1. 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

2. 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, ,
... infinite possibilities
Possible values:
0, 1, 2, 3, or 4.
That’s it. 4
value
time
value
time 3 2 1

3. 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

4. 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 -->
01 -->
1X --> X

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

6. Digital Encoding for All Information
We can represent any digital data using only binary digits (0 and 1), or bits.
ASCII encoding: A B … …
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)

7. Process Stored Digital Information

10. 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

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

12. 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

13. 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

14. 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’

15. 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!