1. Today’s Topics Digital Logic Design Digital Logic Design Boolean Logic Boolean Logic Digital Logic Circuits Digital Logic Circuits

2. What is Digital Logic? Discrete circuitry, where the only valid values are a 1 or 0. Discrete circuitry, where the only valid values are a 1 or 0. Represented by building blocks of defined functionality. Represented by building blocks of defined functionality. An application of boolean logic. An application of boolean logic.

3. Variables(e.g.: A, B, …)‏ Variables(e.g.: A, B, …)‏ can have value of 0 or 1 (false or true)‏ can have value of 0 or 1 (false or true)‏ FunctionalLogicalBoolean FunctionalLogicalBoolean NOT(A)~AA AND(A, B)A AND BAB OR(A, B)A OR B A+B XOR(A, B)A XOR B A B NAND(A, B)A NAND BAB NOR(A, B)A NOR B A + B XNOR(A, B)A XNOR B A B Boolean notation + +

4. Truth tables: 0 = false, 1 = true XYR000 010 101 110XYZR0000 0010 0100 0111 1001 1011 1101 1111 2 input, 1 output R = XY 3 input, 1 output R = X+(YZ)‏

5. Using truth tables Define a Boolean function Define a Boolean function Specify an output for each possible combination of inputs Specify an output for each possible combination of inputs Prove that two functions are equivalent Prove that two functions are equivalent Make a truth table for each function Make a truth table for each function Identical outputs implies equivalent functions Identical outputs implies equivalent functions

6. Internal (electric) representation of binary codes Power source Power source Gates Gates Made of one or more transistors Made of one or more transistors Only 2 voltages are permitted Only 2 voltages are permitted Low (e.g., 0.5v) represents binary 0 Low (e.g., 0.5v) represents binary 0 High (e.g., 1.0v) represents binary 1 High (e.g., 1.0v) represents binary 1 Can "convert" low  high using gates Can "convert" low  high using gates For any given set of inputs (0s and 1s), gates can be combined to produce either output (0 or 1). For any given set of inputs (0s and 1s), gates can be combined to produce either output (0 or 1). These combinations of gates are called digital circuits These combinations of gates are called digital circuits

7. Gates are used to implement Boolean algebra "inverter"

8. Any desired output can be produced by combining these three "functions" to create a circuit.

9. These additional gates can simplify and/or reduce cost of a circuit

10. Boolean functions A function of n binary variables has 2 n possible combinations of values for the variables. A function of n binary variables has 2 n possible combinations of values for the variables. e.g., f(A,B,C) has 2 3 = 8 combinations of values for A, B, and C. e.g., f(A,B,C) has 2 3 = 8 combinations of values for A, B, and C. So … a function can be completely described by its truth table. So … a function can be completely described by its truth table.ABC000 001 010 011 100 101 110 111

11. Example: Let f (A,B,C) be defined by the truth table at the right. Let f (A,B,C) be defined by the truth table at the right. To write this in Boolean notation To write this in Boolean notation n select all rows with R = 1 n AND the values in each row, using X for 1, X for 0 n OR the resulting terms n Set R = resulting expression ABCR0000 0011 0101 0110 1001 1010 1100 1111

12. Connection! We have gates that can implement this function as a circuit. Connection! We have gates that can implement this function as a circuit. In its most primitive form: In its most primitive form:

13. Verification Test all possible combinations of input to verify that the circuit implements the function ABCR0000 0011 0101 0110 1001 1010 1100 1111

14. Simplifications: Multiple-input gates Multiple-input gates Equivalent gates Equivalent gates Use Boolean algebra to simplify the function before implementing the circuit. Use Boolean algebra to simplify the function before implementing the circuit.

