1ECET 331 – Digital Integrated CircuitsChapter 1Basic Principles of Digital SystemsECET 331 - Digital Integrated CircuitsAnalog Electronicsz Values are continuously variable within a defined rangez Can have any value within that rangeECET 331 - Digital Integrated CircuitsDigital Electronicsz Values can vary only by distinct or discreet stepsz Can only have two valuesECET 331 - Digital Integrated CircuitsDigital Logic Levelsz Logic HIGH is the higher voltage and represented by binary digit 1z Logic LOW is the lower voltage and represented by binary digit 0ECET 331 - Digital Integrated CircuitsDigital Logic LevelsECET 331 - Digital Integrated CircuitsBinary Number Systemz Uses two digits: 0 and 1z Can represent any value using positional notation2ECET 331 - Digital Integrated CircuitsPositional Notationz The magnitude of a digit depends on its placement within a numberz In base-10, the positional values are (starting to the left of the decimal) –1(100), 10(101), 100(102), 1000(103), etc.z In base-2, the positional values are1(20), 2(21), 4(22), 8(23), etc.ECET 331 - Digital Integrated CircuitsDecimal Equivalence of Binary NumbersExample:1101 = (1x23) + (1x22) + (0x21) + (1x20)= (1x8) + (1x4) + (0x2) + (1x1)= 8 + 4 + 0 + 1= 13ECET 331 - Digital Integrated CircuitsBitz Shorthand for binary digit, a logic 0 or 1z The most significant bit (MSB) is the leftmost bit of a binary numberz The least significant bit (LSB) is the rightmost bit of a binary numberECET 331 - Digital Integrated CircuitsTruth Tablez A list of output logic levels corresponding to all possible input combinations to a logic function (or circuit)z The number of input combinations is 2n, where n is the number of inputsECET 331 - Digital Integrated CircuitsTruth TableA B C D----------------------0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 01 0 1 11 1 0 11 1 1 0LogicCircuitABCDExample:ECET 331 - Digital Integrated CircuitsConstructing a Binary Sequencez Two methods:• One – learn to count in binary• Two – follow a simple repetitive pattern3ECET 331 - Digital Integrated CircuitsConstructing a Binary Sequencez Memorize the binary numbers from 0000 to 1111 and their decimal equivalents (0 to 15)z Use the weighted values of binary bitsECET 331 - Digital Integrated CircuitsFollow a Simple Repetitive Patternz The LSB of any binary number alternates between 0 and 1 with every linez The next bit alternates every two linesz The next bit alternates every four lines, and so onECET 331 - Digital Integrated CircuitsDecimal-to-Binary Conversionz Two methods:• One – sum powers of 2• Two – repeated division by 2ECET 331 - Digital Integrated CircuitsSum of Powers of 2 Methodz Step 1:• Determine the largest power of 2 less than or equal to the number to be converted • Place a 1 in that positional locationECET 331 - Digital Integrated CircuitsSum of Powers of 2 Methodz Step 2:• Subtract the number found in Step 1 from the number to be converted• For the new number, determine if the next lowest power of 2 is less than or equal to that numberECET 331 - Digital Integrated CircuitsSum of Powers of 2 Methodz Step 3:• If the new power of two from Step 2 is larger, place a 0 in that positional location • If the new value is less than or equal, place a 1 in that positional location4ECET 331 - Digital Integrated CircuitsSum of Powers of 2 Methodz Step 4: • Repeat Steps 2 and 3 until there is nothing left to subtract• All remaining bits are set to 0ECET 331 - Digital Integrated CircuitsSum of Powers of 2 MethodExample:(35)10= 32 + 2 + 1= 25+ 21 + 20= 1x25+ 0x24+ 0x23+0x22 + 1x21 + 1x20= (100011)2ECET 331 - Digital Integrated CircuitsRepeated Division by 2 Methodz Step 1:• Divide the number to be converted by 2 • The remainder, 0 or 1, is the LSB of the binary valueECET 331 - Digital Integrated CircuitsRepeated Division by 2 Methodz Step 2:• Divide the quotient from Step 1 by 2• The remainder, 0 or 1, is the next most significant bitECET 331 - Digital Integrated CircuitsRepeated Division by 2 Methodz Step 3: • Continue to execute Step 2 until the quotient is 0• The last remainder is the MSBECET 331 - Digital Integrated CircuitsRepeated Division by 2 MethodExample: Convert 35 to binary numberStep Quotient Remainder Binary[1] 17 1 1 (LSB)[2]811[3]400[4]200[5]100[6]001 (MSB)5ECET 331 - Digital Integrated CircuitsFractional Binary Numbersz Radix point• The generalized decimal point• The dividing line between positive and negative powers for positional multipliersz Binary point• The radix point for binary numbersECET 331 - Digital Integrated CircuitsFractional Binary Numbersz The value immediately to the right of the binary point is 2-1= 1/(21)= 0.5z The next value to the right is 2-2= 1/(22)= 0.25z The next value to the right is 2-3= 1/(23) = 0.125, and so onECET 331 - Digital Integrated CircuitsBinary FractionExample:0.101101 = (1 x 2-1) + (0 x 2-2) +(1 x 2-3) + (1 x 2-4) +(0 x 2-5) + (1 x 2-6)= 1/2 + 0 + 1/8 + 1/16 + 1/64= 45/64= 0.703125ECET 331 - Digital Integrated CircuitsFractional-Decimal-to-Fractional-Binary Conversionz Step 1:•Multiply the decimal fraction by 2 •The integer part, 0 or 1, is the first bit to the right of the binary pointECET 331 - Digital Integrated CircuitsFractional-Decimal-to-Fractional-Binary Conversionz Step 2: • Discard the integer part from Step 1 and repeat Step 1 until the fraction repeats or terminatesECET 331 - Digital Integrated CircuitsFractional-Decimal-to-Fractional-Binary ConversionExample: Convert 0.35 to fractional binary:Integer Part0.35 x 2 = 0.7 00.70 x 2 = 1.4 10.40 x 2 = 0.8 00.80 x 2 = 1.6 10.60 x 2 = 1.2 10.20 x 2 = 0.4 00.40 x 2 = 0.8 Last 4 digits repeat0.80 x 2 = 1.6 …So, (0.35)10= (0.010110)26ECET 331 - Digital Integrated CircuitsHexadecimal Numbersz Base-16 number systemz Primarily used as a shorthand form of binary numbersECET 331 - Digital Integrated CircuitsCounting in Hexadecimalz Values range from 0 to F with the letters A to F used to represent the values 10 to 15 respectivelyz Positional multipliers are powers of 16:160= 1, 161= 16, 162= 256, etc.ECET 331 - Digital Integrated CircuitsHexadecimal-to-decimal Conversionz Multiply each digit by its power-of-16 positional multiplier and add the productsExample:1FD5= (1x163) + (15x162) + (13x161) + (5x160)= (1x4096) + (15x256) + (13x16) + (5x1)= 8149ECET 331 - Digital Integrated