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Combinational Logic ReviewgDigital Devices was a LONG, LONG time ago in a galaxy FAR, FAR, AWAYformanyofyouFAR, AWAYfor many of you.We don’t expect you to remember everything you learned in Digital Devices but you need to remember > 0%Digital Devices, but you need to remember > 0%.We will review some to help you remember. You also need to go back and look at old notes. After a couple of days of review, we will expect you to be up to speed and then we willZOOMwill expect you to be up to speed, and then we will ZOOMalong.Ask QUESTIONS during CLASS to SLOW things down.V 0.5 1s QU S ONS du g C SS o S OW gs dow .Binary Representation• The basis of all digital data is binary representation.• Binary - means ‘two’– 1, 0–True, Falseld–Hot, Cold– On, OffWtbblthdl thjtlf•We must be able to handle more than just values for real world problems–10 561, 0, 56– True, False, Maybe–Hot, Cold, LukeWarm, CoolV 0.5 2,, ,– On, Off, LeakyNumber Systemsy• To talk about binary data, we must first talk about number systemsnumber systems• The decimal number system (base 10) you should be familiar with!be familiar with!– A digit in base 10 ranges from 0 to 9.–A digit in base 2 ranges from 0 to 1 (binary number gg(ysystem). A digit in base 2 is also called a ‘bit’.– A digit in base R can range from 0 to R-1di i i f–A digit in Base 16 can range from 0 to 16-1 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Use letters A-F to represent values 10 to 15. Base 16 is also called V 0.5 3pHexadecimal or just ‘Hex’.Positional NotationValue of number is determined by multiplying each digit by a weight and then summing. The weight of each digit is a POWER of the BASE and is determined by positionPOWER of the BASE and is determined by position.953.78 = 9 * 102+ 5 * 101+ 3 * 100+ 7 * 10-1+ 8 * 10-2= 900 + 50 + 3 + .7 + .08 = 953.780b1011 11 1*23+ 0*22+ 1*21+ 1*20+ 1*21+ 1*220b1011.11 = 1*23+ 0*22+ 1*21+ 1*20+ 1*2-1+ 1*2-2= 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.750xA2F = 10*162+ 2*161+ 15*160=10*256+2*16+15*1V 0.5 4 10 256 2 16 15 1 = 2560 + 32 + 15 = 2607Base 10, Base 2, Base 16The textbook uses subscripts to represent different bases (ie. A2F16, 953.7810, 1011.112)I will use special symbols to represent the different basesI will use special symbols to represent the different bases.The default base will be decimal, no special symbol for base 10.The ‘0x’ will be used for base 16 (0xA2F)The ‘0b’ will be used for base 2 (0b10101111)If ALL numbers on a page are the same base (ie, all in base 16 or base 2 or whatever) then no symbols will be used and )ya statement will be present that will state the base (ie, all numbers on this page are in base 16).V 0.5 5Common Powers2-3= 0.1252-2= 0.252-1=05160= 1 = 20161=16=2421= 0.520= 121= 222=4161= 16 = 24162= 256 = 28163= 4096 = 2122 423= 824 = 1625=3226= 6427= 12828= 256210= 1024 = 1 Ki (kilobinary)220= 1048576 = 1 Mi (1 megabinary) = 1024 K = 210* 21029= 512210 = 1024211= 20482124096(g y)230= 1073741824 = 1 Gi (1 gigabinary) V 0.5 6212= 4096Conversion of Any Base to DecimalConverting from ANY base to decimal is done by multiplying each digit by its weight and summing.Binary to Decimal0b1011.11 = 1*23+ 0*22+ 1*21+ 1*20+ 1*2-1+ 1*2-2= 8 + 0 + 2 + 1 + 0.5 + 0.25 =1175= 11.75Hex to Decimal0xA2F = 10*162+ 2*161+ 15*160= 10 * 256 + 2 * 16 + 15 * 1 V 0.5 7= 2560 + 32 + 15 = 2607Conversion of Decimal Integer To ANY BaseTo ANY BaseDivide Number N by base R until quotient is 0. Remainder at EACH step is a digit in base R, from Least Significant digit to pg , g gMost significant digit.Convert 53 to binary53/2=26 rem=1Least Significant Digit53/2 26, rem 126/2 = 13, rem = 013/2 = 6 , rem = 16/2 3 0gg6 /2 = 3, rem = 03/2 = 1, rem = 11/2 = 0, rem = 1 Most Significant Digit53 = 0b 110101=1*25+1*24+0*23+1*22+0*21+1*20ggV 0.5 8 12+ 12+ 02+ 12+ 02+ 12= 32 + 16 + 0 + 4 + 0 + 1 = 53Least Significant DigitLeast Significant DigitMost Significant Digit53 = 0b 110101Most Significant DigitL t Si ifi t Di itMost Significant Digit (has weight of 25or 32). For base 2, also ll d M Si ifiLeast Significant Digit (has weight of 20or 1). For base 2, also called called Most Significant Bit (MSB). Always LEFTMOST digit.Least Significant Bit (LSB). Always RIGHTMOST digit.V 0.5 9gRIGHTMOST digit.More ConversionsMore ConversionsConvert 53 to Hex53/16 = 3, rem = 53/16 = 0 rem=33 /16 = 0, rem = 353 = 0x35 = 3*161+5*160= 3 * 16+ 5 * 16= 48 + 5 = 53V 0.5 10Hex (base 16) to Binary Conversion() yEach Hex digit represents 4 bits. To convert a Hex number to gpBinary, simply convert each Hex digit to its four bit value.Hex Digits to binary:H Di it t bi ( t)gy0x0 = 0b 00000x1 = 0b 00010x2 = 0b 0010Hex Digits to binary (cont):0x9 = 0b 10010xA = 0b 10100x2 = 0b 00100x3 = 0b 00110x4 = 0b 01000xB = 0b 10110xC = 0b 11000xD = 0b 11010x5 = 0b 01010x6 = 0b 01100x7 = 0b 01110xE = 0b 11100xF = 0b 1111V 0.5 110x8 = 0b 1000Hex to Binary Binary to HexHex to Binary, Binary to Hex0xA2F = 0b 1010 0010 11110x345 = 0b 0011 0100 0101Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not h4bih dih f id bhave 4 bits, then pad with zeros for unsigned numbers.0b 1010001 = 0b 0101 0001 = 0x51 Padded with a zeroV 0.5 12Padded with a zeroA Trick!If faced with a large binary number that has to be converted to decimal, I first convert the binary number to HEX then convert the HEX to decimal Less work!to HEX, then convert the HEX to decimal. Less work!0b 110111110011 = 0b 1101 1111 0011 =DF3= D F 3= 13 * 162+ 15 * 161+ 3*160= 13 * 256 + 15 * 16 + 3 * 1= 3328 + 240 + 3= 3571Of course, you can also use the binary, hex conversion feature on your calculator. Too bad calculators won’t be allowed on the first test, though…...V 0.5 13s es , oug …...Binary Numbers AgainygRecall than N binary digits (N bits) can represent unsigned integers from 0 to 2N1integers from 0 to 2N-1. 4 bits = 0 to 158 bits = 0 to 25516 bits = 0 to 65535B id i l t ti ld lik t l dBesides simply representation, we would like to also do arithmetic operations on numbers in binary form. Principle operations are addition and subtraction.V 0.5 14Binary Arithmetic, SubtractionThe rules for binary arithmetic are: The rules for binary subtraction are: 0 + 0 = 0, carry = 01 + 0 =


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MSU ECE 3724 - Combinational Logic Review

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