FINITE STATE MACHINE

FSM = FINITE STATE MACHINE

The mathematical model of a sequential circuit is usually referred to as a sequential machine or a finite state machine.

Since memory elements in sequential circuits are usually flip-flops, it is worth summarising the behaviour of various flip-flop types before proceeding further  All flip-flops can be divided into four basic types: SR, JK, D and T. They differ in the number of inputs and in the response invoked by different value of input signals. The four types of flip-flops are defined in Table 1.

TABLE 1:

Flip FLops  - Characteristic Tables - Characteristic Equations

TABLE 2:

Flip FLops  - Characteristic Tables - Excitation Tables 

 Each of these flip-flops can be uniquely described by its graphical symbol, its characteristic table, its characteristic equation or excitation table. All flip-flops have output signals Q and Q'.

The characteristic table in the third column of Table 1 defines the state of each flip-flop as a function of its inputs and previous state. Q refers to the present state and Q(next) refers to the next state after the occurrence of the clock pulse. The characteristic table for the RS flip-flop shows that the next state is equal to the present state when both inputs S and R are equal to 0. When R=1, the next clock pulse clears the flip-flop. When S=1, the flip-flop output Q is set to 1. The equation mark (?) for the next state when S and R are both equal to 1 designates an indeterminate next state.

The characteristic table for the JK flip-flop is the same as that of the RS when J and K are replaced by S and R respectively, except for the indeterminate case. When both J and K are equal to 1, the next state is equal to the complement of the present state, that is, Q(next) = Q'.

The next state of the D flip-flop is completely dependent on the input D and independent of the present state.

The next state for the T flip-flop is the same as the present state Q if T=0 and complemented if T=1.

The characteristic table is useful during the analysis of sequential circuits when the value of flip-flop inputs are known and we want to find the value of the flip-flop output Q after the rising edge of the clock signal. As with any other truth table, we can use the map method to derive the characteristic equation for each flip-flop, which are shown in the third column of Table 1.

During the design process we usually know the transition from present state to the next state and wish to find the flip-flop input conditions that will cause the required transition. For this reason we will need a table that lists the required inputs for a given change of state. Such a list is called the excitation table, which is shown in the 3RD column of Table 2. There are four possible transitions from present state to the next state. The required input conditions are derived from the information available in the characteristic table. The symbol X in the table represents a "don't care" condition, that is, it does not matter whether the input is 1 or 0.

Characteristic Equation

Specify next state as a function of its current state and inputs

• Q(t) = current state

• Q(t+1) = next state

Therefore 

• SR latch:        Q(t+1) = S + R’Q(t)

• D flip-flop:     Q(t+1) = D

• JK flip-flop:   Q(t+1) = JQ’(t)+K’Q(t)

• T flip-flop:     Q(t+1) = T⊕Q(t)= TQ’(t)+T’Q(t)

 State Tables and State Diagrams

We have examined a general model for sequential circuits. In this model the effect of all previous inputs on the outputs is represented by a state of the circuit. Thus, the output of the circuit at any time depends upon its current state and the input. These also determine the next state of the circuit. The relationship that exists among the inputs, outputs, present states and next states can be specified by either the state table or the state diagram.

State Table

The state table is a tabular representation of a sequential circuit consists of three sections labelled present state, next state and output. The present state designates the state of flip-flops before the occurrence of a clock pulse. The next state shows the states of flip-flops after the clock pulse, and the output section lists the value of the output variables during the present state.

State Diagram

In addition to graphical symbols, tables or equations, flip-flops can also be represented graphically by a state diagram. In this diagram, a state is represented by a circle, and the transition between states is indicated by directed lines (or arcs) connecting the circles. An example of a state diagram is shown in Figure 3 below.

The binary number inside each circle identifies the state the circle represents. The directed lines are labelled with two binary numbers separated by a slash (/). The input value that causes the state transition is labelled first. The number after the slash symbol / gives the value of the output. For example, the directed line from state 00 to 01 is labelled 1/0, meaning that, if the sequential circuit is in a present state and the input is 1, then the next state is 01 and the output is 0. If it is in a present state 00 and the input is 0, it will remain in that state. A directed line connecting a circle with itself indicates that no change of state occurs. The state diagram provides exactly the same information as the state table and is obtained directly from the state table.

Example: This example is taken from P. K. Lala, Practical Digital Logic Design and Testing, Prentice Hall, 1996, p.155.

Consider a sequential circuit shown in Figure 4. It has one input x, one output Z and two state variables Q1Q2 (thus having four possible present states 00, 01, 10, 11).

