Combinational Circuits
Digital Logic
(Materials taken primarily from:
http://www.facstaff.bucknell.edu/mastascu/eLessonsHTML/EEIndex.html
http://www.cs.Princeton.EDU/~cos126 )
Digital Systems
What is a digital system?
• Digital: signals are 0 or I
• Analog: signals vary continuously
Why digital systems?
• Accuracy and reliability
• Fast and cheap
Basic abstractions.
• On, off
• Switch that con turn something on or off
Characteristics
• The signals can and usually will change with time
Digital circuits you know
• Computer microprocessors
• Cell phones
• Thermostat
• Anti-lock brakes
All digital systems are binary but not all binary systems are digital!
Signals
• A digital signal takes on values of 0 (FALSE, OFF) or
1 (TRUE, ON).
– That signal might be a voltage, a switch closure, etc.
• We think in terms of zeros and ones, not in terms
of the values of the voltage---Logic Values
• Wires propagate these logic values
– Signals “flow” from right to left
Operations on Signals
Assume 2 signals are input to a circuit that has 1 output signal
•The output, C, depends upon the inputs, A and B
•It‘s a function of A & B
A signal can be viewed as the truth value of a logic proposition.
•A & B could each be either TRUE (a logical 1) or FALSE (a logical 0)
C value depends on the values of A and B
•Example: C is TRUE only when A is TRUE and B is TRUE.
We can construct a truth table describing this relationship.
•All possible combinations of A & B and the corresponding
values of C
Truth Tables
Logic Gates
Multi-way Gates
One row for each possible input combination
N inputs => 2N rows
Boolean Algebra
• An algebra in which the variables can take only
one of two values.
• Any Boolean function can be expressed using
AND, OR, NOT
– "Universal"
– E.g., XOR(x,y) = ~x & y | x & ~y
• Notation:
– AND: &, xy,
– OR: +, |
– NOT: ~, x, ‘
Sum Of Products
Any Boolean function can be expressed using AND,
OR, NOT.
• Sum-of-products is systematic procedure.
– form AND term for each 1 in truth table of Boolean
function
– OR terms together
Expressing any Boolean Function Using
AND, OR and NOT
• Ingredients.
– AND gates.
– OR gates.
– NOT gates.
– Wire.
• Instructions.
– Step 1: represent input and output signals with Boolean
variables.
– Step 2: construct truth table to carry out computation.
– Step 3: derive (simplified) Boolean expression using sumof products.
– Step 4: transform Boolean expression into circuit.
Translate Boolean Formula to Boolean
Circuits
Majority:
Y = ~ABC + A~BC + AB~C + ABC
Axioms
Translate Boolean Formula to Boolean
Circuits
DeMorgan’s Theorem
A + B = A B and A B = A + B
DeMorgan’s Law
Translate Boolean Formula to Boolean
Circuits (NAND-NAND form)
Many representations for Each
Function
Many possible circuits for each Boolean function.
• Sum-of-products not necessarily optimal in:
– number of gates (space)
– depth of circuit (time)
• Majority
Y = A‘BC + AB‘C + ABC' + ABC = AB + BC + AC
Simplifying Boolean Expressions
• A Karnaugh map is a method of grouping together
expressions with common factors used to eliminate
unwanted variables
– one square in the map for every row in the truth table
– any two adjacent cells differ by a change in one variable
– each single cell with a 1 represents a minterm in the
function
Karnaugh Maps
Examine the map:
The term on the left in the gray area of the map corresponds to:
A~BC
•The term on the right in the gray area of the map corresponds to:
•ABC
•These two terms can be combined to give
•AC
•When only one variable changes, you can eliminate that variable!
Simplifying Boolean Expressions Using Karnaugh
Maps
• Given a Boolean function described by a truth table or logic
function
– Draw the Karnaugh Map for the function.
– Use the information in the Karnaugh Map to determine
the smallest sum-of-products function
• crossing a vertical or horizontal cell boundary is a
change of only one variable
• each single cell that contains a 1 represents a minterm
in the function
• to combine variables, use groups of 2, 4, 8, etc.
– never use a group of 3, a group of 5, etc.
f(A,B,C) = A B + B C + A C