Introduction to Digital Logic Missouri S&T University CPE 2210 Number Systems

Transcription

1 Introduction to Digital Logic Missouri S&T University CPE 2210 Number Systems Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology 22 January 2018 rev Egemen K. Çetinkaya

2 Number Systems Outline Signals and representations Number systems Summary 22 January 2018 MST CPE2210 Number Systems 2

3 value Egemen K. Çetinkaya Digital vs. Analog Analog Signal Signal: physical phenomenon has unique value at every instant of time Analog signal (aka continuous signal) Infinite set of possible values Examples: temperature: F degrees human speech pressure light time Possible values: 1.00, 1.01, ,... infinite possibilities 22 January 2018 MST CPE2210 Number Systems 3

4 value Egemen K. Çetinkaya Digital vs. Analog Digital Signal Signal: physical phenomenon has unique value at every instant of time Digital signal (aka discrete signal) Finite set of possible values Examples: pressing a button on keypad switch on/off time Possible values: 0, 1, 2, 3, or 4. That s it. 22 January 2018 MST CPE2210 Number Systems 4

5 value Egemen K. Çetinkaya Digital Systems Representations Digital signals represented by two values on/off, 0/1 Two-value representation: binary representation A single value is called bit (binary digit) Digital system: take digital inputs generates digital outputs Digital circuits: connection of digital components Embedded systems: for a particular purpose 1 0 time 22 January 2018 MST CPE2210 Number Systems 5

6 Why binary system? Digital Systems Representations 22 January 2018 MST CPE2210 Number Systems 6

7 Why binary system? Digital Systems Representations Ease of operation compared to 3 digits or more ease of storage, computing, transmission Transistors operate on two-value logic transistor is a basic electrical circuit component 22 January 2018 MST CPE2210 Number Systems 7

8 Digital-Analog Conversion ADC (A2D) and DAC (D2A) Egemen K. Çetinkaya 22 January 2018 MST CPE2210 Number Systems 8

9 Digital vs. Analog Pros and Cons What are the pros and cons of analog vs. digital? 22 January 2018 MST CPE2210 Number Systems 9

10 Digital vs. Analog Pros and Cons What are the pros and cons of analog vs. digital? Analog signal is prone to noise amplified during transmission, storage, processing Digitized analog signal is never exact due to sampling Digital signal can be compressed repetitive patterns can be encoded in other way January 2018 MST CPE2210 Number Systems 10

11 Number Systems Overview Type Natural numbers N Explanation {0, 1, 2, } Integers Z {, -2, -1, 0, 1, 2, } Rational numbers Q m/n where m and n are integers and n 0: e.g. 5/4, -8/3 Irrational numbers J Any real number that can t be expressed as ratio of integers e.g.: π, e, 2 Real numbers R Rational & irrational numbers, +, 0, or Complex numbers C a+bi, where i 2 = 1 and a and b are real numbers 22 January 2018 MST CPE2210 Number Systems 11

12 Number Systems Representation Type Explanation positive x > 0 negative x < 0 non-negative x 0 non-positive x 0 signed (in computing) represents both negative and positive numbers unsigned (in computing) represents only non-negative numbers 22 January 2018 MST CPE2210 Number Systems 12

13 Number Systems Representations Important bases throughout the class: Decimal (base 10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] Binary (base 2) [0, 1] Octal (base 8) [0, 1, 2, 3, 4, 5, 6, 7] Hexadecimal (base 16) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F] For the number 3205, what is the minimum base? 22 January 2018 MST CPE2210 Number Systems 13

14 Number Systems Floating Point vs. Fixed Point Representation Floating point approximates real number = x 10 3 exponent sign mantissa (significand) base IEEE 754 standard is followed Fixed point: radix is fixed at a point less costly to represent compared to floating point Other real number representations: binary-coded decimal (BCD) logarithmic number systems 22 January 2018 MST CPE2210 Number Systems 14

15 Number Systems Decimal Representation position weight b 3 b 2 b 1 b 0 b -1 b -2 digit a 3 a 2 a 1 a 0 a -1 a -2 decimal example weight decimal example digit whole part fractional part radix (decimal) point 22 January 2018 MST CPE2210 Number Systems 15

16 Number Systems Binary Representation position weight b 3 b 2 b 1 b 0 b -1 b -2 digit a 3 a 2 a 1 a 0 a -1 a -2 binary example weight binary example digit binary point 22 January 2018 MST CPE2210 Number Systems 16

