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**Chapter 3 Digital Logic Structures**

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**State Machine Another type of sequential circuit**

Combines combinational logic with storage “Remembers” state, and changes output (and state) based on inputs and current state

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**Combinational vs. Sequential**

Two types of “combination” locks 30 15 5 10 20 25 4 1 8 Combinational Success depends only on the values, not the order in which they are set. Sequential Success depends on the sequence of values (e.g, R-13, L-22, R-3).

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State The state of a system is a snapshot of all the relevant elements of the system at the moment the snapshot is taken. Examples: The state of a basketball game can be represented by the scoreboard. Number of points, time remaining, possession, etc. The state of a tic-tac-toe game can be represented by the placement of X’s and O’s on the board.

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**State of Sequential Lock**

Our lock example has four different states, labelled A-D: A: The lock is not open, and no relevant operations have been performed. B: The lock is not open, and the user has completed the R-13 operation. C: The lock is not open, and the user has completed R-13, followed by L-22. D: The lock is open. (user has completed R-13, L-22 and then R-3)

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State Diagram Shows states and actions that cause a transition between states.

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**Definition of a Finite State Machine**

A set of input events A set of output events A set of states A function that maps states and input to output A function that maps states and inputs to states (which is called a state transition function) Must be complete A description of the initial state A finite state machine is one that has a limited or finite number of possible states.

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**Example 2: A Door Combination Lock**

entry code is the 4-bit sequence “0110” Partial Complete

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**Example 3: Odd Parity Checker**

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**Why Finite State Machines (FSMs)**

A FSM is simple and intuitive way of describing a system which has discrete dynamics (State Transition Diagrams). An FSM is an “abstract machine.” That means that we use a mathematical description of the machine to reason about it without actually building it. A FSM can be directly and unambiguously converted into a digital electronic circuits.

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**Step 1: Form the State Transition Table (STT)**

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**Step 2: Code STT in numbers**

(NS) (NS)

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Step 3: Implement STT

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**Step 3a: Next State Logic for 1 D-Latch**

Inputs Output NS= (~PS S) + (PS ~S)

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Step 3b: Output Logic Inputs Output R= PS

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**Step 4: Implement Circuit**

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The Clock Frequently, a clock circuit triggers transition from one state to the next. At the beginning of each clock cycle, state machine makes a transition, based on the current state and the external inputs. Not always required. In lock example, the input itself triggers a transition. “1” “0” One Cycle time

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**Master-Slave Flipflop**

A pair of gated D-latches, to isolate next state from current state. PS NS PS During 1st phase (clock=1), previously-computed state becomes current state and is sent to the logic circuit. During 2nd phase (clock=0), next state, computed by logic circuit, is stored in Latch A.

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