# Digital Logic Design. Basics Combinational Circuits Sequential Circuits. Pu-Jen Cheng

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1 Digital Logic Design Basics Combinational Circuits Sequential Circuits Pu-Jen Cheng Adapted from the slides prepared by S. Dandamudi for the book, Fundamentals of Computer Organization and Design.

2 Introduction to Digital Logic Basics Hardware consists of a few simple building blocks These are called logic gates AND, OR, NOT, NAND, NOR, XOR, Logic gates are built using transistors t NOT gate can be implemented by a single transistor AND gate requires 3 transistors Transistors are the fundamental devices Pentium consists of 3 million transistors Compaq Alpha consists of 9 million transistors Now we can build chips with more than 100 million transistors

3 Basic Concepts Simple gates AND OR NOT Functionality can be expressed by a truth table A truth table lists output for each possible input combination Precedence NOT > AND > OR F = A B + A B = (A (B)) + ((A) B)

4 Basic Concepts (cont.) Additional useful gates NAND NOR XOR NAND = AND + NOT NOR = OR + NOT XOR implements exclusive-or function NAND and NOR gates require only 2 transistors AND and OR need 3 transistors!

5 Basic Concepts (cont.) Number of functions With N logical variables, we can define 2 2N functions Some of them are useful AND, NAND, NOR, XOR, Some are not useful: Output is always 1 Output is always 0 Number of functions definition is useful in proving completeness property

6 Basic Concepts (cont.) Complete sets A set of gates is complete If we can implement any logical function using only the type of gates in the set You can uses as many gates as you want Some example complete sets {AND, OR, NOT} Not a minimal complete set {AND, NOT} {OR, NOT} {NAND} {NOR} Minimal complete set A complete set with no redundant elements.

7 Basic Concepts (cont.) Proving NAND gate is universal NAND gate is called universal gate

8 Basic Concepts (cont.) Proving NOR gate is universal NOR gate is called universal gate

9 Logic Chips

10 Logic Chips (cont.) Integration levels SSI (small scale integration) Introduced in late 1960s 1-10 gates (previous examples) MSI (medium scale integration) Introduced in late 1960s gates LSI (large scale integration) Introduced in early 1970s ,000 gates VLSI (very large scale integration) Introduced in late 1970s More than 10,000 gates

11 Logic Functions Logical functions can be expressed in several ways: Truth table Logical expressions Graphical form Example: Majority function Output is 1 whenever majority of inputs is 1 We use 3-input majority function

12 Logic Functions (cont.) 3-input majority function A B C F Logical expression form F = A B + B C + A C

13 Logical Equivalence All three circuits implement F = A B function

14 Logical Equivalence (cont.) Proving logical equivalence of two circuits Derive the logical expression for the output of each circuit Show that these two expressions are equivalent Two ways: You can use the truth table method For every combination of inputs, if both expressions yield the same output, they are equivalent Good for logical expressions with small number of variables You can also use algebraic manipulation Need Boolean identities

15 Logical Equivalence (cont.) Derivation of logical expression from a circuit Trace from the input to output Write down intermediate logical expressions along the path

16 Logical Equivalence (cont.) Proving logical equivalence: Truth table method A B F1 = A B F3 = (A + B) (A + B) (A + B)

17 Boolean Algebra

18 Boolean Algebra (cont.)

19 Boolean Algebra (cont.) Proving logical equivalence: Boolean algebra method To prove that two logical functions F1 and F2 are equivalent Start with one function and apply Boolean laws to derive the other function Needs intuition as to which laws should be applied and when Practice helps Sometimes it may be convenient to reduce both functions to the same expression Example: F1= A B and F3 are equivalent A B= (A + B) (A + B) (A + B)

20 Logic Circuit Design Process A simple logic design process involves Problem specification Truth table derivation Derivation of logical expression Simplification of logical expression Implementation ti

21 Deriving Logical Expressions Derivation of logical expressions from truth tables sum-of-products (SOP) form product-of-sums (POS) form SOP form Wi Write an AND term for each hinput combination i that produces a 1 output Write the variable if its value is 1; complement otherwise OR the AND terms to get the final expression POS form Dual of the SOP form

22 Deriving Logical Expressions (cont.) 3-input majority function A B C F SOP logical expression Four product terms Because there are 4 rows with a 1 output F = A B C + A B C + A B C + A B C

23 Deriving Logical Expressions (cont.) 3-input majority function A B C F POS logical expression Four sum terms Because there are 4 rows with a 0 output F = (A + B + C) (A + B + C) (A + B + C) (A + B + C)

24 Logical Expression Simplification Two basic methods Algebraic manipulation Use Boolean laws to simplify the expression Difficult to use Don t know if you have the simplified form Karnaugh map (K-map) method Graphical method Easy to use Can be used to simplify logical expressions with a few variables

