Working with DIGITAL Information Represent Knowledge Digitally Perform Simple Reasoning Perform Logical Operations Write Formulas in Boolean Logic How Boolean Formulas are implemented in machines Perform Arithmetic Operations

What Does “Digital” Mean? Analog signal Inifinite possible values Ex: voltage on a wire created by microphone Digital signal Finite possible values Ex: button pressed on a keypad 3 4 2 1 digital signal analog signal Possible values: 1.00, 1.01, 2.0000009, ... infinite possibilities Possible values: 0, 1, 2, 3, or 4. That’s it. 4 value time value time 3 2 1

Example of Digitization Benefit Analog signal (e.g., audio) may lose quality Voltage levels not saved/copied/transmitted perfectly Digitized version enables near-perfect save/cpy/trn. “Sample” voltage at particular rate, save sample using bit encoding Voltage levels still not kept perfectly But we can distinguish 0s from 1s Volts 1 2 3 original signal time 1 2 3 received signal How fix -- higher, lower, ? 11 11 Digitized signal not perfect re-creation, but higher sampling rate and more bits per encoding brings closer. 10 10 lengthy transmission (e.g, cell phone) 01 time Volts digitized signal time 1 a2d time Can fix -- easily distinguish 0s and 1s, restore 1 lengthy transmission (e.g, cell phone) 01 10 11 10 11 same Let bit encoding be: 1 V: “01” 2 V: “10” 3 V: “11” Volts 1 2 3 d2a time

Digitized Audio: Compression Benefit Digitized audio can be compressed e.g., MP3s A CD can hold about 20 songs uncompressed, but about 200 compressed Compression also done on digitized pictures (jpeg), movies (mpeg), and more Digitization has many other benefits too Example compression scheme: 00 --> 0000000000 01 --> 1111111111 1X --> X 0000000000 0000000000 0000001111 1111111111 00 00 10000001111 01

Benefits of Digital Reliable storage (CD, DVD, …) Compression (MP3, JPEG, …) Reliable transmission (cell phones, digital TVs, …) Conversion from Analog to Digital Technology

Digital Encoding for All Information We can represent any digital data using only binary digits (0 and 1), or bits. ASCII encoding: A 01000001 B 01000010 … … Why binary numbers? Base ten: decimal numbers (0,1,2,3,4,5,6,7,8,9) Base two: binary numbers (0,1) Base eight: octal numbers (0,1,2,3,4,5,6,7) Base sixteen: hexadecimal numbers (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)

Process Stored Digital Information

Combinational Logic Circuits A digital circuit whose output depends solely on the present combination of input values is called a combinational circuit Logic gates – building blocks of logic circuits AND OR NOT Boolean Algebra Boolean algebra is a branch of mathematics that uses variables whose values can only be 1 or 0 (“true” or “false”, respectively) and whose operators, like AND, OR, NOT, operate on such variables and return 1 or 0. We can build circuits by doing math

Relating Boolean Algebra to Digital Information Logic Gates Truth Tables 1 y x F 1 x y F 1 F x Transistor Circuits

Boolean Algebra Notation By defining logic gates based on Boolean algebra, we can use algebraic methods to manipulate digital circuits Start with notation: Writing a AND b, a OR b, and NOT(a) is cumbersome Use symbols: a * b, a + b, and a’ (in fact, a * b can be just ab). Original: w = (p AND NOT(s) AND k) OR t New: w = ps’k + t Spoken as “w equals p and s prime and k, or t” Or even just “w equals p s prime k, or t” s’ known as “complement of s” While symbols come from regular algebra, don’t say “times” or “plus” Boolean algebra precedence, highest precedence first. Symbol Name Description ( ) Parentheses Evaluate expressions nested in parentheses first ’ NOT Evaluate from left to right * AND Evaluate from left to right + OR Evaluate from left to right

Boolean Algebra Properties Commutative a + b = b + a a * b = b * a Distributive a * (b + c) = a * b + a * c a + (b * c) = (a + b) * (a + c) Associative (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Identity 0 + a = a + 0 = a 1 * a = a * 1 = a Complement a + a’ = 1 a * a’ = 0

Boolean Algebra: Additional Properties Null elements a + 1 = 1 a * 0 = 0 Idempotent Law a + a = a a * a = a Involution Law (a’)’ = a De Morgan’s Law (a + b)’ = a’ b’ (a b)’ = a’ + b’ f = c’ (h + p) f = h c’ + h’ p c’

Boolean Functions Converting a truth table to an equation Boolean function is a mapping of each possible combination of input values to either 0 or 1. Boolean function can be represented as an equation, a circuit, and as a truth table. Converting a truth table to an equation F = a b + a’ F = a’ b’ + a’ b + a b For any function, there may be many equivalent equations, and many equivalent circuits, but there is only one truth table!