1. Charles Kime & Thomas Kaminski © 2008 Pearson Education, Inc. (Hyperlinks are active in View Show mode) Chapter 1 – Digital Systems and Information Logic and Computer Design Fundamentals

2. Chapter 1 2 Overview  Digital Systems, Computers, and Beyond  Information Representation  Number Systems [binary, octal and hexadecimal]  Arithmetic Operations  Base Conversion  Decimal Codes [BCD (binary coded decimal)]  Alphanumeric Codes  Parity Bit  Gray Codes

3. Chapter 1 3 D IGITAL & C OMPUTER S YSTEMS - Digital System  Takes a set of discrete information inputs and discrete internal information (system state) and generates a set of discrete information outputs. System State Discrete Information Processing System Discrete Inputs Discrete Outputs

4. Chapter 1 4 Types of Digital Systems  No state present Combinational Logic System Output = Function(Input)  State present State updated at discrete times => Synchronous Sequential System State updated at any time => Asynchronous Sequential System State = Function (State, Input) Output = Function (State) or Function (State, Input)

5. Chapter 1 5 I NFORMATION R EPRESENTATION - Signals  Information variables represented by physical quantities.  Two level, or binary values are the most prevalent values in digital systems.  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

6. Chapter 1 6 Signal Example – Physical Quantity: Voltage Threshold Region

7. Chapter 1 7  What are other physical quantities represent 0 and 1? CPU Voltage Disk CD Dynamic RAM Binary Values: Other Physical Quantities Magnetic Field Direction Surface Pits/Light Electrical Charge

8. Chapter 1 8 NUMBER SYSTEMS – Representation  Positive radix, positional number systems  A number with radix r is represented by a string of digits: A n - 1 A n - 2 … A 1 A 0. A - 1 A - 2 … A - m  1 A - m in which 0  A i < r and. is the radix point.  The string of digits represents the power series:  (Number) r =   j = - m j j i i = 0 i rArA (Integer Portion) + (Fraction Portion) i = n - 1 j = - 1

9. N UMBER S YSTEMS – Representation  Example The decimal number 724.5 represents 7 hundreds + 2 tens + 4 units + 5 tenths The hundreds, tens, units and tenths are powers of 10 implied by the position of the digits The value of the number is computed as 724.5 = 7x10 2 + 2x10 1 + 4x10 0 + 5x10 -1 Chapter 1 9

10. Chapter 1 10 N UMBER S YSTEMS – Representation 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 A, B, C, D, E, and F represent the digits for values 10, 11, 12, 13, 14, 15 (given in decimal), respectively, in hexadecimal

11. Chapter 1 11 Number Systems – Examples GeneralDecimalBinary Radix (Base)r102 Digits0 => r - 10 => 90 => 1 0 1 2 3 Powers of 4 Radix 5 -2 -3 -4 -5 r 0 r 1 r 2 r 3 r 4 r 5 r -1 r -2 r -3 r -4 r -5 1 10 100 1000 10,000 100,000 0.1 0.01 0.001 0.0001 0.00001 1 2 4 8 16 32 0.5 0.25 0.125 0.0625 0.03125

12. Chapter 1 12 Special Powers of 2  2 10 (1024) is Kilo, denoted "K"  2 20 (1,048,576) is Mega, denoted "M"  2 30 (1,073, 741,824)is Giga, denoted "G"  2 40 (1,099,511,627,776 ) is Tera, denoted “T"

13. Chapter 1 13  Value = 2 Exponent ExponentValue ExponentValue 0 1 11 2,048 1 2 12 4,096 2 4 13 8,192 3 8 14 16,384 4 16 15 32,768 5 3216 65,536 6 64 17 131,072 7 128 18 262,144 19 524,288 20 1,048,576 21 2,097,152 8 256 9 512 10 1024 B ASE C ONVERSION - Positive Powers of 2