25 Algebraic Manipulation Majority function example A B C + A B C + A B C + A B C = Added extra A B C + A B C + A B C + A B C + A B C + A B C We can now simplify this expression as B C + A C + A B A difficult method to use for complex expressions

26 Karnaugh Map Method Note the order

27 Karnaugh Map Method (cont.) Simplification examples

28 Karnaugh Map Method (cont.) First and last columns/rows are adjacent

29 Karnaugh Map Method (cont.) Minimal expression depends on groupings

30 Karnaugh Map Method (cont.) No redundant groupings

31 Karnaugh Map Method (cont.) Example Seven-segment display Need to select the right LEDs to display a digit

32 Karnaugh Map Method (cont.)

33 Karnaugh Map Method (cont.) Don t cares simplify the expression a lot

34 Implementation Using NAND Gates Using NAND gates Get an equivalent expression A B + C D = A B + C D Using de Morgan s law A B + C D = A B. C D Can be generalized Majority function A B + B C + AC = A B. BC. AC Idea: NAND Gates: Sum-of-Products, NOR Gates: Product-of-Sums

35 Implementation Using NAND Gates (cont.) Majority function

36 Introduction to Combinational Circuits Combinational circuits Output depends only on the current inputs Combinational circuits provide a higher level of abstraction Help in reducing design complexity Reduce chip count We look at some useful combinational circuits

37 Multiplexers Multiplexer 2 n data inputs n selection inputs a single output 4-data input MUX Selection input determines the input that should be connected to the output

38 Multiplexers (cont.) 4-data input MUX implementation

39 Multiplexers (cont.) MUX implementations

40 Multiplexers (cont.) Example chip: 8-to-1 MUX

41 Multiplexers (cont.) Efficient implementation: Majority function

42 Demultiplexers (DeMUX) Demultiplexer a single input n selection inputs 2 n outputs

43 Decoders Decoder selects one-out-of-n inputs

44 Decoders (cont.) Logic function implementation

45 Comparator Used to implement comparison operators (=, >, <,, )

46 Comparator (cont.) A=B: O x = I x (x=a<b, A=B, & A>B) 4-bit magnitude comparator chip

47 Comparator (cont.) Serial construction of an 8-bit comparator

48 1-bit Comparator x CMP y x>y x=y x<y x y x>y x=y x<y

49 8-bit comparator x n >yn xn=yn xn<yn CMP x y x>y x=y x<y

50 Adders Half-adder Adds two bits Produces a sum and carry Problem: Cannot use it to build larger inputs Full-adder Adds three 1-bit values Like half-adder, produces a sum and carry Allows building N-bit adders Simple technique Connect C out of one adder to C in of the next These are called ripple-carry adders

52 Adders (cont.) A 16-bit ripple-carry adder

53 Adders (cont.) Ripple-carry adders can be slow Delay proportional to number of bits Carry lookahead adders Eliminate the delay of ripple-carry adders Carry-ins are generated independently C 0 = A 0 B 0 C 1 = A 0 B 0 A 1 + A 0 B 0 B 1 + A 1 B 1... Requires complex circuits Usually, a combination carry lookahead and ripple-carry techniques are used

54 Programmable Logic Arrays PLAs Implement sum-of-product expressions No need to simplify the logical expressions Take N inputs and produce M outputs Each input represents a logical variable Each output represents a logical function output Internally uses An AND array Each AND gate receives 2N inputs N inputs and their complements An OR array

55 Programmable Logic Arrays (cont.) A blank PLA with 2 inputs and 2 outputs

56 Programmable Logic Arrays (cont.) Implementation examples

57 Programmable Logic Arrays (cont.) Simplified notation

58 1-bit Arithmetic and Logic Unit Preliminary ALU design 2 s complement Required 1 is added via C in

59 1-bit Arithmetic and Logic Unit (cont.) Final design

60 Arithmetic and Logic Unit (cont.) 16-bit ALU

61 Arithmetic and Logic Unit (cont d) 4-bit ALU

62 Introduction to Sequential Circuits Output depends on current as well as past inputs Depends on the history Have memory property Sequential circuit consists of Combinational i circuit i Feedback circuit Past input is encoded into a set of state variables Uses feedback (to feed the state variables) Simple feedback Uses flip flops

63 Introduction (cont.) Main components of a sequential circuit

64 Clock Signal

65 Clock Signal (cont.) Clock serves two distinct purposes Synchronization point Start of a cycle End of a cycle Intermediate point at which the clock signal changes levels Timing information Clock period, ON, and OFF periods Propagation delay Time required for the output to react to changes in the inputs

