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DIGITAL LOGIC DESIGN by Dr. Fenghui Yao Tennessee State University Department of Computer Science DIGITAL LOGIC DESIGN by Dr. Fenghui Yao Tennessee State University Department of Computer Science Nashville, TN Binary Systems 1

Digital Systems u They manipulate discrete information (A finite number of elements) Ø Example Digital Systems u They manipulate discrete information (A finite number of elements) Ø Example discrete sets ü u u 10 decimal digits, the 26 letters of alphabet Information is represented in binary form Examples Ø Ø Binary Systems Digital telephones, digital television, and digital cameras The most commonly used one is DIGITAL COMPUTERS 2

Digital Computers CENTRAL PROCESSING UNIT Control Unit Arithmetic Logic Unit (ALU) Registers R 1 Digital Computers CENTRAL PROCESSING UNIT Control Unit Arithmetic Logic Unit (ALU) Registers R 1 R 2 Rn Bus Main Memory Binary Systems Disk Keyboard Printer I/O Devices 3

Binary Signals u It means two-states Ø Ø Ø u u 1 and 0 Binary Signals u It means two-states Ø Ø Ø u u 1 and 0 true and false on and off A single “on/off”, “true/false”, “ 1/0” is called a bit Example: Toggle switch Binary Systems 4

Byte u u Computer memory is organized into groups of eight bits Each eight Byte u u Computer memory is organized into groups of eight bits Each eight bit group is called a byte Binary Systems 5

Why Computers Use Binary u They can be represented with a transistor that is Why Computers Use Binary u They can be represented with a transistor that is relatively easy to fabricate (in silicon) Ø u Millions of them can be put in a tiny chip Unambiguous signal (Either 1 or 0) Ø Binary Systems This provides noise immunity 6

Analog Signal Binary Systems 7 Analog Signal Binary Systems 7

Binary Signal u A voltage below the threshold Ø u off A voltage above Binary Signal u A voltage below the threshold Ø u off A voltage above threshold Ø Binary Systems on 8

Binary Signal Binary Systems 9 Binary Signal Binary Systems 9

Noise on Transmission u When the signal is transferred it will pick up noise Noise on Transmission u When the signal is transferred it will pick up noise from the environment Binary Systems 10

Recovery u Even when the noise is present the binary values are transmitted without Recovery u Even when the noise is present the binary values are transmitted without error Binary Systems 11

Binary Numbers u A number in a base-r system x = xn-1 xn-2. . Binary Numbers u A number in a base-r system x = xn-1 xn-2. . . x 1 x 0. x-1 x-2. . . X-(m-1) x-m Binary Systems 12

Radix Number System u Base – 2 (binary numbers) Ø u Base – 8 Radix Number System u Base – 2 (binary numbers) Ø u Base – 8 (octal numbers) Ø u 01 01234567 Base – 16 (hexadecimal numbers) Ø Binary Systems 0123456789 ABCDEF 13

Radix Operations u The same as for decimal numbers 11001011 101 +10011101 - 10011101 Radix Operations u The same as for decimal numbers 11001011 101 +10011101 - 10011101 * 110 101101000 00101110 000 1010 +10100 11110 Binary Systems 14

Conversion From one radix to another u From decimal to binary Binary Systems 15 Conversion From one radix to another u From decimal to binary Binary Systems 15

Conversion From one radix to another u From decimal to base-r Ø Ø Separate Conversion From one radix to another u From decimal to base-r Ø Ø Separate the number into an integer part and a fraction part For the integer part ü Divide the number and all successive quotients by r ü Accumulate the remainders 165 23 3 2 0 Binary Systems 4 3 0. 6875 x 2 = 1 + 0. 3750 x 2 = 0 + 0. 7500 x 2 = 1 + 0. 5000 x 2 = 1 + 0. 0000 16

Different Bases Binary Systems 17 Different Bases Binary Systems 17

Conversion From one radix to another u From binary to octal Ø Ø Divide Conversion From one radix to another u From binary to octal Ø Ø Divide into groups of 3 bits Example ü u 11001101001000. 1011011 = 31510. 554 From octal to binary Ø Ø Replace each octal digit with three bits Example ü Binary Systems 75643. 5704 = 11110100011. 101111000100 18

Conversion From one radix to another u From binary to hexadecimal Ø Ø Divide Conversion From one radix to another u From binary to hexadecimal Ø Ø Divide into groups of 4 bits Example ü u 11001101001000. 1011011 = 3348. B 6 From hexadecimal to binary Ø Ø Replace each digit with four bits Example ü Binary Systems 7 BA 3. BC 4 = 11110100011. 101111000100 19

Complements u u They are used to simplify the subtraction operation Two types (for Complements u u They are used to simplify the subtraction operation Two types (for each base-r system) Ø Ø Diminishing radix complement (r-1 complement) Radix complement (r complement) For n-digit number N r-1 complement r complement Binary Systems 20

9’s and 10’s Complements u 9’s complement of 674653 Ø u 9’s complement of 9’s and 10’s Complements u 9’s complement of 674653 Ø u 9’s complement of 023421 Ø u 999999 -023421 = 976578 10’s complement of 674653 Ø u 999999 -674653 = 325346+1 = 325347 10’s complement of 023421 Ø Binary Systems 976578+1=976579 21

