Understanding State Machines: Transitions, Inputs, and Truth Tables

1. S0 S3 Ames Chicago S1 S2 Ceder Rapids DeKalb State machine • Thus far what we only dealt with a pure combination circuit • That means output was simply dependent on the current input • However, output may depend on the input sequence • System state changes with input and a set of input applied in a different sequence produce different output • Consider the following state transition • For example, going from Ames to Chicago, one may pass through the following states while driving, but may fly back • One is in Ames • One is in Ceder Rapids • One is in DeKalb • One is in Chicago

2. Truth tables to represent state transitions • States can be coded as binary combinations of variables • Each state is represented by n=log N bits • N is total number of states • 2 bits will represent 4 state, 3 bits will represent eight, and so on • For 4 states we use two bits X and Y • A truth table will then give the next state • Xn and Yn can be specified in terms Xo and Yo xn = xo’ yo + xo yo’ yn = xo’ yo’ + xo yo’ = yo’

3. S0 S3 Ames Chicago S1 S2 Ceder Rapids DeKalb State Change, with clock and/or input • In a state transition diagram, state may change with time • A clock signal represents passage of time • Each time a clock arrives, state changes to next state • There is no other explicit input (or there is an implicit input) • There may be an explicit input, say i • Next state may depend on current state and the value of input • Let us assume a binary input • Thus i can be 0 or 1 • The state changes are shown 0 1 1 0 0 1 1 0

4. Truth tables with input for state transitions • States transition table will have two sets of inputs • Current state variable and input variables • Total number of row in table is 2(n+m) • n is number of variables representing states • m is number of input variables xn=xo’ yo’ i+xo’ yo i’+xo’ yo i+xo yo’ i’ =xo’ i + xo’ yo + xo yo’ i’ yn=xo’ yo’ i’+xo’ yo i+xo yo’ i’+xo yo i = yo’ i’ + yo i

5. Determining number of states • Identify how many different things we need to keep track of • This is critical to know • Otherwise the number of states (and their meaning) may get out of hand very quickly • This is different than what is the output of interest (in each state we may have some outputs) • For example, if we are to process a sequence of inputs • Depending on interest, the number of states may be different • If we need to know how many 1’s are there, we need states corresponding to count • If we need to know if we have even or odd number of 1’s, we may need only two states

6. Steps in designing a state machine • Start writing a state diagram • It has an initial state • It has other states to keep track of various activities • Generate a state table • Generate state table in binary • Needs state assignment, i.e., what state will have what code • State assignment is a complex process • For the time being assume straightforward combinations • Derive canonical sum-of-products form equations • You can simplify the equations • When the next state depend upon the inputs, the inputs are examined at the clock ticks