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DIGITAL LOGIC DEVICES

BINARY NUMBER SYSTEM. Digital logic devices. Number ….in whatever base Decimal value of the given number

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DIGITAL LOGIC DEVICES

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  1. BINARY NUMBER SYSTEM Digital logic devices Number ….in whatever base Decimal value of the given number ___________________________________________________________________________________________________________________________________________________________________ Decimal: 1,998 = 1x103 +9x102 +9x101 +8x100 =1,000+900+90+8 =1,998 Binary: 11111001110 = 1x210 +1x29 +1x28 +1x27 +1x26 +1x23 +1x22 +1x21 = 1,024+512+258+128+64+8+4+2 = 1,998 DIGITAL LOGIC DEVICES electronic circuits that handle information encoded in binary form (deal with signals that have only two values, 0 and 1) Digital …. computers, watches, controllers, telephones, cameras, ...

  2. ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ N 2N Comments _____________________________________________________________________________________________________________________________________________________________________ 0 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1,024 “Kilo”as 210 is the closest power of 2 to 1,000 (decimal) 11 2,048 ………………………………………………………..... 15 32,768 215 Hz often used as clock crystal frequency in digital watches ……………………………………………………..... 20 1,048,576 “Mega” as 220 is the closest power of 2 to 1,000,000 (decimal) …………………………………………………... 30 1,073,741,824 “Giga” as 230 is the closest power of 2 to 1,000,000,000(decimal) ____________________________________________________________________________________________________________________________________________________________________ Powers of 2

  3. __________________________________________________________________________________________________________________________ N <0 2N ____________________________________________________________ -1 2-1 = 0.5 -2 2-2 = 0.25 -3 2-3 = 0.125 -4 2-4 = 0.0625 -5 2-5 = 0.03125 -6 2-6 = 0.015625 -7 2-7 = 0.0078125 -8 2-8 = 0.00390625 -9 2-9 = 0.001953125 -10 2-10 = 0.0009765625 … _____________________________________________________________ Binary numbers less than 1 Binary Decimal value ----------------------------------------------------------------------------------------------------------------- 0.101101 Negative Powers of 2 = 1x2-1 +1x2-3 + 1x2-4 + 1x2-6 = 0.703125

  4. HEXADECIMAL ---------------------------------------------------------------------------------------------------- Binary Decimal Hexadecimal ---------------------------------------------------------------------------------------------------- 0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 1110 14 E 1111 15 F ---------------------------------------------------------------------------------------------------- Binary: 11111001110 7 12 14 <== Decimal = 7x162 +12x161 +14x160 = 1998 1111100 1110 Hexadecimal: 7CE

  5. LOGIC OPERATIONS AND TRUTH TABLES Binary logic dealing with “true” and “false” comes in handy to describe the behaviour of these circuits: 0 is usually associated with “false” and 1 with “true.” Quite complex digital logic circuits (e.g. entire computers) can be built using a few types of basic circuits called gates, each performing a single elementary logic operation : NOT, AND, OR, NAND, NOR, etc.. Boole’s binary algebra is used as a formal (mathematical) tool to describe and design complex binary logic circuits. More info on the digital logic big names Boole, Turing, Shannon is available in the Annex F of the textbook. Digital logic circuits handle data encoded in binary form, i.e. signals that have only two values, 0 and 1.

  6. GATES A A ______________________ 0 1 1 0 A F =A . B B A B A . B _____________________________________ 0 0 0 0 1 0 1 0 0 1 1 1 _____________________________________ F =A A A B A + B _____________________________________ 0 0 0 0 1 1 1 0 1 1 1 1 _____________________________________ A F = A + B B AND NOT OR

  7. … more GATES A B A . B _____________________________________ 0 0 1 0 1 1 1 0 1 1 1 0 _____________________________________ A F =A . B B A B A +B _____________________________________ 0 0 1 0 1 0 1 0 0 1 1 0 _____________________________________ A F = A + B B NAND NOR

  8. … and more GATES A B A B _____________________________________ 0 0 0 0 1 1 1 0 1 1 1 0 _____________________________________ A F = A B F = A B B A B A B _____________________________________ 0 0 1 0 1 0 1 0 0 1 1 1 _____________________________________ A B XOR EQU or XNOR

  9. GATES … with more inputs A+B+C ___________________ 0 1 1 1 1 1 1 1 ___________________ A.B.C ___________________ 1 1 1 1 1 1 1 0 ___________________ A+B+C ___________________ 1 0 0 0 0 0 0 0 ___________________ A B C A. B.C ______________________________________________ 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 _______________________________________________ F =A.B.C A A A A F = A+B+C F = A+B+C B B B B F =A.B.C C C C C EXAMPLES OF GATES WITH THREE INPUTS AND OR NAND NOR

  10. Logic Gate Array that Produces an Arbitrarily Chosen Output A B C A . B . C + A . B . C + A . B . C + A . B . C A B C F ________________________________________ 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 _________________________________________ A . B . C A . B . C F A . B . C A . B . C A B C A B C F =

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