\[
\newcommand{\bx}{\bar{x}}
\newcommand{\by}{\bar{y}}
\newcommand{\bz}{\bar{z}}
\newcommand{\bA}{\bar{A}}
\newcommand{\bB}{\bar{B}}
\newcommand{\bC}{\bar{C}}
\newcommand{\bD}{\bar{D}}
\newcommand{\bE}{\bar{E}}
\]
ECE 275: State assignment
Website: https://vikasdhiman.info/ECE275SequentialLogic/
Announcemnts

Midterm 2 postponed on Nov 15th.

HW 8 is posted is due next Wednesday (Nov 10th) at 9 AM before class.
Ex2: Parity Checker using T flipflops
Ex2: Parity Checker using JK flipflops
Ex2: Parity Checker
State reduction: Ex1
A sequential circuit has one input (X) and one output (Z). The circuit examines groups
of four consecutive inputs and produces an output \( Z = 1 \) if the input sequence 0101 or
1001 occurs. The circuit resets after every four inputs. Find the Mealy state graph.
Equivalent states: Defn

Let \( N_1 \) and \( N_2 \) be sequential circuits (not necessarily
different).

Let \( \underline{X} \) represent a sequence of inputs of
arbitrary length.

Let \( \underline{Z}_1 = \lambda_1(p, \underline{X} ) \) the sequence of outputs corresponding to \( \underline{X} \) starting at state state \(p \) for circuit \( N_1 \). Let \( \underline{Z}_2 = \lambda_2(p, \underline{X} ) \) denote the same for circuit \( N_2 \).

Then state \( p \) in \( N_1 \) is equivalent to state \( q \) in \(
N_2 \) iff \( \lambda_1(p, \underline{X} ) = \lambda_2 (q, \underline{X}) \)
for every possible input sequence.

Symbol: \( p \equiv q \)
Equivalent states: Thm
Two states \( p \) and \( q \) of a sequential circuit are equivalent iff for every
single input \( X \), the outputs \( \lambda(p, X) \) are the same and the next states \( p_{t+1} = \phi(p, X) \) are equivalent, that is,

\( \lambda(p, X) = \lambda(q, X) \)

and \( \phi(p, X) = \phi(q, X) \).
Implication tables
Implication tables algorithm

Construct a chart which contains a square for each pair of states.

Compare each pair of rows in the state table. If the outputs associated with states i
and j are different, place an X in square ij to indicate that \( i\equiv j \).

If the outputs are the
same, place the implied pairs in square ij. (If the next states of i and j are m and n
for some input x, then mn is an implied pair.)

If the outputs and next states are the
same (or if ij only implies itself), place a check (√) in square ij to indicate that \( i \equiv j \)

Go through the table squarebysquare. If square ij contains the implied pair mn, and square mn contains an X, then i [ j, and an X should be placed in square \(i \equiv j \).

If any X’s were added in step 3, repeat step 3 until no more X’s are added.

For each square ij which does not contain an X, \( i \equiv j \).
Equivalent State assignments

Swapping flipflops are equivalent.

Inverting flipflop are equivalent.
Equivalent State assignments: Ex1
Distict state assignments
Distict state assignments
Guidelines for state assignment

(Group inneigbours) States which have the same next state for a given input should be given
adjacent assignments.

(Group outneighbors) States which are the next states of the same state should be given
adjacent assignments.

States which have the same output for a given input should be given adjacent assignments.