1. Computer Architecture Undergrad review CSC321

2. Terminology • “Digital” • Discrete, well defined values/steps • Opposite of analog • Analogy: digital is to analog as int is to double • “Binary” • A system consisting of two states • on/off, true/false, yes/no, high/low, 0/1 • Basis for modern computers CSC321

3. Terminology • “Bit” • Binary-digit • Smallest unit of storage in modern computers • “Nibble” – 4 bits • “Byte” – 8 bits • “Word” – typically two bytes but often refers to the “native” bit length of the machine CSC321

4. Data Representation • 1000001 • one million, one • sixteen million, seven hundred seventy seven thousand, two hundred, seventeen • two hundred sixty two thousand, one hundred forty five • sixty five • “A” • AJMP assembly language instruction CSC321

5. Boolean Algebra • Boolean algebra is an algebra that deals with binary variables and logic operations CSC321

6. Boolean Algebra • Boolean algebra consists of… • A set of symbols that represent variables • Use letters just like “regular algebra” • A, B, C, a, b, c… • Variables are binary (2-valued) • 0, 1 • Three basic operators • AND, OR, NOT • Other symbols • ( ) = CSC321

7. Boolean Operators • AND • Notation: A · B, AB, (AB), A(B) • Yields a value of 1 when both A and B are 1 • Yields a value of 0 when either A or B is 0 CSC321

8. Boolean Operators • OR • Notation: A + B • Yields a value of 1 when either A or B is 1 • Yields a value of 0 when both A and B are 0 CSC321

9. Boolean Operators • NOT • Notation: A, A’ • Yields a value of 1 A is 0 • Yields a value of 0 when A is 1 CSC321

10. Boolean Expressions • As in “regular” algebra, variables, operators, and symbols can be combined to form expressions or functions F(x, a, b) = x + (a · b) • F is a boolean function of three variables • Often written as: F = x + (a · b) CSC321

11. Boolean Functions • Typically, we want to exhaustively evaluate a given boolean function • That is, we want to know its functional value for every possible combination of inputs • This leads us to Truth Tables CSC321

12. Truth Tables • List all possible combinations of input values in the left hand columns • List expression result in the right hand column CSC321

13. x + 0 = x x + 1 = 1 x + x = x x + x’ = 1 x + y = y + x x + (y + z) = (x + y) + z x(y + z) = xy + xz (x + y)’ = x’y’ (x’)’ = x x · 1 = x x · 0 = 0 x · x = x x · x’ = 0 xy = yx x(yz) = (xy)z x + yz = (x + y)(x + z) (xy)’ = x’ + y’ Axioms CSC321

14. AND OR NOT Logic Circuits Schematic Symbols • These are the things computers (and other digital devices) are made of • Circuit designers use Boolean algebra to design circuits drawn on schematic drawings • Fabrication facilities use schematic drawings to produce silicon chips CSC321

15. NAND More Gates • NAND • Shortened form of “not and” CSC321

16. NOR More Gates • NOR • Shortened form of “not or” CSC321

17. NAND/NOR • So, what’s so special about NAND and NOR? • NAND and NOR are considered “universal gates” • That is, anything that can be done with AND/OR/NOT can be done with only NAND or NOR gates (one or the other, not both) CSC321

18. NAND/NOR • The universality of NAND/NOR is important because it means you can make many copies of a single gate type on a single piece of silicon and then use it to create complex circuits on a single chip • Exercise • Show how to make AND, OR, and NOT gates using only • NAND gates • NOR gates CSC321

19. XOR XOR Operator • Another specialty gate – “odd” function CSC321

20. BC B A A A 0 1 0 1 00 01 11 10 0 1 0 0 1 0 1 3 2 0 1 3 2 4 5 7 8 1 CD AB 00 01 11 10 0 1 3 2 00 01 4 5 7 8 11 12 13 15 14 10 8 9 11 10 K-Maps • A K-Map is a grid (map) where each square corresponds to a minterm Note the ordering here is Gray code, not binary CSC321

21. Sum-of-Products • This is what we previously called the “sum-of-minterms” • Form the largest power-of-two groupings of 1’s on the K-map • Create the schematic CSC321

22. Product-Of-Sums • Instead of forming large adjacent groups of 1’s (on the K-map), form large adjacent groups of 0’s • What does this mean in terms of the original expression/truth-table? • It means you have simplified F’, not F • To “fix” what you’ve done you need only negate the final result them apply De Morgan’s theorem CSC321

23. Example – Sum-of-Products B’ D’ F C’ A’ D CSC321

24. Example – Product-of-Sums B’ D F A’ C’ D’ CSC321

25. So What? • As it turns out, the sum-of-products can be easily implemented with NAND gates • Similarly, the product-of-sums can be easily implemented with NOR gates • This may greatly simplify the design thus saving us money! CSC321

26. B’ B’ D D’ A’ C’ C’ A’ D’ D NAND/NOR Implementations CSC321

27. Combinational Circuits • Definition: A connected arrangement of logic gates with a set of inputs and outputs • Specifically, they have no memory and no clock! CSC321