14. Chapter 1 14 Commonly Occurring Bases 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:

15. B ASE C ONVERSION 281016 2 8 10 16 Chapter 1 15

16.  To convert to decimal, use decimal arithmetic to form  (digit × respective power of 2).  Example:Convert 11010 2 to N 10 : 11010 2 =1 X 2 4 +1 X 2 3 +0 X 2 2 +1 X 2 1 + 0 X 2 0 = 26 10 11010 2 = 16 + 8 + 0 + 2 + 0 = 26 10 Converting Binary to Decimal 168421 11010 Assignment : Convert (101010) 2 and (110011) 2 to decimal

17. Chapter 1 17  Method 1 Subtract the largest power of 2 that gives a positive remainder and record the power. Repeat, subtracting from the prior remainder and recording the power, until the remainder is zero. Place 1’s in the positions in the binary result corresponding to the powers recorded; in all other positions place 0’s.  Example: Convert 625 10 to N 2 Converting Decimal to Binary 625 – 512 = 113 => 9 113 – 64 = 49 => 6 49 – 32 = 17 => 5 17 – 16 = 1 => 4 1 – 1 = 0 => 0 9 8 7 6 5 4 3 2 1 0 1 0 0 1 1 1 0 0 0 1 01 12 24 38 416 532 664 7128 8256 9512 101024

18. Chapter 1 18 Decimal (Base 10) Binary (Base 2) Octal (Base 8) Hexadecimal (Base 16) 00 0000000 01 0000101 02 0001002 03 0001103 04 0010004 05 0010105 06 0011006 07 0011107 08 0100010 08 09 0100111 09 10 0101012 0A 11 0101113 0B 12 0110014 0C 13 0110115 0D 14 0111016 0E 15 0111117 0F 16 1000020 10  Good idea to memorize! Numbers in Different Bases

19. Chapter 1 19 Conversion Between Bases  Method 2 To convert from one base to another: 1) Convert the Integer Part 2) Convert the Fraction Part 3) Join the two results with a radix point

20. Chapter 1 20 Conversion Details  To Convert the Integral Part: Repeatedly divide the number by the new radix and save the remainders. The digits for the new radix are the remainders in reverse order of their computation. If the new radix is > 10, then convert all remainders > 10 to digits A, B, …  To Convert the Fractional Part: Repeatedly multiply the fraction by the new radix and save the integer digits that result. The digits for the new radix are the integer digits in order of their computation. If the new radix is > 10, then convert all integers > 10 to digits A, B, …

21. Chapter 1 21 Example: Convert 46.6875 10 To Base 2  Convert 46 to Base 2 101110  Convert 0.6875 to Base 2: 0.1011 2  Join the results together with the radix point: 101110. 1011 2 46 2230 2111 251 221 210 01 0.68750.37500.7500.50 1.3750.7501.51.0 1011

22. Chapter 1 22 Additional Issue - Fractional Part  Note that in this conversion, the fractional part can become 0 as a result of the repeated multiplications.  In general, it may take many bits to get this to happen or it may never happen.  Example Problem: Convert 0.65 10 to N 2 0.65 = 0.1010011001001 … The fractional part begins repeating every 4 steps yielding repeating 1001 forever!  Solution: Specify number of bits to right of radix point and round or truncate to this number.

23. 01 12 24 38 416 532 664 7128 8256 9512 101024 B ASE C ONVERSION - Powers of 2 0.5 -20.25 -30.125 -40.0625 -50.03125

24. Example:  Convert the binary number (1011.1101) 2 to decimal  Convert the decimal number 45.73 to binary up to 4  Convert the binary number (1101011.11101) 2 to octal Chapter 1 24

25. Chapter 1 25 Octal to Hexadecimal via Binary  Convert octal to binary.  Use groups of four bits and convert as above to hexadecimal digits.  Example: Octal to Binary to Hexadecimal 6 3 5. 1 7 7 8  Why do these conversions work?

