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

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

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

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)

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

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

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

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

N UMBER S YSTEMS – Representation  Example The decimal number 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 = 7x x x x10 -1 Chapter 1 9

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

Chapter 1 11 Number Systems – Examples GeneralDecimalBinary Radix (Base)r102 Digits0 => r - 10 => 90 => Powers of 4 Radix r 0 r 1 r 2 r 3 r 4 r 5 r -1 r -2 r -3 r -4 r , ,

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"

Chapter 1 13  Value = 2 Exponent ExponentValue ExponentValue , , , , , , , , , ,048, ,097, B ASE C ONVERSION - Positive Powers of 2

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:

B ASE C ONVERSION Chapter 1 15

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

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 to N 2 Converting Decimal to Binary 625 – 512 = 113 => – 64 = 49 => 6 49 – 32 = 17 => 5 17 – 16 = 1 => 4 1 – 1 = 0 =>

Chapter 1 18 Decimal (Base 10) Binary (Base 2) Octal (Base 8) Hexadecimal (Base 16) A B C D E F  Good idea to memorize! Numbers in Different Bases

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

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, …

Chapter 1 21 Example: Convert To Base 2  Convert 46 to Base  Convert to Base 2:  Join the results together with the radix point:

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 to N = … 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.

B ASE C ONVERSION - Powers of

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

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  Why do these conversions work?

where r=16 (2AE0B) 16 =2 x x x x =(175627) 10 Example : What is the decimal expansion of the hexadecimal expansion of (2AE0B) 16

Example : Find the base 8 expansion of (12345) = 8 x = 8 x = 8 x = 8 x = 8 x (12345)10 =(30071)8

Chapter 1 28 Binary Coded Decimal (BCD) Decimal

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

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.  = (This is conversion)  13  0001|0011 (This is coding)

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

Chapter 1 32 Single Bit Binary Addition with Carry

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

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 X Y BS Z X Y BS

Chapter 1 35  Extending this to two multiple bit examples: Borrows 0 0 Minuend Subtrahend 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

Chapter 1 36 Binary Multiplication

Chapter 1 37 BCD Arithmetic  Given a BCD code, we use binary arithmetic to add the digits: Eight Plus 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 Eight Plus is 13 (> 9) so add 6 carry = leaving 3 + cy 0001 | 0011 Final answer (two digits)  If the digit sum is > 9, add one to the next significant digit

Chapter 1 38 BCD Addition Example  Add 2905 BCD to 1897 BCD showing carries and digit corrections

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

Chapter 1 40 ASCII Properties ASCII has some interesting properties:  Digits 0 to 9 span Hexadecimal values to  Upper case A -Z span to 5A 16.  Lower case a -z span to 7A 16. Lower to upper case translation (and vice versa) occurs byflipping bit 6.

Chapter 1 41 D ECIMAL C ODES - Binary Codes for Decimal Digits Decimal8,4,2,1 Excess3 8,4,-2,-1 Gray  There are over 8,000 ways that you can chose 10 elements from the 16 binary numbers of 4 bits. A few are useful:

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, –

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