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 cetinkayae@mst.edu http://web.mst.edu/~cetinkayae/teaching/cpe2210fall2016 26 August 2016 rev. 16.0 2014 2016 Egemen K. Çetinkaya
Number Systems Outline Signals and representations Number systems Summary 26 August 2016 MST CPE2210 Number Systems 2
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: 72.16 F degrees human speech pressure light value time Possible values: 1.00, 1.01, 2.0000009,... infinite possibilities 26 August 2016 MST CPE2210 Number Systems 3
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 value 4 3 2 1 0 time Possible values: 0, 1, 2, 3, or 4. That s it. 26 August 2016 MST CPE2210 Number Systems 4
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 value Embedded systems: for a particular purpose 1 0 time 26 August 2016 MST CPE2210 Number Systems 5
Why binary system? Digital Systems Representations 26 August 2016 MST CPE2210 Number Systems 6
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 26 August 2016 MST CPE2210 Number Systems 7
Digital-Analog Conversion (a) ADC (A2D) and DAC (D2A) wire microphone analog-todigital converter Volts 3 2 1 0 samples Egemen K. Çetinkaya analog signal on wire 00 01 10 10 11 11 11 01 10 10 00 time digitized signal 0001101011111101101000 (b) 0001101011111101101000 read from tape, CD, etc. wire speaker digital-toanalog converter Volts 3 2 1 0 00 01 10 10 11 11 11 01 10 10 00 analog signal reproduced from digitized signal time 26 August 2016 MST CPE2210 Number Systems 8
Digital vs. Analog Pros and Cons What are the pros and cons of analog vs. digital? 26 August 2016 MST CPE2210 Number Systems 9
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 00000000 00 26 August 2016 MST CPE2210 Number Systems 10
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 JJ 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 26 August 2016 MST CPE2210 Number Systems 11
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 26 August 2016 MST CPE2210 Number Systems 12
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? 26 August 2016 MST CPE2210 Number Systems 13
Number Systems Floating Point vs. Fixed Point Representation Floating point approximates real number 2015.82814 = + 2.01582814 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 26 August 2016 MST CPE2210 Number Systems 14
Number Systems Decimal Representation position 3 2 1 0-1 -2 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 10 3 10 2 10 1 10 0 10-1 10-2 decimal example digit 2 0 1 5 0 8 whole part fractional part radix (decimal) point 26 August 2016 MST CPE2210 Number Systems 15
Number Systems Binary Representation position 3 2 1 0-1 -2 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 2 3 2 2 2 1 2 0 2-1 2-2 binary example digit 1 0 1 1 1 0 binary point 26 August 2016 MST CPE2210 Number Systems 16
Number Systems Octal Representation position 3 2 1 0-1 -2 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 8 3 8 2 8 1 8 0 8-1 8-2 octal example digit 1 7 5 3 6 2 26 August 2016 MST CPE2210 Number Systems 17
Number Systems Hexadecimal Representation position 3 2 1 0-1 -2 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 16 3 16 2 16 1 16 0 16-1 16-2 hex example digit 2 0 A F 0 0 26 August 2016 MST CPE2210 Number Systems 18
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 = 26 August 2016 MST CPE2210 Number Systems 19
Binary Systems Powers of Two 2 0 = 1 2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 16 2 5 = 32 2 6 = 64 2 7 = 128 2 8 = 256 2 9 = 512 2 10 = 1024 26 August 2016 MST CPE2210 Number Systems 20
Number System Conversion Binary to Decimal binary 1 binary weight 2 0 multiply weights and add decimal 1 binary 1 0 binary weight 2 1 2 0 decimal 2 + 0 = 2 binary 1 0 1 binary weight 2 2 2 1 2 0 decimal 4 + 0 + 1 = 5 Egemen K. Çetinkaya 26 August 2016 MST CPE2210 Number Systems 21
Number System Conversion Binary to Decimal Egemen K. Çetinkaya binary 0 1 1 binary weight 2 2 2 1 2 0 decimal 0 + 2 + 1 = 3 binary 1 0 1. 1 binary weight 2 2 2 1 2 0 2-1 decimal 4 + 0 + 1 + 0.5 = 5.5 26 August 2016 MST CPE2210 Number Systems 22
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 0 0 16 8 4 2 1 (b) 8 <= 12, so put 1 in 8 s place, current sum is 8 8 0 1 16 8 4 2 1 (c) 8+4=12 <= 12, so put 1 in 4 s place, current sum is 12 12 0 1 1 16 8 4 2 1 a (d) Reached desired 12, so put 0s in remaining places done 0 1 1 0 0 16 8 4 2 1 26 August 2016 MST CPE2210 Number Systems 23
Number Systems Base 16 System hex binary hex binary 0 8 1 9 2 A 3 B 4 C 5 D 6 E 7 F 26 August 2016 MST CPE2210 Number Systems 24
Number Systems Base 16 System hex binary 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 hex binary 8 1000 9 1001 A 1010 B 1011 C 1100 D 1101 E 1110 F 1111 26 August 2016 MST CPE2210 Number Systems 25
Number System Conversion Hex to Binary Examples Egemen K. Çetinkaya 26 August 2016 MST CPE2210 Number Systems 26
Number System Conversion Binary to Hex Examples Egemen K. Çetinkaya 26 August 2016 MST CPE2210 Number Systems 27
Number System Conversion Hex to Decimal Examples Egemen K. Çetinkaya 26 August 2016 MST CPE2210 Number Systems 28
Number System Conversion Decimal to Hex Examples Egemen K. Çetinkaya 26 August 2016 MST CPE2210 Number Systems 29
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? 1 0 1 0 1 1 0 1 26 August 2016 MST CPE2210 Number Systems 30
LSB: Least Significant Bit right-most bit MSB: Most Significant Bit higher-order bit left-most bit Example: LSB MSB Number Systems Representations 1 0 1 0 1 1 0 1 26 August 2016 MST CPE2210 Number Systems 31
bit: binary digit (b) Byte: 8-bits (B) nibble: 4-bits Number Systems Representations 1 0 1 0 1 1 0 1 high nibble low nibble 26 August 2016 MST CPE2210 Number Systems 32
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 µ 10 6 Mega M 2 20 mebi Mi 10 9 nano n 10 9 Giga G 2 30 gibi Gi 10 12 pico p 10 12 Tera T 2 40 tebi Ti 10 15 femto f 10 15 Peta P 2 50 pebi Pi 10 18 atto a 10 18 Exa E 2 60 exbi Ei 10 21 zepto z 10 21 Zetta Z 10 24 yocto y 10 24 Yotta Y Egemen K. Çetinkaya 26 August 2016 MST CPE2210 Number Systems 33
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 2 0-2 10 by heart 26 August 2016 MST CPE2210 Number Systems 34
References and Further Reading [V2011] Frank Vahid, Digital Design with RTL Design, VHDL, and Verilog, 2nd edition, Wiley, 2011. 26 August 2016 MST CPE2210 Number Systems 35
End of Foils 26 August 2016 MST CPE2210 Number Systems 36