15. Boolean identities Definition: A B = AB + AB+

16. Simplify Distributive Definition, distributive Definition

17. Representation of arithmetic / algebra circuits Truth table table Boolean Circuitfunction diagram

18. Conversions Circuit diagram to Boolean function Circuit diagram to Boolean function Circuit diagram to truth table Circuit diagram to truth table Truth table to diagram Truth table to diagram Simplification by eliminating the "don't cares" Simplification by eliminating the "don't cares" Truth table to Boolean function Truth table to Boolean function Simplification by eliminating the "don't cares" Simplification by eliminating the "don't cares" Boolean function to truth table Boolean function to truth table Simplification by Boolean identities Simplification by Boolean identities Boolean function to circuit diagram Boolean function to circuit diagram Simplification by Boolean identities Simplification by Boolean identities

19. CMOS Inverter at the transistor level

20. Integrated circuits Scale Scale SSI: 1956-1960’s (10’s of transistors)‏ SSI: 1956-1960’s (10’s of transistors)‏ MSI: 1960’s -1970’s (100’s of transistors)‏ MSI: 1960’s -1970’s (100’s of transistors)‏ LSI: 1970’s – 1980’s (10,000’s of transistors)‏ LSI: 1970’s – 1980’s (10,000’s of transistors)‏ VLSI: 1980’s – present (100k – billions)‏ VLSI: 1980’s – present (100k – billions)‏ Gate to I/O ratio Gate to I/O ratio I/O take much more space than gates I/O take much more space than gates

21. VLSI: Pentium 3

22. Multiplexers

23. Decoder (demultiplxer) – select one output line – select one output line

24. Multiplexers “All-purpose” implementation of a truth table. “All-purpose” implementation of a truth table. 2 n combinations of data inputs, n control inputs, one output 2 n combinations of data inputs, n control inputs, one output 1 0In B In A Select Output

25. Show the truth table for this multiplexer (“Homework” #3, problem #1) 4 data bits, 2 control bits D3D3 D0D0 Output D1D1 D2D2 S0S0 S1S1 S0S0 S1S1 00 D0D0 01 D1D1 10 D2D2 11 D3D3

26. What does this circuit do? (“Homework” #3, problem #2) ABSC 0000 0110 1010 1101

27. What does this circuit do? (“Homework” #3, problem #3)

28. Comparators

29. Shifters (right/left) controlled by C

30. Adders Bitwise addition implements “sum bit” and “carry bit” Bitwise addition implements “sum bit” and “carry bit” Carry digits “ripple” to next adder Carry digits “ripple” to next adder Half adder: 2 inputs (corresponding bits of two numbers), 2 outputs (sum bit and carry-out bit)‏ Half adder: 2 inputs (corresponding bits of two numbers), 2 outputs (sum bit and carry-out bit)‏ Full adder: 3 inputs (same as above, plus carry-in bit), 2 outputs (same as above)‏ Full adder: 3 inputs (same as above, plus carry-in bit), 2 outputs (same as above)‏

31. Half adder

32. Full adder

33. Ripple carry adder

34. Questions on gates, circuits, etc.? Questions on gates, circuits, etc.?

35. Quiz #4 Tuesday in class Calculator and one 4x6 two-sided notecard permitted No sharing Short answer, multiple-choice 40 minutes, 25 points Can be dropped as lowest quiz Coverage: A few repeat questions from Quiz #3 Lectures #15,16,17,18 Program #4 Homeworks #2, 3

36. Quiz #4 Topics Two of the problems will be very similar to problems #2 and #4 on Quiz #3 Solutions are posted System stack How procedure call/return works call, ret, ret n Stack frame Parameters Value/Reference Accessing Arrays Declaration Accessing elements (address calculation) Passing as parameters (reference) Array processing loops Array addressing modes

37. Quiz #4 Topics Data-related operators OFFSET, PTR, TYPE, LENGTHOF, SIZEOF Address calculation (2-D arrays) String operations lodsb, stosb, std, cld Macros Parameters Local labels Compared to procedures NOTE: local variables will NOT be covered

38. Quiz #4 Topics Reverse Polish Notation (RPN) Convert in-fix  post-fix Post-fix expression evaluation Boolean logic Truth tables Gates and circuits Tracing only

39. Questions? Hand in “Homework” #3 Quiz #4 Tuesday Get moving on Program #5