Thus far what we only dealt with a pure combination circuit

1
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 nlog 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
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(nm)
• n is number of variables representing states
• m is number of input variables

xnxo yo ixo yo ixo yo ixo yo i
xo i xo yo xo yo i ynxo yo ixo
yo ixo yo ixo 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 1s are there, we
  need states corresponding to count
• If we need to know if we have even or odd number
  of 1s, 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