28. Augend Sum Half-Adder Addend Carry Combinational Circuit Design • Design a Half-Adder • A combinational circuit that adds 2 bits • Input 1 is call the “Augend” • Input 2 is called the “Addend” • Output 1 is called the “Sum” • Output 2 is called the “Carry” CSC321

29. Carry-in Sum Augend Full-Adder Addend Carry-out Combinational Circuit Design • Design a Full-Adder • A combinational circuit that adds 3 bits • Input 1 is call the “Augend” • Input 2 is called the “Addend” • Input 3 is call the “Carry-in” • Output 1 is called the “Sum” • Output 2 is called the “Carry-out” CSC321

30. Sequential Circuits • Two primary differences between combinational circuits and sequential circuits • Sequential circuits are synchronous (use a clock) • Sequential circuits have memory (current state) CSC321

31. Rising edge 1 0 Falling edge One period Clock • A series of pulses • Sometimes referred to as a pulse train • Basically, it’s an digital signal just oscillates between 0 and 1 CSC321

32. Clock • Clock period is specified in units of time • Seconds, milliseconds, microseconds… • Clock frequency is specified in units of frequency – 1/period – pulses per time-unit • Hertz, Megahertz, Gigahertz… CSC321

33. Memory Devices • Flip flops • Four basic types • SR, D, JK, T • Each type stores 1-bit (two states: 0/1) • Each maintains its current state until a clock pulse arrives • i.e. Ignores input lines until a clock pulse reaches the clock input CSC321

34. S Q C R Q’ Clock symbol means that this flip-flop responds to the rising edge of a clock pulse SR Flip-Flop • “Set/Reset” • Two input lines • One clock input line • Two output lines Characteristic Table • Q(t) refers to the current state • Q(t+1) refers to the next state CSC321

35. “Delay” One input line One clock input line Two output lines D Q C Q’ Clock symbol means that this flip-flop responds to the rising edge of a clock pulse D Flip-Flop Characteristic Table • Q(t+1) refers to the next state CSC321

36. “JK” Two input lines One clock input line Two output lines J Q C K Q’ Clock symbol means that this flip-flop responds to the rising edge of a clock pulse JK Flip-Flop Characteristic Table • Q(t) refers to the current state • Q(t+1) refers to the next state CSC321

37. “Toggle” One input line One clock input line Two output lines T Q C Q’ Clock symbol means that this flip-flop responds to the rising edge of a clock pulse T Flip-Flop Characteristic Table • Q(t+1) refers to the next state CSC321

38. Clock symbol means that this flip-flop responds to the rising edge of a clock pulse Clock symbol means that this flip-flop responds to the falling edge of a clock pulse Edge-Triggering • Output of the flip-flop occurs on the edge of a pulse • Rising edge (0 to 1 transition) • Falling edge (1 to 0 transition) CSC321

39. 1 1 0 0 Edge-Triggering Positive (rising) edge-triggered output frozen positive clock transition Negative (falling) edge-triggered negative clock transition output frozen CSC321

40. Edge-Triggering • Setup time • This is the minimum time that the inputs must remain constant before the edge transition • Hold time • This is the amount of time in which the inputs must not change after the edge transition • These values are not to interesting from a theoretical point of view but can make or break a circuit in practice CSC321

41. Master-Slave Flip-Flops • Two flip-flops of the same type wired together • Master • Rising edge triggered • Receives inputs from “outside world” • Sends outputs to the slave • Slave is falling edge triggered • Falling edge triggered • Receives inputs from master • Sends outputs to the “outside world” • This set-up basically creates a more stable flip-flop in terms of set-up and hold times CSC321

42. Master Slave J Q J Q C C K Q’ K Q’ Master-Slave JK Flip-Flop CSC321

43. Sequential Circuits • Combination of logic gates, flip-flops (memory elements), and a clock signal • The circuit can be described in two parts • The combinational part • The sequential part CSC321

44. The Combinational Part • Describe outputs in terms of logic gates and flip-flop outputs CSC321

45. The Sequential Part • Flip-flop input equations • Describe the inputs to flip-flop elements in terms of logic gates and other flip-flop elements CSC321

46. Designing a Circuit • The goal of circuit design is to convert a specification (a bunch of words) into a circuit • No different than software development! • Example • Design a circuit that counts modulo 4 every time it receives a 1 on the input line CSC321

47. State Diagram X = 0 X = 0 X = 1 X = 1 X = 1 X = 1 X = 0 X = 0 CSC321

48. State Table CSC321

49. Flip-Flop Usage • We know we need 2 flip-flops since the counter must count modulo 4 (2 bits) • We can choose any type we want • D, T, SR, JK • By looking at the excitation table for the chosen type we can create flip-flop equations CSC321

50. JK Flip-Flop Based Design State Table Excitation Table JK Excitation Table CSC321