Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines Basic Digital Logic Gates ...

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Chapter 2 Basic Digital Logic 1 Basic Digital Logic Chapter 2

Transcript of Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines Basic Digital Logic Gates ...

Page 1: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 1

Basic Digital Logic

Chapter 2

Page 2: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 2

Outlines Basic Digital Logic Gates Two types of digital logic circuits

Combinational logic circuits Sequential logic circuits

Combinational logic circuit design Using sum of product Using product of sum

Karnaugh map: Minimization of logic circuits Without don’t care term With don’t care term

Construction from smaller components Sequential logic circuit

Page 3: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 3

Switches

Abstraction of building block of digital computers

Basic logic functions AND OR NOT

B

A

A

A B

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2Chapter Basic Digital Logic 4

Basic Digital Logic Gates

NOT gate

AND gate OR gate XOR gate

NAND gate NOR gate XNOR gate

Page 5: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 5

Two types of digital logic circuits Combinational logic circuits

Output depends only on its input at that time.

Sequential logic circuits Output depends both on

its previous output and its input at that time

CombinationalLogic Circuit

SequentialLogic Circuit

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2Chapter Basic Digital Logic 6

Combinational Logic Circuit

Page 7: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 7

How to design a combinational logic circuit

Decide how to encode input and output in 0 and 1

Describe each bit of the output in term of input Truth table Logical function

Construct a logic circuit from the logical function

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2Chapter Basic Digital Logic 8

Truth table and logical function

out = f (in1, in2, …)

in1 in2 … out

0 0 0 1

0 0 1 0

0 1 0 …

… … … …

1 1 1 0

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2Chapter Basic Digital Logic 9

Boolean Logic Basic Boolean operations

AND: X Y, X & Y, X Y, X Y OR: X Y, X + Y NOT: ~X, X’, X

Any Boolean expressions can be described in: Disjunctive normal form (DNF) /sum of product Conjunctive normal form (CNF) / product of

sum

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2Chapter Basic Digital Logic 10

Example: Exclusive OR (XOR)

output = X Y + XY

X Youtpu

t

0 0 0

0 1 1

1 0 1

1 1 0

Page 11: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 11

Construct Sum of Products

Find out all conditions when the function is true X Y = T(1) when

X=0, Y=1 => X Y =1 X=1, Y=0 => XY = 1

OR the conditions X Y + XY

Sum of products

X Youtpu

t

0 0 0

0 1 1

1 0 1

1 1 0

minterm

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2Chapter Basic Digital Logic 12

Exclusive OR (XOR): Sum of products

output =X Y + XY mintern

X Y output

XY 0 0 0

X Y 0 1 1

XY 1 0 1

X Y 1 1 0

X

Y

output

Page 13: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 13

Construct Product of Sums

Find out all conditions when the negation of the function is true X Y = F(0) when

X=0, Y=0 => X + Y = 0 X=1, Y=1 =>X +Y = 0

AND the conditions ( X + Y )(X +Y)

Product of sum

X Y output

0 0 0

0 1 1

1 0 1

1 1 0

maxterm

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2Chapter Basic Digital Logic 14

Exclusive OR (XOR) output = ( X + Y )(X +Y )

X

Y

output

maxterm

X Youtpu

t

X + Y 0 0 0

X +Y 0 1 1

X +Y 1 0 1

X +Y 1 1 0

Page 15: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 15

Karnaugh Map (K-map)

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2Chapter Basic Digital Logic 16

2-variable K-map A

B0 1

0

1

A B X

0 0

0 1

1 0

1 1

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2Chapter Basic Digital Logic 17

3-variable K-mapA B C X

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

AB

C00 01 11 10

0

1

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2Chapter Basic Digital Logic 18

4-variable K-map

A B C D X0 0 0 00 0 0 10 0 1 00 0 1 10 1 0 00 1 0 10 1 1 00 1 1 11 0 0 01 0 0 11 0 1 01 0 1 11 1 0 01 1 0 11 1 1 01 1 1 1

AB

CD00 01 11 10

00

01

11

10

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2Chapter Basic Digital Logic 19

Full Adder

Full

ad

der

ABCi

SCo

A B Ci S Co

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

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2Chapter Basic Digital Logic 20

Full AdderA B C S Co

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

ABC 00 01 11 10

0 0 1 0 1

1 1 0 1 0

AB

C 00 01 11 10

0 0 0 1 0

1 0 1 1 1

S=A’B’C+A’BC’+ABC+AB’C’

Co=AB+BC+AC

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2Chapter Basic Digital Logic 21

Encoder

Encoder

A3

A2

A1

X1

A0X0

A3 A2 A1 A0 X1 X0

0 0 0 0 X X

0 0 0 1 0 0

0 0 1 0 0 1

0 0 1 1 X X

0 1 0 0 1 0

… X X

1 0 0 0 1 1

… X X

1 1 1 1 X X

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2Chapter Basic Digital Logic 22

Encoder

A3 A2 A1 A0 X1 X0

0 0 0 0 X X

0 0 0 1 0 0

0 0 1 0 0 1

0 0 1 1 X X

0 1 0 0 1 0

… X X

1 0 0 0 1 1

… X X

1 1 1 1 X X

A3A2

A1A0

00 01 11 10

00 X 1 X 1

01 0 X X X

11 X X X X

10 0 X X X

X1 = A1A0

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2Chapter Basic Digital Logic 23

