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Summary Page Chapter 1: Digital Computers and Information Illustration at beginning of each Chapter Base 10 Binary Base 2 Octal Base 8 Hex bas 16 08 1000 10 8 15 1111 17 F -Addition -Subtraction BCD Binary Coded Decimal 4 bit code represents number 0-9 Base 10 BCD 0 0000 1 0001 9 1001 Parity Bit (checks for transmission errors Checks if total number of bits is even or odd Number even parity 1000001 01000001 1010100 11010100 Digital Computers Chapter 1: Logic Design deals with the basic concepts and tools used to design digital hardware consisting of logic circuits. Computer Design deals with the additional concepts and tools used to design computers and other complex digital hardware. Computers and digital hardware in general are referred to as digital systems. Characteristics of a digital system is the manipulation of discrete elements of information. Any set that is restricted to a finite number of elements contains discrete information. Examples of discrete sets are the 10 decimal digits, the 26 letters of the alphabet etc. Discrete elements of information are represented in a digital system by physical quantities called signals. Electrical signals such as voltages and currents are most common. Transistors dominate the circuitry that implements these signals. Signals in most present day electronic digitals systems ase just two discrete values and are therefore said to be binary. Orange Boxes include information not in your text A Bipolar Transistor is a 3 terminal semiconductor sevice in which a small current at one terminal can control a much larger current flowing between the 2nd and 3rd terminal. Transistors can function both as amplifiers ans switches. +5V +5V R2 1K Hi R1 1K +5V And Gate A 10K Lo Off Hi On LED Lo Out A +5V 10K S1 LED B B +5V Out +5V 4.7K +5V +5V Digital Computers Chapter 1: We typically represent two discrete values by ranges of voltages values called HIGH and LOW. The HIGH output voltage value ranges between 4.0 and 5.5 Volts The LOW output voltages ranges between -0.5 and 5.5 voltages The HIGH input range allows 3.0-5.5 volts to be recognized The LOW input ranges allow -0.5 to 2.0 volts. The fact that the input ranges are longer than the output ranges allows the circuits to function correctly in spite of variation in their behavior and undesirable noise voltages that may be added or subtracted from their outputs. HIGH (H) or True (T) or 1 LOW (L) or False (F) or 0 Parity Bits: Used to detect errors Output INPUT 5.0 4.0 3.0 2.0 1.0 0.0 HIGH LOW (if there is excessive noise or errors, how would you detect it?) An additional bit is sometimes added to a binary code to make the total number of 1’s in the resulting code word even or odd. Original message(7 bits) Modified with Even Parity (8 Total bits) 1000001 (two 1’s) 01000001 (total # bits is even no change) 1010100 (three 1’s) 11010100 (total # bits is odd, so add a 1 so total is even four 1’s) Digital Computers Chapter 1: Why is Binary used? Consider a system with 10 values. The voltages between 0 and 5.0 volts would be divided into 10 ranges. Each of length 0.5 volt. A circuit with have to provide an output with each of these ranges. An input circuit would have to determine which of these belonged to each of these 10 ranges. If we wanted to compensate for noise that each range would be 0.25 volts. And the boundaries would be less than 0.25 volts This would require costly and complex electronic circuits and still would be disturbed by small noise voltages. Instead binary circuits are used with significant variation in output and input ranges. The resulting transistor circuit is simple, easy to design and extremely reliable. Information Representation: A Binary Digit is referred to as a bit. Information is represented as groups of bits. By using various coding schemes groups of bits can represent discrete symbols. American Standard Code for Information Interchange (7-bit code) 0000 0001 0010 0011 0100 0101 0110 (pg 25 text) Digital Computers Chapter 1: Codes Unicode: A 16 bit code for representing the symbols and ideographs for the worlds languages. Gray Code: A code having the property that only one bit at a time changes between codes during counting is a Gray Code. Binary Coded Decimal: (BCD) Most commonly used code to represent decimal digits: (binary combinations 1010-1111 not used) Decimal BCD 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 Digital Computers Chapter 1: Number Systems Octal: (base 8) 83 Use symbols 0,1,2,3,4,5,6,7 82 81 80 Hexadecimal: (base 16): Base 10 Base 2 Base 8 Base 16 Value 100 10 1 8 4 2 1 64 8 1 256 16 1 Power 102 101 100 23 22 21 20 82 81 80 163 161 160 1 0 0 0 1 0 0 1 0 0 1 10 1 0 1 0 0 1 2 0 0 A Use symbols 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 163 162 161 160 Decimal: (base 10) Use symbols 0,1,2,3,4,5,6,7,8,9 104 103 102 100 Arithmetic Operations: Example 2: Example 1: Base 10 Power 10100 Base 2 10 1 16 8 4 2 1 Carries Base 10 Base 2 100 10 1 32 16 8 4 2 1 Carries 0 0000 12 0 1100 + 17 +1 0001 ------- ------------- 29 1 1101 10 1100 22 1 0110 + 23 +1 0111 ------45 ------------1 0 1101 Digital Computers Chapter 1: Number Systems The Rules for subtraction are the same in decimal. A borrow here adds 2 (in the decimal system a borrow adds 10) Example: (Base 10) Borrow 1 1 1 13 15 245 -1 - 9 7 - 197 ---- ---- ---- 4 8 ------48 Explanation Column 3 Example 1: Base 10 Power 10100 Base 2 10 1 16 8 4 2 1 Borrows Starting Problem 0 0110 22 1 0110 - 19 -1 0011 ------- ------------- 03 0 0011 Column 2 0 1 -1 -1 Column 1 Column 1 0 -1 But we needed to borrow due to the second column, so This would normally be 0 (1-1) 10 borrow 1 = 1 But we needed to borrow due to the first column, so Cant take 1 from 0 so we borrow from the next column becomes 11 borrow 1 = 10 Result 1 10 10 -1 -1 - 1 ---- ---- ------ 0 1 1 Digital Computers Chapter 1: Number Systems Two’s Complement: (used to subtract two numbers by adding) (Hardware simpler) Subtract a number by converting the subtrahend to a two complement form then adding. Take the boolean complement of each bit, including the sign bit. -That is set each 1 to 0 and each 0 to 1. Then add 1 Example: Result positive 25 00011001 -18 00010010 = B Register 25=00011001 2s compliment Complementer 11101110 2s compliment +18 B Register 18=00010010 00011001 adder 00010010 Reverse the digits 11101101 Then add 1 + 11101110 Overflow ignored +1 ------------11101110 Example: 18 -25 00010010 = -18 00010010 00011001 2s complement 11100111 + 11100111 -------------Have to reverse process 011111001 --------------7=100000111 11111001 reverse digits 00000110 +1 Final answer 00000111 =-7