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Digital System Ch5-1 Chapter 5 Synchronous Sequential Logic Ping-Liang Lai ( 賴秉樑 ) Digital System 數位系統

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Digital System Ch5-2 Outline of Chapter 5 5.1Introduction 5.2Sequential Circuits 5.3 Storage Element: Latches 5.4Storage Element: Flip-Flops 5.5Analysis of Clocked Sequential Circuits 5.7State Reduction and Assignment 5.8Design Procedure

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Digital System Ch Introduction Combinational circuits Contains no memory elements The outputs depends on the inputs

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Digital System Ch Sequential Circuits Sequential circuits A feedback path The state of the sequential circuit (inputs, current state) (outputs, next state) Synchronous: the transition happens at discrete instants of time Asynchronous: at any instant of time

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Digital System Ch5-5 Synchronous sequential circuits A master-clock generator to generate a periodic train of clock pulses The clock pulses are distributed throughout the system Clocked sequential circuits Most commonly used No instability problems The memory elements: flip-flops » Binary cells capable of storing one bit of information. » Two outputs: one for the normal value and one for the complement value. » Maintain a binary state indefinitely until directed by an input signal to switch states.

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Digital System Ch5-6 Fig. 5.2 Synchronous clocked sequential circuit

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Digital System Ch Latches ( 栓鎖器 ) Basic flip-flop circuit Two NOR gates More complicated types can be built upon it Directed-coupled RS flip-flop: the cross-coupled connection An asynchronous sequential circuit (S, R)= (0, 0): no operation (S, R)=(0, 1) : reset (Q=0, the clear state) (S, R)=(1, 0) : set (Q=1, the set state) (S, R)=(1, 1) : indeterminate state (Q=Q'=0) Consider (S, R) = (1, 1) (0, 0)

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Digital System Ch5-8 SR latch with NAND gates Fig. 5.4 SR latch with NAND gates

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Digital System Ch5-9 SR latch with control input En=0, no change En=1, 0/1 1/S' 1/R' Fig. 5.5 SR latch with control input

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Digital System Ch5-10 D Latch Eliminate the undesirable conditions of the indeterminate state in the RS flip-flop D: data Gated D-latch D Q when En=1; no change when En=0 Fig. 5.6 D latch S R 0/1 1/D' 1/D

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Digital System Ch5-11 Fig. 5.7 Graphic symbols for latches

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Digital System Ch Flip-Flops ( 正反器 ) A trigger The state of a latch or flip-flop is switched by a change of the control input. Level triggered – Latches Edge triggered – Flip-Flops Fig. 5.8 Clock response in latch and flip-flop

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Digital System Ch5-13 If level-triggered flip-flops are used ( 準位觸發 ) The feedback path may cause instability problem Edge-triggered flip-flops ( 邊緣觸發 ) The state transition happens only at the edge Eliminate the multiple-transition problem

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Digital System Ch5-14 Edge-triggered D flip-flop Master-slave D flip-flop Two separate flip-flops A master flip-flop (positive-level triggered) A slave flip-flop (negative-level triggered) Fig. 5.9 Master-slave D flip-flop

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Digital System Ch5-15 CP = 1: (S, R) (Y, Y'); (Q, Q') holds. CP = 0: (Y, Y') holds; (Y, Y') (Q, Q'). (S, R) could not affect (Q, Q') directly. The state changes coincide with the negative-edge transition of CP. 第三版內容，參考用 !

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Digital System Ch5-16 Edge-triggered flip-flops The state changes during a clock-pulse transition A D-type positive-edge-triggered flip-flop Fig D-type positive-edge-triggered flip-flop

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Digital System Ch5-17 Three basic flip-flops (S, R) = (0, 1): Q = 1 (S, R) = (1, 0): Q = 0 (S, R) = (1, 1): no operation (S, R) = (0, 0): should be avoided Fig D-type positive-edge-triggered flip-flop

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Digital System Ch Holding data Set Reset Clk=0 Clk=1 positive-edge-triggered

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Digital System Ch5-19 Setup and Hold Time for FF The setup time D input must be maintained at a constant value prior to the application of the positive CP pulse. Data to the internal latches. The hold time D input must not changes after the application of the positive CP pulse. Clock to the internal latch. Q D CK D T setup T hold D Q T propagation delay

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Digital System Ch5-20 Summary CP=0: (S, R) = (1, 1), no state change CP= ↑ : state change once CP=1 and ↓ : state holds Eliminate the feedback problems in sequential circuits All flip-flops must make their transition at the same time

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Digital System Ch5-21 Other Flip-Flops The edge-triggered D flip-flops The most economical and efficient Positive-edge and negative-edge Fig Graphic symbols for edge-triggered D flip-flop

