010 + 00001101) 00000111 (+ 7)

38. The BCD code is the 8,4,2,1 code. This code is the simplest, most intuitive binary code for decimal digits and uses the same powers of 2 as a binary number, but only encodes the first ten values from 0 to 9. Example: 1001 (9) = 1000 (8) + 0001 (1) How many “invalid” code words are there? What are the “invalid” code words? Binary Coded Decimal (BCD) 6 1010, 1011, 1100, 1101, 1110, 1111

39. Binary Codes ■ BCD Code 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. A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9. A BCD number greater than 10 looks different from its equivalent binary number, even though both contain 1's and 0's. Moreover, the binary combinations 1010 through 1111 are not used and have no meaning in BCD.

40. Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE. 1310 = 11012 (This is conversion) 13  0001|0011 (This is coding) Warning: Conversion or Coding?

41. BCD Arithmetic • Given a BCD code, we use binary arithmetic to add the digits: 8 1000 Eight +5 +0101 Plus 5 13 1101 is 13 (> 9) • Note that the result is MORE THAN 9, so must be represented by two digits! • To correct the digit, subtract 10 by adding 6 modulo 16. 8 1000 Eight +5 +0101 Plus 5 13 1101 is 13 (> 9) +0110 so add 6 carry = 1 0011 leaving 3 + cy 0001 | 0011 Final answer (two digits) • If the digit sum is > 9, add one to the next significant digit

42. Binary Codes Example: Consider decimal 185 and its corresponding value in BCD and binary: ■BCD Addition

43. Binary Codes Example: Consider the addition of 184 + 576 = 760 in BCD: ■Decimal Arithmetic

44. Add 2905BCD to1897BCD showing carries and digit corrections. 2 4 8 0 BCD Addition Example 0001 1000 1001 0111 + 0010100100000101 0011 10001 1001 1100 +0110 +0001 +0110 +0001 1010 10111 10010 0100 +0110 +0001 11000 10000

45. Binary Codes ■ Other Decimal Codes

46. Binary Codes ■ Gray Code 0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000

47. What special property does the Gray code have in relation to adjacent decimal digits? Gray Code Decimal 8,4,2,1 Gray 0 0000 0000 1 0001 0001 2 0010 0011 3 0011 0010 4 0100 0110 5 0101 0111 6 0110 0101 7 0111 0100 8 1000 1100 9 1001 1101

48. Does this special Gray code property have any value? An Example: Optical Shaft Encoder 111 000 100 000 B 0 B 1 101 001 001 110 B 2 G 2 G 1 G 111 0 010 011 101 100 011 110 010 (b) Gray Code for Positions 0 through 7 (a) Binary Code for Positions 0 through 7 Gray Code (Continued)

49. Binary Codes ■ ASCII Character Code

50. Binary Codes ■ ASCII Character Code