The behaviour of the circuit is determined by the following Boolean expressions:

These equations can be used to form the state table. Suppose the present state (i.e. Q1Q2) = 00 and input x = 0. Under these conditions, we get Z = 0, D1 = 1, and D2 = 1. Thus the next state of the circuit D1D2 = 11, and this will be the present state after the clock pulse has been applied. The output of the circuit corresponding to the present state Q1Q2 = 00 and x = 1 is Z = 0. This data is entered into the state table as shown in Table 2.

Table 2. State table for the sequential circuit in Figure 4.

The state diagram for the sequential circuit in Figure 4 is shown in Figure 5.

State Diagrams of Various Flip-flops

Table 3 shows the state diagrams of the four types of flip-flops.

Table 3. State diagrams of the four types of flip-flops.

You can see from the table that all four flip-flops have the same number of states and transitions. Each flip-flop is in the set state when Q=1 and in the reset state when Q=0. Also, each flip-flop can move from one state to another, or it can re-enter the same state. The only difference between the four types lies in the values of input signals that cause these transitions.

A state diagram is a very convenient way to visualise the operation of a flip-flop or even of large sequential components.

Analysis of Sequential Circuits

The behaviour of a sequential circuit is determined from the inputs, the outputs and the states of its flip-flops. Both the output and the next state are a function of the inputs and the present state.

The suggested analysis procedure of a sequential circuit is set out in Figure 6 below.

Now let's look at some examples, using these procedures to analyse a sequential circuit.

This example is taken from D. D. Gajski, Principles of Digital Design, Prentice Hall, 1997, p.230.

Example 1.1. Modulo-4 counter

Derive the state table and state diagram for the sequential circuit shown in Figure 7.

SOLUTION:

STEP 1:     First we derive the Boolean expressions for the inputs of each flip-flops in the schematic, in terms of external input Cnt and the flip-flop outputs Q1 and Q0. Since there are two D flip-flops in this example, we derive two expressions for D1 and D0:

These Boolean expressions are called excitation equations since they represent the inputs to the flip-flops of the sequential circuit in the next clock cycle.

STEP 2:     Derive the next-state equations by converting these excitation equations into flip-flop characteristic equations. In the case of D flip-flops, Q(next) = D. Therefore the next state equal the excitation equations.

STEP 3:     Now convert these next-state equations into tabular form called the next-state table.

Each row is corresponding to a state of the sequential circuit and each column represents one set of input values. Since we have two flip-flops, the number of possible states is four - that is, Q1Q0 can be equal to 00, 01, 10, or 11. These are present states as shown in the table.

For the next state part of the table, each entry defines the value of the sequential circuit in the next clock cycle after the rising edge of the Clk. Since this value depends on the present state and the value of the input signals, the next state table will contain one column for each assignment of binary values to the input signals. In this example, since there is only one input signal,Cnt, the next-state table shown has only two columns, corresponding to Cnt = 0 and Cnt = 1.

Note that each entry in the next-state table indicates the values of the flip-flops in the next state if their value in the present state is in the row header and the input values in the column header.

Each of these next-state values has been computed from the next-state equations in STEP 2.

STEP 4:     The state diagram is generated directly from the next-state table, shown in Figure 8.

Each arc is labelled with the values of the input signals that cause the transition from the present state (the source of the arc) to the next state (the destination of the arc).

In general, the number of states in a next-state table or a state diagram will equal 2m , where m is the number of flip-flops. Similarly, the number of arcs will equal 2m x 2k , where k is the number of binary input signals. Therefore, in the state diagram, there must be four states and eight transitions. Following these transition arcs, we can see that as long as Cnt = 1, the sequential circuit goes through the states in the following sequence: 0, 1, 2, 3, 0, 1, 2, ....  On the other hand, when Cnt = 0, the circuit stays in its present state until Cnt changes to 1, at which the counting continues.

Since this sequence is characteristic of modulo-4 counting, we can conclude that the sequential circuit in Figure 7 is a modulo-4 counter with one control signal, Cnt, which enables counting when Cnt = 1 and disables it when Cnt = 0.

To see how the states changes corresponding to the input signals Cnt, click on this image.

Below, we show a timing diagram, representing four clock cycles, which enables us to observe the behaviour of the counter in greater detail.