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Digital System Ch5-22 JK FF JK flip-flop D=JQ ’ +K ’ Q J=0, K=0: D=Q, no change J=0, K=1: D=0 Q=0 J=1, K=0: D=1 Q=1 J=1, K=1: D=Q' Q=Q' Fig JK flip-flop JKQ(t+1) 00Q(t) (No change) 010 (Reset) 101 (Set) 11Q’(t) (Complement)

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Digital System Ch5-23 T FF T (toggle) flip-flop D = T ⊕ Q = TQ'+T'Q » T=0: D=Q, no change » T=1: D=Q' Q=Q' Fig T flip-flop T Q ( t +1) 0 Q ( t ) (No change) 1 Q’ ( t ) (Complement)

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Digital System Ch5-24 Summary Characteristic tables

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Digital System Ch5-25 Characteristic equations D flip-flop » Q(t+1) = D JK flip-flop » Q(t+1) = JQ'+K'Q T flop-flop » Q(t+1) = T ⊕ Q

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Digital System Ch5-26 Direct inputs Asynchronous set and/or asynchronous reset Fig D flip-flop with asynchronous reset

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Digital System Ch Analysis of Clocked Sequential Circuits A sequential circuit (inputs, current state) (output, next state). A state equation (also called transition equation) specified the next state as a function of the present state and inputs. A state transition table or state transition diagram.

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Digital System Ch5-28 State Equations A(t+1) = A(t)x(t) + B(t)x(t) B(t+1) = A'(t)x(t) A compact form A(t+1) = Ax + Bx B(t+1) = A'x The output equation y(t) = (A(t)+B(t))x'(t) y = (A+B)x' Fig Example of sequential circuit

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Digital System Ch5-29 State Table State transition table = State equations A ( t + 1) = Ax + Bx B ( t + 1) = Ax y = Ax + Bx

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Digital System Ch5-30 State Equation A ( t + 1) = Ax + Bx B ( t + 1) = Ax y = Ax + Bx

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Digital System Ch5-31 State Diagram State transition diagram A circle: a state. A directed lines connecting the circles: the transition between the states » Each directed line is labeled “ inputs/outputs ”. A logic diagram a state table a state diagram. Fig State diagram of the circuit of Fig. 5.15

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Digital System Ch5-32 Flip-Flop Input Equations The part of circuit that generates the inputs to flip-flops Also called excitation functions D A = Ax +Bx D B = A'x The output equations To fully describe the sequential circuit y = (A+B)x' Note Q(t+1)=D Q DADA DBDB y

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Digital System Ch5-33 Analysis with D FFs The input equation D A =A ⊕ x ⊕ y The state equation A(t+1)=A ⊕ x ⊕ y Fig Sequential circuit with D flip-flop

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Digital System Ch5-34 Analysis with JK flip-flops The next-state can be derived by following procedure 1.Determine the flip-flop equations in terms of the present state and input variables ( 用目前狀態和輸入變數的觀點決定正反器的輸入方程式 ); 2.List the binary values of each input equation ( 列出每一個輸入方程式的 二進位值 ); 3.Use the corresponding flip-flop characteristic table to determine the next state values in the state table ( 使用相對應的的正反器特性表，決定狀 態表中的次一狀態值 ).

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Digital System Ch5-35 Analysis with JK Flip-Flops An sequential circuit with JK FFs Determine the flip-flop input function in terms of the present state and input variables; Used the corresponding flip-flop characteristic table to determine the next state. Fig Sequential circuit with JK flip-flop J A =B, K A =Bx’ J B =x’, K B =A’x+Ax’

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Digital System Ch5-36 J A = B, K A = Bx' J B = x', K B = A'x + Ax' Derive the state table Or, derive the state equations using characteristic equation. JKQ(t+1) 00Q(t) (No change) 010 (Reset) 101 (Set) 11Q’(t) (Complement)

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Digital System Ch5-37 State transition diagram State equation for A and B: Fig State diagram of the circuit of Fig J A =B, K A =Bx’ J B =x’, K B =A’x+Ax’

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Digital System Ch5-38 Analysis with T Flip-Flops The characteristic equation Q(t+1)= T ⊕ Q = TQ'+T'Q Fig Sequential circuit with T flip-flop

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Digital System Ch5-39 The input and output functions T A =Bx T B = x y = AB The state equations A(t+1) = (Bx)'A+(Bx)A' =AB'+Ax'+A'Bx B(t+1) = x ⊕ B

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Digital System Ch5-40 State Table T Q ( t +1) 0 Q ( t ) (No change) 1 Q’ ( t ) (Complement)

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Digital System Ch5-41 Mealy and Moore Models The Mealy model ( 密利模型 ): the outputs are functions of both the present state and inputs (Fig. 5-15). The outputs may change if the inputs change during the clock pulse period. » The outputs may have momentary false values unless the inputs are synchronized with the clocks. The Moore model ( 莫爾模型 ): the outputs are functions of the present state only (Fig. 5-20). The outputs are synchronous with the clocks.