1’s and 2’s Complements u 1’s complement of 10111001 Ø Ø u 1’s complement 1’s and 2’s Complements u 1’s complement of 10111001 Ø Ø u 1’s complement of 10100010 Ø u 01011101 2’s complement of 10111001 Ø Ø u 1111 – 10111001 = 01000110 Simply replace 1’s and 0’s 01000110 + 1 = 01000111 Add 1 to 1’s complement 2’s complement of 10100010 Ø Binary Systems 01011101 + 1 = 01011110 22

Subtraction with Complements of Unsigned u M–N Ø Add M to r’s complement of Subtraction with Complements of Unsigned u M–N Ø Add M to r’s complement of N ü Ø Ø If M > N, Sum will have an end carry rn , discard it If M

Subtraction with Complements of Unsigned u 65438 - 5623 65438 10’s complement of 05623 Subtraction with Complements of Unsigned u 65438 - 5623 65438 10’s complement of 05623 +94377 159815 Discard end carry 105 Answer Binary Systems -100000 59815 24

Subtraction with Complements of Unsigned u 5623 - 65438 05623 10’s complement of 65438 Subtraction with Complements of Unsigned u 5623 - 65438 05623 10’s complement of 65438 +34562 40185 There is no end carry => -(10’s complement of 40185) -59815 Binary Systems 25

Subtraction with Complements of Unsigned u 10110010 - 10011111 10110010 2’s complement of 10011111 Subtraction with Complements of Unsigned u 10110010 - 10011111 10110010 2’s complement of 10011111 Discard end carry 2^8 Answer Binary Systems +01100001 100010011 -1000010011 26

Subtraction with Complements of Unsigned u 10011111 -10110010 10011111 2’s complement of 10110010 +010011101101 Subtraction with Complements of Unsigned u 10011111 -10110010 10011111 2’s complement of 10110010 +010011101101 There is no end carry => -(2’s complement of 11101101) Answer = -00010011 Binary Systems 27

Signed Binary Numbers u u Unsigned representation can be used for positive integers How Signed Binary Numbers u u Unsigned representation can be used for positive integers How about negative integers? Ø Ø Binary Systems Everything must be represented in binary numbers Computers cannot use – or + signs 28

Negative Binary Numbers u Three different systems have been used Ø Ø Ø Signed Negative Binary Numbers u Three different systems have been used Ø Ø Ø Signed magnitude One’s complement Two’s complement NOTE: For negative numbers the sign bit is always 1, and for positive numbers it is 0 in these three systems Binary Systems 29

Signed Magnitude u u The leftmost bit is the sign bit (0 is + Signed Magnitude u u The leftmost bit is the sign bit (0 is + and 1 is - ) and the remaining bits hold the absolute magnitude of the number Examples ü ü -47 = 1 0 1 1 47 = 0 0 1 1 1 1 For 8 bits, we can represent the signed integers – 128 to +127 How about for N bits? Binary Systems 30

One’s complement u u Replace each 1 by 0 and each 0 by 1 One’s complement u u Replace each 1 by 0 and each 0 by 1 Example (-6) Ø Ø Binary Systems First represent 6 in binary format (00000110) Then replace (11111001) 31

Two’s complement u u u Find one’s complement Add 1 Example (-6) Ø Ø Two’s complement u u u Find one’s complement Add 1 Example (-6) Ø Ø Ø Binary Systems First represent 6 in binary format (00000110) One’s complement (11111001) Two’s complement (11111010) 32

Arithmetic Addition u Usually represented by 2’s complement Discard +5 00000101 -5 11111011 +11 Arithmetic Addition u Usually represented by 2’s complement Discard +5 00000101 -5 11111011 +11 00001011 +16 00010000 +6 100000110 +5 00000101 -5 11111011 -11 11110101 -6 11111010 -16 111110000 Binary Systems Discard 33

Registers u u They can hold a groups of binary data Data can be Registers u u They can hold a groups of binary data Data can be transferred from one register to another Binary Systems 34

Processor-Memory Registers Binary Systems 35 Processor-Memory Registers Binary Systems 35

Operations Binary Systems 36 Operations Binary Systems 36

Logic Gates - 1 Binary Systems 37 Logic Gates - 1 Binary Systems 37

Logic Gates - 2 Binary Systems 38 Logic Gates - 2 Binary Systems 38

Ranges The gate input Binary Systems The gate output 39 Ranges The gate input Binary Systems The gate output 39

Study Problems u Course Book Chapter – 1 Problems Ø Ø Ø Ø Binary Study Problems u Course Book Chapter – 1 Problems Ø Ø Ø Ø Binary Systems 1– 2 1– 7 1– 8 1 – 20 1 – 34 1 – 35 1 – 36 40

Sneak Preview u Next time Ø ASSIGNMENT ü Ø Will be given QUIZ……. ü Sneak Preview u Next time Ø ASSIGNMENT ü Ø Will be given QUIZ……. ü Expect a question from each one of the following Ø Ø Binary Systems Convert decimal to any base Convert between binary, octal, and hexadecimal Binary add, subtract, and multiply Negative numbers 41

Questions Binary Systems 42 Questions Binary Systems 42