Encoder

A3 A2 A1 A0 X1 X0

0 0 0 0 X X

0 0 0 1 0 0

0 0 1 0 0 1

0 0 1 1 X X

0 1 0 0 1 0

… X X

1 0 0 0 1 1

… X X

1 1 1 1 X X

A3A2

A1A0

00 01 11 10

00 X 0 X 1

01 0 X X X

11 X X X X

10 1 X X X

X0 = A1+A3

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2Chapter Basic Digital Logic 24

Decoder

2 - 4 decoder

b3

b2

b1

b0

a1

a0

If a1 a0 is a binary i,

•bi is 1 and

•bj is 0 when ji.

a1 a0 b3 b2 b1 b0

0 0 0 0 0 1

0 1 0 0 1 0

1 0 0 1 0 0

1 1 1 0 0 0

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2Chapter Basic Digital Logic 25

Decoder

a1 a0 b3 b2 b1 b0

0 0 0 0 0 1

0 1 0 0 1 0

1 0 0 1 0 0

1 1 1 0 0 0

b3 = a1 a0

b2 = a1a0

b1 =a1 a0

b0 =a1a0

b3

b2

b0

b1

a0

a1

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2Chapter Basic Digital Logic 26

Multiplexor

2-1 multiplexor

D1

D0

x

S

X = Ds

S D1 D0 x

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 1

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2Chapter Basic Digital Logic 27

MultiplexorS D1 D0 x

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 1

D1D0

S

00 01 11 10

0 0 1 1 0

1 0 0 1 1

X = S D0 + S D1

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2Chapter Basic Digital Logic 28

Multiplexor

X = S D0 + S D1

D1

D0

S

X

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2Chapter Basic Digital Logic 29

Construction from Smaller Components

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2Chapter Basic Digital Logic 30

2-bit Full Adder

1 1 A1 A0

0 1 + B1 B0 +

1 0 0 C S1 S0FA0FA1

A1 B1 A0 B0 Ci=0

C S1 S0

A B Ci

Co S

A B Ci

Co S

2-bit full adder

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2Chapter Basic Digital Logic 31

4-bit Full Adder

1 0 1 1 0 1 1 1 + 1 0 0 1 0

A1 A0

B1 B0 +

C S1 S0

2FA02FA1

A1 B1 A0 B0 Ci=0

C S1 S0

A B Ci

Co S

A B Ci

Co S

2

22

2

2 2

4-bit full adder

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2Chapter Basic Digital Logic 32

Sequential Circuit

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2Chapter Basic Digital Logic 33

ClockSignal oscillating between 0 and 1

clock cycle time / clock period

Rising edge Falling edge

Edge-triggered clockingThe state of the sequential circuits changes on the clock edge.

Page 34: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 34

Types of sequential circuits Synchronous circuits

With clock Use in digital computers

Asynchronous circuits Without clock

Page 35: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 35

SR LatchR

Q’S

Q0

1

1

0

0

1

set

R

Q’S

Q1

0

0

1

1

0

reset

R

Q’S

Q0

0

1/0

0/1

0/1

1/0

hold

R

Q’S

Q1

1

?

?

?

?

unstable

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2Chapter Basic Digital Logic 36

D LatchC

D

Q

Q

0

0

Q0

0

0

Q0

Q0

C

D

Q

Q

0

1

Q0

0

0

Q0

Q0

C

D

Q

Q

1

0

0

0

1

0

C

D

Q

Q

1

1

1

1

0

0

Q0 0

CDQ

C=0 hold dataC=1 load data

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2Chapter Basic Digital Logic 37

D Flip-flop

D latch D latchD

C

D

C

Q QD

C

QQ

10

xD QD

D latch D latchD

C

D

C

Q QD

C

QQ

01

xx QQ

D latch D latchD

C

D

C

Q QD

C

QQ

00

xx xx

D latch D latchD

C

D

C

Q QD

C

QQ

11

xD DD

C

D

Q

C=1 0 load data

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2Chapter Basic Digital Logic 38

Registers

D flip-flop

D flip-flop

D flip-flop

D flip-flop

D flip-flop

D flip-flop

D flip-flop

D flip-flop

clkD7 D

6 D

5 D

4 D3 D2 D1 D0

O7 O

6 O

5 O

4 O3 O2 O1 O0

8-bit REGISTERDin Dout

clk

8 8 C

D

Q

Set-up time

Hold time

Page 39: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 39

Register Files

n-2n

decoder

Register n-1

Register n-2

Register 0

. . .

nn-1n-2

0

Write

Register number

Datain8

n-1MUX

Dataout8

Page 40: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 40

Finite State Machines

Next-state function

(Combinational)

Current State(registers)

Outputfunction

(Combinational)Inputs

Outputs

Page 41: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 41

Counter 0 -> 3

clka

b

Current State

Next State

a b A B

0 0 0 1

0 1 1 0

1 0 1 1

1 1 0 0

A = a bB =b

D Q

Q

D Q

Q

clk

a b

Page 42: Chapter 2Basic Digital Logic1 Chapter 2. Basic Digital Logic2 Outlines  Basic Digital Logic Gates  Two types of digital logic circuits Combinational.

2Chapter Basic Digital Logic 42

Counter with resetR

clk

a

b resetR

Current State

Next State

a b A B

0 0 0 0 1

0 0 1 1 0

0 1 0 1 1

0 1 1 0 0

1 0 0 0 0

1 0 1 0 0

1 1 0 0 0

1 1 1 0 0

A = R (a b)B = Rb

D Q

Q

D Q

QR

clk

a b