# Combinational Circuits Digital Logic (Materials taken primarily from:

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1 Combinational Circuits Digital Logic (Materials taken primarily from: )

2 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!

3 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

4 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

5 Truth Tables

6 Logic Gates

7 Multi-way Gates One row for each possible input combination N inputs => 2 N rows

8 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,

9 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

10 Expressing any Boolean Function Using Ingredients. AND gates. OR gates. NOT gates. Wire. AND, OR and NOT 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.

11 Translate Boolean Formula to Boolean Circuits Majority: Y = ~ABC + A~BC + AB~C + ABC

12 Axioms

13 Translate Boolean Formula to Boolean Circuits

14 DeMorgan s Theorem A + B = A B and A B = A + B

15 DeMorgan s Law

16 Translate Boolean Formula to Boolean Circuits (NAND-NAND form)

17 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

18 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

19 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!

20 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

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