66 Clock Signal (cont.)

67 SR Latches Can remember a bit Level-sensitive (not edge-sensitive) A NOR gate implementation of SR latch

68 SR Latches (cont.) SR latch outputs follow inputs In clocked SR latch, outputs respond at specific instances Uses a clock signal

69 D Latches D Latch Avoids the SR = 11 state

70 Positive Edge-Triggered D Flip-Flops Edge-sensitive devices Changes occur either at positive or negative edges

71 Notation for Latches & Flip-Flops Not strictly followed in the literature Latches Flip-flops Low level High level Positive edge Negative edge

72 Example of Shift Register Using D Flip-Flops shift Register chip

73 Memory Design Using D Flip-Flops Require separate data in and out lines

74 JK Flip-Flops JK flip-flop (master-slave) J K Q n Q n Q n

75 Examples of D & JK Flip-Flops Two example chips D latches JK flip-flops

76 Example of Shift Register Using JK Flip-Flops Shift Registers Can shift data left or right with each clock pulse A 4-bit shift register using JK flip-flops

77 Example of Counter Using JK Flip-Flops Counters Easy to build using JK flip-flops Use the JK = 11 to toggle Binary counters Simple design B bits can count from 0 to 2 B 1 Ripple counter Increased delay as in ripple-carry adders Delay proportional to the number of bits Synchronous counters Output changes more or less simultaneously Additional cost/complexity

78 Modulo-8 Binary Ripple Counter Using JK Flip-Flops LSB

79 Synchronous Modulo-8 Counter Designed using the following simple rule Change output if the preceding count bits are 1 Q1 changes whenever Q0 = 1 Q2 changes whenever Q1Q0 = 11

80 Example Counters

81 Sequential Circuit Design Sequential circuit consists of A combinational circuit that produces output A feedback circuit We use JK flip-flops for the feedback circuit Simple counter examples using JK flip-flops Provides alternative counter designs We know the output Need to know the input combination that produces this output Use an excitation table Built from the truth table

82 Sequential Circuit Design (cont.)

83 Sequential Circuit Design (cont.) Build a design table that consists of Current state output Next state output JK inputs for each flip-flop Binary counter example 3-bit binary counter 3 JK flip-flops are needed Current state and next state outputs are 3 bits each 3 pairs of JK inputs

84 Sequential Circuit Design (cont.) Design table for the binary counter example

85 Sequential Circuit Design (cont.) Use K- maps to simplify expression s for JK inputs

86 Sequential Circuit Design (cont.) Final circuit for the binary counter example Compare this design with the synchronous counter design

87 Sequential Circuit Design (cont.) A more general counter design Does not step in sequence Same design process One significant change Missing states 1, 2, and 4 Use don t cares for these states

88 Design table for the general counter example Sequential Circuit Design (cont.)

89 Sequential Circuit Design (cont.) K-maps to simplify JK input expressions

90 Sequential Circuit Design (cont.) Final circuit for the general counter example

91 General Design Process FSM can be used to express the behavior of a sequential circuit Counters are a special case State transitions are indicated by arrows with labels X/Y X: inputs that cause system state change Y: output generated while moving to the next state Look at two examples Even-parity checker Pattern recognition

92 General Design Process (cont.) Even-parity checker FSM needs to remember one of two facts Number of 1 s is odd or even Need only two states 0 input does not change the state 1 input changes state Simple example Complete the design as an exercise

93 General Design Process (cont.) Pattern recognition example Outputs 1 whenever the input bit sequence has exactly two 0s in the last three input bits FSM requires thee special states to during the initial phase S0 S2 After that we need four states S3: last two bits are 11 S4: last two bits are 01 S5: last two bits are 10 S6: last two bits are 00

94 General Design Process (cont.) State diagram for the pattern recognition example

95 General Design Process (cont.) Steps in the design process 1. Derive FSM 2. State assignment Assign flip-flop states to the FSM states Necessary to get an efficient design 3. Design table derivation Derive a design table corresponding to the assignment in the last step 4. Logical expression derivation Use K-maps as in our previous examples 5. Implementation

96 General Design Process (cont.) State assignment Three heuristics Assign adjacent states for states that have the same next state states that are the next states of the same state States that have the same output for a given input For our example Heuristic 1 groupings: (S1, S3, S5) 2 (S2, S4, S6) 2 Heuristic 2 groupings: (S1, S2) (S3, S4) 3 (S5, S6) 3 Heuristic 1 groupings: (S4, S5)

97 General Design Process (cont.) State table for the pattern recognition example

98 General Design Process (cont.) K-map for state assignment State assignment

99 Design table General Design Process (cont.)

100 General Design Process (cont.) K-maps for JK inputs K-map for the output

101 General Design Process (cont.) Final implementation

102

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