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DIGITAL LOGIC DESIGN by Dr. Fenghui Yao Tennessee State University Department of Computer Science Nashville, TN Binary Systems 1 Digital Systems They manipulate discrete information (A finite number of elements) Example discrete sets 10 decimal digits, the 26 letters of alphabet Information is represented in binary form Examples 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 R1 R2 Rn Bus Main Memory Binary Systems Disk Keyboard Printer I/O Devices 3 Binary Signals It means two-states 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 Computer memory is organized into groups of eight bits Each eight bit group is called a byte Binary Systems 5 Why Computers Use Binary They can be represented with a transistor that is relatively easy to fabricate (in silicon) Millions of them can be put in a tiny chip Unambiguous signal (Either 1 or 0) Binary Systems This provides noise immunity 6 Analog Signal Binary Systems 7 Binary Signal A voltage below the threshold off A voltage above the threshold Binary Systems on 8 Binary Signal Binary Systems 9 Noise on Transmission When the signal is transferred it will pick up noise from the environment Binary Systems 10 Recovery Even when the noise is present the binary values are transmitted without error Binary Systems 11 Binary Numbers A number in a base-r system x = xn-1xn-2 ... x1x0 . x-1 x-2 ... X-(m-1) x-m Value( x) xn 1 r n 1 xn 2 r n 2 ... x0 r 0 x1 r 1 x 2 r 2 ... x m r m (234.26) 6 2 6 2 3 61 4 60 2 6 1 6 6 2 (94.5)10 (45.4)8 4 81 5 80 4 81 (39.5)10 Binary Systems 12 Radix Number System Base – 2 (binary numbers) Base – 8 (octal numbers) 01 01234567 Base – 16 (hexadecimal numbers) Binary Systems 0123456789ABCDEF 13 Radix Operations The same as for decimal numbers 11001011 11001011 101 +10011101 - 10011101 * 110 101101000 00101110 000 1010 +10100 11110 Binary Systems 14 Conversion From one radix to another From decimal to binary Binary Systems 15 Conversion From one radix to another From decimal to base-r Separate the number into an integer part and a fraction part For the integer part Divide the number and all successive quotients by r Accumulate the remainders 165 23 4 3 2 0 3 (165)10 (324)7 Binary Systems 0.6875 x 2 = 1 + 0.3750 0.3750 x 2 = 0 + 0.7500 0.7500 x 2 = 1 + 0.5000 0.5000 x 2 = 1 + 0.0000 (0.6875)10 (0.1011) 2 16 Different Bases Binary Systems 17 Conversion From one radix to another From binary to octal Divide into groups of 3 bits Example 11001101001000.1011011 = 31510.554 From octal to binary Replace each octal digit with three bits Example Binary Systems 75643.5704 = 111101110100011.101111000100 18 Conversion From one radix to another From binary to hexadecimal Divide into groups of 4 bits Example 11001101001000.1011011 = 3348.B6 From hexadecimal to binary Replace each digit with four bits bits Example Binary Systems 7BA3.BC4 = 111101110100011.101111000100 19 Complements They are used to simplify the subtraction operation Two types (for each base-r system) Diminishing radix complement (r-1 complement) Radix complement (r complement) For n-digit number N (r 1) N n r N n Binary Systems r-1 complement r complement 20 9’s and 10’s Complements 9’s complement of 674653 9’s complement of 023421 999999-023421 = 976578 10’s complement of 674653 999999-674653 = 325346 325346+1 = 325347 10’s complement of 023421 Binary Systems 976578+1=976579 21 1’s and 2’s Complements 1’s complement of 10111001 1’s complement of 10100010 01011101 2’s complement of 10111001 11111111 – 10111001 = 01000110 Simply replace 1’s and 0’s 01000110 + 1 = 01000111 Add 1 to 1’s complement 2’s complement of 10100010 Binary Systems 01011101 + 1 = 01011110 22 Subtraction with Complements of Unsigned M–N Add M to r’s complement of N If M > N, Sum will have an end carry rn , discard it If M<N, Sum will not have an end carry and Binary Systems Sum = M+(rn – N) = M – N+ rn Sum = rn – (N – M) (r’s complement of N – M) So M – N = – (r’s complement of Sum) 23 Subtraction with Complements of Unsigned 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 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 10110010 - 10011111 10110010 2’s complement of 10011111 +01100001 Discard end carry 2^8 Answer 100010011 -100000000 000010011 Binary Systems 26 Subtraction with Complements of Unsigned 10011111 -10110010 10011111 2’s complement of 10110010 +01001110 11101101 There is no end carry => -(2’s complement of 11101101) Answer = -00010011 Binary Systems 27 Signed Binary Numbers Unsigned representation can be used for positive integers How about negative integers? Binary Systems Everything must be represented in binary numbers Computers cannot use – or + signs 28 Negative Binary Numbers Three different systems have been used 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 The leftmost bit is the sign bit (0 is + and 1 is - ) and the remaining bits hold the absolute magnitude of the number Examples -47 = 1 0 1 0 1 1 1 1 47 = 0 0 1 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 Replace each 1 by 0 and each 0 by 1 Example (-6) Binary Systems First represent 6 in binary format (00000110) Then replace (11111001) 31 Two’s complement Find one’s complement Add 1 Example (-6) Binary Systems First represent 6 in binary format (00000110) One’s complement (11111001) Two’s complement (11111010) 32 Arithmetic Addition Usually represented by 2’s complement Discard +5 00000101 -5 11111011 +11 00001011 +11 00001011 +16 00010000 +6 100000110 +5 00000101 -5 11111011 -11 11110101 -11 11110101 -6 11111010 -16 111110000 Binary Systems Discard 33 Registers 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 Operations Binary Systems 36 Logic Gates - 1 Binary Systems 37 Logic Gates - 2 Binary Systems 38 Ranges The gate input Binary Systems The gate output 39 Study Problems Course Book Chapter – 1 Problems Binary Systems 1–2 1–7 1–8 1 – 20 1 – 34 1 – 35 1 – 36 40 Sneak Preview Next time ASSIGNMENT Will be given QUIZ……. Expect a question from each one of the following 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