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Digital System Ch5-42 Mealy and Moore Models Fig Block diagram of Mealy and Moore state machine

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Digital System Ch State Reduction and Assignment State Reduction ( 狀態簡化 ) Reductions on the number of flip- flops and the number of gates. A reduction in the number of states may result in a reduction in the number of flip-flops. An example state diagram showing in Fig Fig. 5.25State diagram

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Digital System Ch5-44 State Reduction Only the input-output sequences are important. Two circuits are equivalent » Have identical outputs for all input sequences; » The number of states is not important. Fig. 5.25State diagram State:aabcdeffgfga Input: Output:

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Digital System Ch5-45 Equivalent states Two states are said to be equivalent » For each member of the set of inputs, they give exactly the same output and send the circuit to the same state or to an equivalent state. » One of them can be removed.

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Digital System Ch5-46 Reducing the state table e = g (remove g); d = f (remove f);

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Digital System Ch5-47 The reduced finite state machine State:aabcdeddedea Input: Output:

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Digital System Ch5-48 The checking of each pair of states for possible equivalence can be done systematically (section 9-5). The unused states are treated as don't-care condition fewer combinational gates. Fig Reduced State diagram

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Digital System Ch5-49 State Assignment State Assignment ( 狀態指定 ) To minimize the cost of the combinational circuits. Three possible binary state assignments. (m states need n-bits, where 2 n > m)

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Digital System Ch5-50 Any binary number assignment is satisfactory as long as each state is assigned a unique number. Use binary assignment 1.

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Digital System Ch Design Procedure Design Procedure for sequential circuit ( 設計程序 ) The word description of the circuit behavior to get a state diagram; ( 從文 字敘述及所需的操作規格，獲得電路的狀態圖 ) State reduction if necessary; ( 如果需要，簡化狀態數量 ) Assign binary values to the states; ( 指定狀態的二進位值 ) Obtain the binary-coded state table; ( 獲得二進位編碼的狀態表 ) Choose the type of flip-flops; ( 選擇欲使用的正反器形式 ) Derive the simplified flip-flop input equations and output equations; ( 推導 出已化簡的輸入方程式，及輸出方程式 ) Draw the logic diagram; ( 繪製邏輯圖 )

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Digital System Ch5-52 Synthesis using D flip-flops An example state diagram and state table: design a circuit to detect a sequence of three or more consecutive 1’s. Fig State diagram for sequence detector

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Digital System Ch5-53 The flip-flop input equations A(t+1) = D A (A, B, x) = (3, 5, 7) B(t+1) = D B (A, B, x) = (1, 5, 7) The output equation y(A, B, x) = (6, 7) Logic minimization using the K map D A = Ax + Bx D B = Ax + B'x y = AB

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Digital System Ch5-54 Fig Maps for sequence detector

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Digital System Ch5-55 Sequence Detector The logic diagram Fig Logic diagram of sequence detector

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Digital System Ch5-56 Excitation Tables ( 激勵表 ) A state diagram flip-flop input functions Straightforward for D flip-flops. We need excitation tables for JK and T flip- flops. JKQ(t+1) 00 Q(t) (No change) 01 0 (Reset) 10 1 (Set) 11 Q’(t) (Complement) TQ ( t +1) 0 Q ( t ) (No change) 1 Q’ ( t ) (Complement)

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Digital System Ch5-57 Synthesis using JK FFs The same procedure with D FFs, but a extra excitation table must be used. The state table and JK flip-flop inputs.

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Digital System Ch5-58 J A = Bx'; K A = Bx J B = x; K B = (A ⊕ x) ‘ y = ? Fig Maps for J and K input equations

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Digital System Ch5-59 Fig Logic diagram for sequential circuit with JK flip-flops

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Digital System Ch5-60 Synthesis using T flip-flops A n-bit binary counter The state diagram. No inputs (except for the clock input). Fig State diagram of three-bit binary counter

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Digital System Ch5-61 The state table and the flip-flop inputs

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Digital System Ch5-62 Fig Maps of three-bit binary counter

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Digital System Ch5-63 Logic simplification using the K map T A2 = A 1 A 2 T A1 = A 0 T A0 = 1 The logic diagram Fig Logic diagram of three-bit binary counter

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陳澄波 — 嘉義公園 ‧ 1937 年‧油彩、畫布‧ 130×162cm ‧台灣省立美術 館收藏。 (Source: 陳澄波 [1895 ~ 1947]

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