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Arithmetic Logic Unit - Digital Circuits and Systems - Lecture Notes, Study notes of Computer Science

Chaudhary Charan Singh Haryana Agricultural UniversityComputer Science

These are the Lecture Notes of Digital Circuits and Systems which includes Present State, Principle, Memory Elements, Elementscombinational Logic, Description, Circuit and Create, Number of States, Binary Codes, Letter Symbol etc. Key important points are: Arithmetic Logic Unit, Present State, Principle, Memory Elements, Elementscombinational Logic, Description, Circuit and Create, Number of States, Binary Codes, Letter Symbol

Typology: Study notes

2012/2013

Uploaded on 03/22/2013

dhritiman
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Section 4 – Sequential Circuits
4.1 Overview of Sequential Circuits:
Definition
• The circuit whose outputs and next state
depend on both the input signals and the
present state of the circuit
Principle [spot the error!]
Memory Elements
Combinational Logic
Present
State
Input
Signals O
S
Clock
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Section 4 – Sequential Circuits

4.1 Overview of Sequential Circuits:

Definition

  • The circuit whose outputs and next state

depend on both the input signals and the

present state of the circuit

Principle [spot the error!]

Memory Elements

Combinational Logic

Present State

Input Signals

O
S

Clock

4.2 Flip-Flops

SR Flip-Flop

Q

Q

SET

CLR

S

R

JK Flip-Flop

J

Q

Q

K

SET

CLR

D Flip-Flop

Q

Q

SET

CLR

D

T Flip-Flop

Q

Q

SET

CLR

DT

S R Q’

0 0 Q

1 1 X

Q Q’ S R

0 0 0 X

1 1 X 0

J K Q’

0 0 Q

1 1 !Q

Q Q’ J K

0 0 0 X

0 1 1 X

1 0 X 1

1 1 X 0

D Q’

Q Q’ D

X 0 0

X 1 1

Q Q’ T

T Q’

0 Q

1 !Q

4.4 Example of Design

Design a sequential logic circuit whose output Z is 1 except when the input X = 1 for at least four clock periods. Then the output Z is 0. Use J-K flip-flops.

  • State Diagram [X/Z]
A B C D
  • State Coding

Present State

Code

A 0 0
B 0 1
C 1 0
D 1 1
  • Flip Flops

We require two JK flip-flops. Let’s name them JKA and JKB

  • State Table

Present State

Next State Output Z X=0 X=1 X=0 X= A A B 1 1 B A C 1 1 C A D 1 1 D A D 1 0

  • Excitation Table
QA QB X Q’A Q’B JA KA JB KB Z

0 0 0 0 0 0 x 0 x 1 0 0 1 0 1 0 x 1 x 1 0 1 0 0 0 0 x x 1 1 0 1 1 1 0 1 x x 1 1 1 0 0 0 0 x 1 0 x 1 1 0 1 1 1 x 0 1 x 1 1 1 0 0 0 x 1 x 1 1 1 1 1 1 1 x 0 x 0 0

  • Minimisations and Equations

JA

X\ QA QB 00 01 11 10

0 0 0 x x 1 0 1 x x

J (^) A = XQ B

4.5 State Reduction

Definition of Equivalent States

  • Two or more states of a sequential circuit are equivalent if for the same values for the inputs, have exactly the same output and determine the sequential circuit transition to the same next state or to equivalent states.

Algorithm for State Reduction

  • If more 2 or more states are equivalent, one of them can be substituted with the other one
  • The other states that have transitions to one of the removed states have to have their next states changed into the remaining equivalent state

Advantages of State Reduction

  • By reducing the number of states, it is possible that the number of flip-flops and/or amount of combinational circuitry needed to implement the sequential circuit will decrease, reducing the cost of the circuit

Example of State Reduction

  • Let’s assume that there is the following state table: Present State

Next State Output

X=0 X=1 X=0 X=

A A^ B^0
B C D 0 0
C A D 0 0
D E F 0 1
E A F 0 1
F G F 0 1
G A F 0 1
  • States G and E are equivalent (same next states for the same inputs and same outputs for the same inputs)
  • State reduction => state G will be replaced by E Present State

Next State Output

X=0 X=1 X=0 X=

A A^ B^0
B C D 0 0
C A D 0 0
D E F 0 1
E A F 0 1
F E F 0 1

4.6 State Assignment

Definition of State Assignment

  • The process of assigning a binary code to each state

Possibilities for State Assignment

  • There are many possibilities to assign binary codes to states
  • The number of possible options increases exponentially with the number of states
  • The cost of the combinational circuit strongly depends on the state assignment chosen
  • Although various State Assignment methods have been proposed, there is no assignment procedure that guarantees a minimal cost for the resulting combinational circuit

Examples of State Assignment

States Assign. 1

Assign. 2

Assign. 3 A 0 0 1 0 0 0 0 0 0 B 0 1 0 0 1 0 1 0 0 C 0 1 1 0 1 1 0 1 0 D 1 0 0 1 0 1 1 0 1 E 1 0 1 1 1 1 0 1 1

4.7 Homework

  • Having the following state table, design the sequential circuit following state assignment 1, 2 and 3, respectively.

Present State

Next State Output

X=0 X=1 X=0 X=

A A^ B^0
B C D 0 0
C A D 0 0
D E D 0 1
E A D 0 1