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ECE 3110: Introduction to Digital Systems

ECE 3110: Introduction to Digital Systems. Number Systems. Previous class Summary. Electronics/sw aspects of digital design Integrated Circuits (wafer,die,SSI,MSI,LSI,VLSI) PLDs: PLAs,PALs,CPLD,FPGA ASIC Digital design levels. Binary Representation.

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ECE 3110: Introduction to Digital Systems

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  1. ECE 3110: Introduction to Digital Systems Number Systems

  2. Previous class Summary • Electronics/sw aspects of digital design • Integrated Circuits (wafer,die,SSI,MSI,LSI,VLSI) • PLDs: PLAs,PALs,CPLD,FPGA • ASIC • Digital design levels Dr. Xubin He ECE 3110: Introduction to Digital systems

  3. Binary Representation • The basis of all digital data is binary representation. • Binary - means ‘two’ • 1, 0 • True, False • Hot, Cold • On, Off • We must be able to handle more than just values for real world problems • 1, 0, 56 • True, False, Maybe • Hot, Cold, LukeWarm, Cool • On, Off, Leaky

  4. Number Systems • To talk about binary data, we must first talk about number systems • The decimal number system (base 10) you should be familiar with! • Positional number system

  5. Positional Notation Value 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 position. dp-1dp-2…d1d0.d-1­d-2…d-n Radix point n, p, r>=2 Sum of each digit multiplied by the corresponding power of the radix.

  6. The decimal number system (base 10) you should 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 system). A digit in base 2 is also called a ‘bit’. • A digit in base R can range from 0 to R-1 • 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 Hexadecimal or just ‘Hex’.

  7. Base 10, Base 2, Base 8, Base 16 953.7810= 9 x 102 + 5 x 101 + 3 x 100 + 7 x 10-1 + 8 x 10-2 = 900 + 50 + 3 + .7 + .08 = 953.78 1011.112= 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2 = 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75 43168= 4 x 83 + 3 x 82 + 1 x 81 + 6 x 80 = 2048 + 192 + 8 + 6 = 2254 A2F16 = 10x162 + 2x161 + 15x160 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607

  8. 2-3 = 0.1252-2 = 0.252-1 = 0.520 = 121 = 222 = 423 = 824 = 1625 =3226 = 6427 = 12828 = 25629 = 512210 = 1024211 = 2048212 = 4096 Common Powers 160 = 1 = 20161 = 16 = 24162 = 256 = 28163 = 4096 = 212 210 = 1024 = 1 K220 = 1048576 = 1 M (1 Megabits) = 1024 K = 210 x 210230 = 1073741824 = 1 G (1 Gigabits)

  9. Least Significant DigitMost Significant Digit 5310 = 1101012 Most Significant Digit (has weight of 25 or 32). For base 2, also called Most Significant Bit (MSB). Always LEFTMOST digit. Least Significant Digit (has weight of 20 or 1). For base 2, also called Least Significant Bit (LSB). Always RIGHTMOST digit. Dr. Xubin He ECE 3110: Introduction to Digital systems

  10. Hex (base 16) to Binary Conversion Each Hex digit represents 4 bits. To convert a Hex number to Binary, simply convert each Hex digit to its four bit value. Hex Digits to binary:016 = 00002116 = 00012216 = 00102316 = 00112416 = 01002516 = 01012616 = 01102716 = 01112816 = 10002 Hex Digits to binary (cont):916 = 10012A16 = 10102B16 = 10112C16 = 11002D16 = 11012E16 = 11102F16 = 11112

  11. Hex to Binary, Binary to Hex A2F16 = 1010 0010 1111234516 = 0011 0100 01012 Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers.10100012 = 010100012= 5116 Padded with a zero

  12. Hex to Binary, Binary to Hex A2F16 = 1010 0010 1111234516 = 0011 0100 01012 Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers.10100012 = 010100012= 5116 Padded with a zero

  13. Conversion of Any Base to Decimal Converting from ANY base to decimal is done by multiplying each digit by its weight and summing. Binary to Decimal 1011.112= 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2 = 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75 Hex to Decimal A2F16 = 10x162 + 2x161 + 15x160 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607

  14. A 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! 1101111100112 = 1101 1111 00112 = D16 F16 316 = 13 x 162 + 15 x 161 + 3x160 = 13 x 256 + 15 x 16 + 3 x 1 = 3328 + 240 + 3 = 357110 Of course, you can also use the binary, hex conversion feature on your calculator. Calculators won’t be allowed on the first test, though…...

  15. Conversion of Decimal Integer To ANY Base Divide Number N by base R until quotient is 0. Remainder at EACH step is a digit in base R, from Least Significant digit to Most significant digit. Dr. Xubin He ECE 3110: Introduction to Digital systems

  16. Conversion of Decimal Integer To ANY BaseExample Convert 53 to binary 53/2 = 26, rem = 1 26/2 = 13, rem = 0 13/2 = 6 , rem = 1 6 /2 = 3, rem = 0 3/2 = 1, rem = 1 1/2 = 0, rem = 1 5310 = 1101012 = 1x25 + 1x24 + 0x23 + 1x22 + 0x21 + 1x20 = 32 + 16 + 0 + 4 + 0 + 1 = 53 Least Significant Digit Most Significant Digit

  17. More Conversions Convert 53 to Hex 53/16 = 3, rem = 5 3 /16 = 0 , rem = 35310 = 3516 = 3 x 161 + 5 x 160 = 48 + 5 = 53 341710=??? 16 Dr. Xubin He ECE 3110: Introduction to Digital systems

  18. Binary Numbers Again Recall that N binary digits (N bits) can represent unsigned integers from 0 to 2N-1. 4 bits = 0 to 158 bits = 0 to 25516 bits = 0 to 65535 Besides simply representation, we would like to also do arithmetic operations on numbers in binary form. Principle operations are addition and subtraction. Dr. Xubin He ECE 3110: Introduction to Digital systems

  19. Next… • Addition/Subtraction • Representation of Negative Numbers Dr. Xubin He ECE 3110: Introduction to Digital systems

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