finite state machine: FSM

Definition

A machine consisting of a set of states, a start state, an input, and a transition function that maps input and current states to a next state. Machine begins in the start state with an input. It changes to new states depending on the transition function. The transition function depends on current states and inputs. The output of the machine depends on input and/or current state.

There are two types of FSMs which are popularly used in the digital design. They are

• Moore machine
• Mealy machine

Moore machine

In Moore machine the output depends only on current state.The advantage of the Moore model is a simplification of the behavior.

Mealy machine

In Mealy machine the output depend on both current state and input.The advantage of the Mealy model is that it may lead to reduction of the number of states.

In both models the next state depends on current state and input. Some times designers use mixed models. States will be encoded for representing a particular state.

Representation of a FSM

A FSM can be represented in two forms:

• Graph Notation
• State Transition Table

Graph Notation

• In this representation every state is a node. A node is represented using a circular shape and the state code is written within the circular shape.
• The state transitions are represented by an edge with arrow head. The tail of the edge shows current state and arrow points to next state, depending on the input and current state. The state transition condition is written on the edge.
• The initial/start state is sometime represented by a double lined circular shape, or a different colour shade.

The following image shows the way of graph notation of FSM. The codes 00and 11 are the state codes. 00 is the value of initial/starting/reset state. The machine will start with 00 state. If the machine is reseted then the next state will be 00 state.


State Transition Table

The State Transition Table has the following columns:

• Current State: Contains current state code
• Input: Input values of the FSM
• Next State: Contains the next state code
• Output: Expected output values

An example of state transition table is shown below.


Mealy FSM

In Mealy machine the output depend on both current state and input.The advantage of the Mealy model is that it may lead to reduction of the number of states.


The block diagram of the Mealy FSM is shown above. The output function depends on input also. The current state function updates the current state register (number of bits depends on state encoding used).


The above FSM shows an example of a Mealy FSM, the text on the arrow lines show (condition)/(output). 'a' is the input and 'x' is the output.

Moore FSM

In Moore machine the output depends only on current state.The advantage of the Moore model is a simplification of the behavior.


The above figure shows the block diagram of a Moore FSM. The output function doesn't depend on input. The current state function updates the current state register.


The above FSM shows an example of a Moore FSM. 'a' is the input. Inside every circle the text is (State code)/(output). Here there is only one output, in state '11' the output is '1'.

In both the FSMs the reset signal will change the contents of current state register to initial/reset state.

State Encoding

In a FSM design each state is represented by a binary code, which are used to identify the state of the machine. These codes are the possible values of the state register. The process of assigning the binary codes to each state is known as state encoding.
The choice of encoding plays a key role in the FSM design. It influences the complexity, size, power consumption, speed of the design. If the encoding is such that the transitions of flip-flops (of state register) are minimized then the power will be saved. The timing of the machine are often affected by the choice of encoding.
The choice of encoding depends on the type of technology used like ASIC, FPGA, CPLD etc. and also the design specifications.

State encoding techniques

The following are the most common state encoding techniques used.

• Binary encoding
• One-hot encoding
• Gray encoding

In the following explanation assume that there are N number of states in the FSM.

Binary encoding
The code of a state is simply a binary number. The number of bits is equal to log₂(N) rounded to next natural number. Suppose N = 6, then the number of bits are 3, and the state codes are:
S0 - 000
S1 - 001
S2 - 010
S3 - 011
S4 - 100
S5 - 101

One-hot encoding
In one-hot encoding only one bit of the state vector is asserted for any given state. All other state bits are zero. Thus if there are N states then N state flip-flops are required. As only one bit remains logic high and rest are logic low, it is called as One-hot encoding. If N = 5, then the number of bits (flip-flops) required are 5, and the state codes are:
S0 - 00001
S1 - 00010
S2 - 00100
S3 - 01000
S4 - 10000

To know more about one-hot encoding click here.

Gray encoding
Gray encoding uses the Gray codes, also known as reflected binary codes, to represent states, where two successive codes differ in only one digit. This helps is reducing the number of transition of the flip-flops outputs. The number of bits is equal to log₂(N) rounded to next natural number. If N = 4, then 2 flip-flops are required and the state codes are:
S0 - 00
S1 - 01
S2 - 11
S3 - 10

Designing a FSM is the most common and challenging task for every digital logic designer. One of the key factors for optimizing a FSM design is the choice of state coding, which influences the complexity of the logic functions, the hardware costs of the circuits, timing issues, power usage, etc. There are several options like binary encoding, gray encoding, one-hot encoding, etc. The choice of the designer depends on the factors like technology, design specifications, etc.