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Lecture #14 EGR 277 – Digital Logic

Reading Assignment: Chapter 5 in Digital Design, 3rd Edition by Mano

Self-starting counters

Counters are considered to be self-starting if all unused counts eventually lead to the correct counting sequence. Since the initial state for a flip-flop is unpredictable upon powering up the IC, a counter that is not self-starting could possibly power up into an unused state that would not eventually go into the correct counting sequence (so the counter might “lock up” in an incorrect count or counting pattern.

Recall that the next states for unused counts were sometimes treated as “don’t cares.” With this method it is difficult to predict what will happen if the counter powers up into an unused count (although it can be later determined by analyzing the circuit). A safer technique it to let all unused counts have a valid count for their next state.

7

5

6

7

5

0

0

4

1

4

1

2

3

2

3

Lecture #14 EGR 277 – Digital Logic

Example: Consider the state diagrams for two modulo-5 counters below. Are they self-starting?

Case 1: Counter is NOT self-starting.

Next states for unused counts 5, 6, and 7 were perhaps treated as don’t cares.

Case 2: Counter is self-starting.

Next states for unused counts 5, 6, and 7 were all set to count 0.

JA

KA

JB

KB

C

JC

KC

A

B

C

Lecture #14 EGR 277 – Digital Logic

Example:Determine the counting sequence

for the counter shown (begin with count 0).

Use a timing diagram to display the values

of Clock, JA, KA, JB, KB, JC, KC, A, B,

and C. Is the counter self-starting?

Code Converter

EBCDIC Code

Gray Code

Code Converter

BCD Code

Output = 1 if correct

Sequence Detector

Input Sequence

input sequence detected;

(Digital Lock)

Output = 0 otherwise

Lecture #14 EGR 277 – Digital Logic

State Reduction

In some sequential circuits, the numeric values of the states is not important.

Perhaps the circuit simply needs to produce a certain output sequence.

As long as the output sequence is correct, then if we can reduce the number of states (and thus the number of flip-flops), then the final circuit can be simplified.

Shown below are examples of such circuits.

Note: Reducing the number of states may not always reduce the number of flip-flops. For example, if the number of states is reduced from 13 to 9, then 4 flip-flops are still required. However, the associated combinational logic circuitry may be reduced.

Lecture #14 EGR 277 – Digital Logic

State Reduction – Procedure

1) Form the state table.

2) If 2 states are equivalent, eliminate one and replace all references to it with the equivalent state.

Note: Two states are equivalent if for each input combination they give identical outputs and have the same next state or an equivalent next state.

3) Redraw the state table. Repeat steps 2 and 3 as many times as possible.

4) Draw the final (reduced) state diagram.

5) Test the original and reduced state diagrams to insure that they produce the same result.

Lecture #14 EGR 277 – Digital Logic

Example: (reference: Digital Design, by Mano, p. 199)

Use state reduction to reduce the state diagram below if possible. Draw the reduced state diagram.Test the original state diagram and the reduced state diagram with the input sequence 01010110100 to see if they produce the same output sequence.

State

a

a

Input

Input

0

0

1

1

0

0

1

1

0

0

1

1

1

1

0

0

1

1

0

0

0

0

Output

Output

Lecture #14 EGR 277 – Digital Logic

Example: (continued)

Test the original state diagram and the reduced state diagram with the input sequence 01010110100 to see if they produce the same output sequence.

Original state diagram:

Reduced state diagram:

Lecture #14 EGR 277 – Digital Logic

Sequence Detector: An example of a circuit whose output sequence is critical and the numeric value of the states is unimportant is a “sequence detector”. Such a circuit might be used to detect a certain bit pattern (such as in synchronizing two signals) or for a digital lock – where the lock is unlocked when a correct combination (sequence) is entered.

Example: Design a sequence detector to detect the sequence 1010. The sequence detector should also detect overlapping sequences. The circuit should output a binary 1 when a valid sequence is detected.

Lecture #14 EGR 277 – Digital Logic

Example: (continued) Use state reduction on the last example to see if the number of states can be reduced. If so, draw the new state diagram and discuss how it differs from the original design.

State

a

a

Input

Input

0

0

1

1

0

0

1

1

0

0

1

1

1

1

0

0

0

0

1

1

0

0

1

1

0

0

1

1

0

0

0

0

Output

Output

Lecture #14 EGR 277 – Digital Logic

Example: (continued)

Test the original state diagram and the reduced state diagram with the input sequence 0101011001010100 to see if they produce the same output sequence.

Original state diagram:

Reduced state diagram:

Lecture #14 EGR 277 – Digital Logic

Example: (Problem 5-17 from Digital Design, 3rd Edition, by Mano) Design a one input, one output serial 2’s complementer. The circuit accepts a string of bits from the input and generates the 2’s complement at the output. The circuit can be reset asynchronously to start and stop the operation.

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