In this timing diagram we have assumed that Cnt is asserted in clock cycle 0 at t0 and is disasserted in clock cycle 3 at time t4. We have also assumed that the counter is in state Q1Q0 = 00 in the clock cycle 0. Note that on the clock's rising edge, at t1, the counter will go to state Q1Q0 = 01 with a slight propagation delay; in cycle 2, after t2, to Q1Q0 = 10; and in cycle 3, after t3 to Q1Q0 = 11. Since Cnt becomes 0 at t4, we know that the counter will stay in state Q1Q0 = 11 in the next clock cycle. To see the timing behaviour of the circuit click on this image 

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In Example 1.1 we demonstrated the analysis of a sequential circuit that has no outputs by developing a next-state table and state diagram which describes only the states and the transitions from one state to the next. In the next example we complicate our analysis by adding output signals, which means that we have to upgrade the next-state table and the state diagram to identify the value of output signals in each state.

This example is taken from D. D. Gajski, Principles of Digital Design, Prentice Hall, 1997, p.234.

Example 1.2

Derive the next state, the output table and the state diagram for the sequential circuit shown in Figure 10.

SOLUTION:

The input combinational logic in Figure 10 is the same as in Example 1.1, so the excitation and the next-state equations will be the same as in Example 1.1.

Excitation equations:     

Next-state equations:    

In addition, however, we have computed the output equation.

Output equation:              Y = Q1Q0

As this equation shows, the output Y will equal to 1 when the counter is in state Q1Q0 = 11, and it will stay 1 as long as the counter stays in that state.

Next-state and output table:

State diagram:

Timing diagram:

Note that the counter will reach the state Q1Q0 = 11 only in the third clock cycle, so the output Y will equal 1 after Q0 changes to 1. Since counting is disabled in the third clock cycle, the counter will stay in the state Q1Q0 = 11 and Y will stay asserted in all succeeding clock cycles until counting is enabled again.

Design of Sequential Circuits

The design of a synchronous sequential circuit starts from a set of specifications and culminates in a logic diagram or a list of Boolean functions from which a logic diagram can be obtained. In contrast to a combinational logic, which is fully specified by a truth table, a sequential circuit requires a state table for its specification. The first step in the design of sequential circuits is to obtain a state table or an equivalence representation, such as a state diagram.

A synchronous sequential circuit is made up of flip-flops and combinational gates. The design of the circuit consists of choosing the flip-flops and then finding the combinational structure which, together with the flip-flops, produces a circuit that fulfils the required specifications. The number of flip-flops is determined from the number of states needed in the circuit.

The recommended steps for the design of sequential circuits are set out below.

For information on state reduction.

State Reduction

Any design process must consider the problem of minimising the cost of the final circuit. The two most obvious cost reductions are reductions in the number of flip-flops and the number of gates.

The number of states in a sequential circuit is closely related to the complexity of the resulting circuit. It is therefore desirable to know when two or more states are equivalent in all aspects. The process of eliminating the equivalent or redundant states from a state table/diagram is known as state reduction.

Example: Let us consider the state table of a sequential circuit shown in Table 6.

Table 6. State table

It can be seen from the table that the present state A and F both have the same next states, B (when x=0) and C (when x=1). They also produce the same output 1 (when x=0) and 0 (when x=1). Therefore states A and F are equivalent. Thus one of the states, A or F can be removed from the state table. For example, if we remove row F from the table and replace all F's by A's in the columns, the state table is modified as shown in Table 7.

Table 7. State F removed

It is apparent that states B and E are equivalent. Removing E and replacing E's by B's results in the reduce table shown in Table 8.

Table 8. Reduced state table

The removal of equivalent states has reduced the number of states in the circuit from six to four. Two states are considered to be equivalent if and only if for every input sequence the circuit produces the same output sequence irrespective of which one of the two states is the starting state.

Design of Sequential Circuits

This example is taken from M. M. Mano, Digital Design, Prentice Hall, 1984, p.235.

Example 1.3   We wish to design a synchronous sequential circuit whose state diagram is shown in Figure 13. The type of flip-flop to be use is J-K.

From the state diagram, we can generate the state table shown in Table 9. Note that there is no output section for this circuit. Two flip-flops are needed to represent the four states and are designated Q0Q1. The input variable is labelled x.

Table 9. State table.

We shall now derive the excitation table and the combinational structure. The table is now arranged in a different form shown in Table 11, where the present state and input variables are arranged in the form of a truth table. Remember, the excitable for the JK flip-flop was derive in Table 1.

Table 10. Excitation table for JK flip-flop

Table 11. Excitation table of the circuit

In the first row of Table 11, we have a transition for flip-flop Q0 from 0 in the present state to 0 in the next state. In Table 10 we find that a transition of states from 0 to 0 requires that input J = 0 and input K = X. So 0 and X are copied in the first row under J0 and K0 respectively. Since the first row also shows a transition for the flip-flop Q1 from 0 in the present state to 0 in the next state, 0 and X are copied in the first row under J1 and K1. This process is continued for each row of the table and for each flip-flop, with the input conditions as specified in Table 10.