17 Number Systems Octal Representation position weight b 3 b 2 b 1 b 0 b -1 b -2 digit a 3 a 2 a 1 a 0 a -1 a -2 octal example weight octal example digit January 2018 MST CPE2210 Number Systems 17

18 Number Systems Hexadecimal Representation position weight b 3 b 2 b 1 b 0 b -1 b -2 digit a 3 a 2 a 1 a 0 a -1 a -2 hex example weight hex example digit 2 0 A F January 2018 MST CPE2210 Number Systems 18

19 Binary Systems Powers of Two 2 0 = 2 1 = 2 2 = 2 3 = 2 4 = 2 5 = 2 6 = 2 7 = 2 8 = 2 9 = 2 10 = 22 January 2018 MST CPE2210 Number Systems 19

20 Binary Systems Powers of Two 2 0 = = = = = = = = = = = January 2018 MST CPE2210 Number Systems 20

21 Number System Conversion Binary to Decimal binary 1 binary weight 2 0 multiply weights and add decimal 1 binary 1 0 binary weight decimal = 2 binary binary weight decimal = 5 Egemen K. Çetinkaya 22 January 2018 MST CPE2210 Number Systems 21

22 Number System Conversion Binary to Decimal Egemen K. Çetinkaya binary binary weight decimal = 3 binary binary weight decimal = January 2018 MST CPE2210 Number Systems 22

23 Number System Conversion Decimal to Binary Egemen K. Çetinkaya Desired decimal number: 12 Current sum Binary number (a) 16 > 12, too big; Put 0 in 16 s place (b) 8 <= 12, so put 1 in 8 s place, current sum is (c) 8+4=12 <= 12, so put 1 in 4 s place, current sum is a (d) Reached desired 12, so put 0s in remaining places done January 2018 MST CPE2210 Number Systems 23

24 Number Systems Base 16 System hex binary hex binary A 3 B 4 C 5 D 6 E 7 F 22 January 2018 MST CPE2210 Number Systems 24

25 Number Systems Base 16 System hex binary hex binary A 1010 B 1011 C 1100 D 1101 E 1110 F January 2018 MST CPE2210 Number Systems 25

26 Number System Conversion Hex to Binary Examples Egemen K. Çetinkaya 22 January 2018 MST CPE2210 Number Systems 26

27 Number System Conversion Binary to Hex Examples Egemen K. Çetinkaya 22 January 2018 MST CPE2210 Number Systems 27

28 Number System Conversion Hex to Decimal Examples Egemen K. Çetinkaya 22 January 2018 MST CPE2210 Number Systems 28

29 Number System Conversion Decimal to Hex Examples Egemen K. Çetinkaya 22 January 2018 MST CPE2210 Number Systems 29

30 LSB: Least Significant Bit right-most bit MSB: Most Significant Bit higher-order bit left-most bit Number Systems Representations Example: where is LSB and MSB? January 2018 MST CPE2210 Number Systems 30

31 LSB: Least Significant Bit right-most bit MSB: Most Significant Bit higher-order bit left-most bit Example: LSB MSB Number Systems Representations January 2018 MST CPE2210 Number Systems 31

32 bit: binary digit (b) Byte: 8-bits (B) nibble: 4-bits Number Systems Representations high nibble low nibble 22 January 2018 MST CPE2210 Number Systems 32

33 Performance Metrics Unit Multipliers SI decimal 10 1 deci d 10 1 deka da 10 2 centi c 10 2 hecto h EIC binary 10 3 milli m 10 3 kilo k 2 10 kibi Ki 10 6 micro 10 6 Mega M 2 20 mebi Mi 10 9 nano n 10 9 Giga G 2 30 gibi Gi pico p Tera T 2 40 tebi Ti femto f Peta P 2 50 pebi Pi atto a Exa E 2 60 exbi Ei zepto z Zetta Z yocto y Yotta Y Egemen K. Çetinkaya 22 January 2018 MST CPE2210 Number Systems 33

34 Signals can be: analog: continuous digital: discrete Important terminology: bit, byte, nibble, LSB, MSB Number Systems Summary Important number systems: decimal, binary, hex, octal Conversions will be needed throughout your careers: know by heart 22 January 2018 MST CPE2210 Number Systems 34

35 References and Further Reading [V2011] Frank Vahid, Digital Design with RTL Design, VHDL, and Verilog, 2nd edition, Wiley, [S2017] John Seiffertt, Digital Logic for Computing, 1st edition, Springer, January 2018 MST CPE2210 Number Systems 35

36 End of Foils 22 January 2018 MST CPE2210 Number Systems 36