26. where r=16 (2AE0B) 16 =2 x 16 4 + 10 x 16 3 + 14 x 16 2 + 0 x16 +11 =(175627) 10 Example : What is the decimal expansion of the hexadecimal expansion of (2AE0B) 16

27. Example : Find the base 8 expansion of (12345)10 12345 = 8 x 1543 + 1 812345 815431 81927 8240 830 03 1543 = 8 x 192 + 7 192 = 8 x 24 + 0 24 = 8 x 3 + 0 3 = 8 x 0 + 3 (12345)10 =(30071)8

28. Chapter 1 28 Binary Coded Decimal (BCD) Decimal8421 00000 10001 20010 30011 40100 50101 60110 70111 81000 91001

29. Binary Coded Decimal (BCD) Example: Represent (396) 10 in BCD (396) 10 = (0011 1001 0110) BCD Decimals are written with symbols 0, 1, …., 9 BCD numbers use the binary codes 0000, 0001, 0010, …., 1001 Chapter 1 29

30. Chapter 1 30 Warning: Conversion or Coding?  Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE.  13 10 = 1101 2 (This is conversion)  13  0001|0011 (This is coding)

31. Chapter 1 31 A RITHMETIC O PERATIONS - Binary Arithmetic  Single Bit Addition with Carry  Multiple Bit Addition  Single Bit Subtraction with Borrow  Multiple Bit Subtraction  Multiplication  BCD Addition

32. Chapter 1 32 Single Bit Binary Addition with Carry

33. Chapter 1 33  Extending this to two multiple bit examples: Carries 0 0 Augend 01100 10110 Addend +10001 +10111 Sum  Note: The 0 is the default Carry-In to the least significant bit. Multiple Bit Binary Addition

34. Chapter 1 34  Given two binary digits (X,Y), a borrow in (Z) we get the following difference (S) and borrow (B):  Borrow in (Z) of 0:  Borrow in (Z) of 1: Single Bit Binary Subtraction with Borrow Z 1 1 1 1 X 0 0 1 1 - Y -0 -0 BS 11 1 0 0 1 Z 0 0 0 0 X 0 0 1 1 - Y -0 -0 BS 0 1 0 1 0

35. Chapter 1 35  Extending this to two multiple bit examples: Borrows 0 0 Minuend 10110 10110 Subtrahend - 10010 - 10011 Difference  Notes: The 0 is a Borrow-In to the least significant bit. If the Subtrahend > the Minuend, interchange and append a – to the result. Multiple Bit Binary Subtraction

36. Chapter 1 36 Binary Multiplication

37. Chapter 1 37 BCD Arithmetic  Given a BCD code, we use binary arithmetic to add the digits: 81000 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

38. Chapter 1 38 BCD Addition Example  Add 2905 BCD to 1897 BCD showing carries and digit corrections. 0001 1000 1001 0111 + 0010 1001 0000 0101 0

39. Chapter 1 39 A LPHANUMERIC C ODES - ASCII Character Codes  American Standard Code for Information Interchange  This code is a popular code used to represent information sent as character-based data. It uses 7-bits to represent 128 characters

40. Chapter 1 40 ASCII Properties 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 byflipping bit 6.

41. Chapter 1 41 D ECIMAL C ODES - Binary Codes for Decimal Digits Decimal8,4,2,1 Excess3 8,4,-2,-1 Gray 0 0000 0011 0000 1 0001 0100 0111 0100 2 0010 0101 0110 0101 3 0011 0110 0101 0111 4 0100 0111 0100 0110 5 0101 1000 1011 0010 6 0110 1001 1010 0011 7 0111 1010 1001 0001 8 1000 1011 1000 1001 9 1100 1111 1000  There are over 8,000 ways that you can chose 10 elements from the 16 binary numbers of 4 bits. A few are useful:

42. Chapter 1 42  What interesting property is common to these two codes? Excess 3 Code and 8, 4, –2, –1 Code DecimalExcess 38, 4, –2, –1 000110000 101000111 201010110 3 0101 401110100 510001011 610011010 7 1001 810111000 911001111

43. Chapter 1 43 UNICODE  UNICODE extends ASCII to 65,536 universal characters codes For encoding characters in world languages Available in many modern applications 2 byte (16-bit) code words See Reading Supplement – Unicode on the Companion Website http://www.prenhall.com/mano http://www.prenhall.com/mano