The simplified Boolean functions for the combinational circuit can now be derived. The input variables are Q0, Q1, and x; the output are the variables J0, K0, J1 and K1. The information from the truth table is plotted on the Karnaugh maps shown in Figure 14.

Figure 14. Karnaugh Maps

The flip-flop input functions are derived:

Note: the symbol � is exclusive-NOR.

The logic diagram is drawn in Figure 15.

Design of Sequential Circuits

This example is taken from P. K. Lala, Practical Digital Logic Design and Testing, Prentice Hall, 1996, p.176.

Example 1.4     Design a sequential circuit whose state tables are specified in Table 12, using D flip-flops.

Table 12. State table of a sequential circuit.

Table 13. Excitation table for a D flip-flop.

Next step is to derive the excitation table for the design circuit, which is shown in Table 14. The output of the circuit is labelled Z.

Table 14. Excitation table

Now plot the flip-flop inputs and output functions on the Karnaugh map to derive the Boolean expressions, which is shown in Figure 16.

Figure 16. Karnaugh maps

The simplified Boolean expressions are:

Finally, draw the logic diagram.

Figure 17. Logic diagram of the sequential circuit.

Design of Counters

A sequential circuit that goes through a prescribed sequence of states upon the application of input pulses is called a counter. The input pulses, called count pulses, may be clock pulses. In a counter, the sequence of states may follow a binary count or any other sequence of states. Counters are found in almost all equipment containing digital logic. They are used for counting the number of occurrences of an even and are useful for generating timing sequences to control operations in a digital system.

Of the various sequences a counter may follow, the straight binary sequence is the simplest and most straight forward. A counter that follows the binary sequence is called a binary counter. An n-bit binary counter consists of n flip-flops and can count in binary from 0 to 2n - 1.

Let's look at some examples of counter designs.

Design of Counters

This example is taken from T. L. Floyd, Digital Fundamentals, Fourth Edition, Macmillan Publishing, 1990, p.395.

Example 1.5   A counter is first described by a state diagram, which is shows the sequence of states through which the counter advances when it is clocked. Figure 18 shows a state diagram of a 3-bit binary counter.

The circuit has no inputs other than the clock pulse and no outputs other than its internal state (outputs are taken off each flip-flop in the counter). The next state of the counter depends entirely on its present state, and the state transition occurs every time the clock pulse occurs. Figure 19 shows the sequences of count after each clock pulse.

Once the sequential circuit is defined by the state diagram, the next step is to obtain the next-state table, which is derived from the state diagram in Figure 18 and is shown in Table 15.

Table 15. State table

Since there are eight states, the number of flip-flops required would be three. Now we want to implement the counter design using JK flip-flops.

Next step is to develop an excitation table from the state table, which is shown in Table 16.

Table 16. Excitation table

Now transfer the JK states of the flip-flop inputs from the excitation table to Karnaugh maps to derive a simplified Boolean expression for each flip-flop input. This is shown in Figure 20.

The 1s in the Karnaugh maps of Figure 20 are grouped with "don't cares" and the following expressions for the J and K inputs of each flip-flop are obtained:

The final step is to implement the combinational logic from the equations and connect the flip-flops to form the sequential circuit. The complete logic of a 3-bit binary counter is shown in Figure 21.

Design of Counters

This example is taken from M. M. Mano, Digital Design, Prentice Hall, 1984, p.243.

Example 1.6     Design a counter specified by the state diagram in Example 1.5 using T flip-flops. The state diagram is shown here again in Figure 22.

The state table will be the same as in Example 1.5.

Now derive the excitation table from the state table, which is shown in Table 17.

Table 17. Excitation table.

Next step is to transfer the flip-flop input functions to Karnaugh maps to derive a simplified Boolean expressions, which is shown in Figure 23.

The following expressions are obtained:

T0 = 1;        T1 = Q0;         T2 = Q1*Q0

Finally, draw the logic diagram of the circuit from the expressions obtained. The complete logic diagram of the counter is shown in Figure 24.

To see the timing and state transitions of the counter, click on this image.

Example . Design the sequential circuit for the given state diagram using J-K F/Fs. The sequential circuit has one input (X) and one output (Z). 

Now that you have reached the end of the tutorial, you should be able to understand the basic concept of sequential circuits. You should be able to analyse and design a basic sequential circuit. Now you can practice some of the exercises using the analysis and design procedures shown in the examples.

END of FSM - 1