csee 3827: fundamentals of computer systemsmartha/courses/3827/sp09/slides/lecture6.pdf · csee...

CSEE 3827: Fundamentals of Computer Systems

Lecture 6

February 9, 2009

Martha Kimmartha@cs.columbia.edu

CSEE 3827, Spring 2009 Martha Kim

Combinational circuits

• Combinational circuits are stateless

• The outputs are functions only of the intputs

2

Combinational circuit OutputsInputs

CSEE 3827, Spring 2009 Martha Kim

Sequential circuits

• Sequential circuits have state

• The outputs are functions of both the current inputs and the current state of the circuit

3

Sequential circuit

OutputsInputs Combinational circuit

Storage elements

next state

current state

CSEE 3827, Spring 2009 Martha Kim

Design procedure

1.Specification

(define desired behavior)

2.Formulation

(derive a truth table or boolean equations that relate outputs to inputs)

3.Optimization

4.Technology Mapping

(transform circuit to use available gates (e.g., NANDs))

5.Verification

(verify that final design behaves correctly)

4

CSEE 3827, Spring 2009 Martha Kim

Hierarchical design

5

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

3-4

(a)

A0 N0MX

MX

MX

MX

ME

N1

N2

N3

E

B0

A1

B1

A2

B2

A3

B3

Ai

Ni

MXBi

N0N1 E

MEN2N3

(b) (c)

(4-bit equality comparator)

Enabling

CSEE 3827, Spring 2009 Martha Kim

Enabled circuits

7

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

3-15

XF

EN

(a)

ENX

F

(b)

Output is enabled only when input ‘ENABLE’ signal is asserted

CSEE 3827, Spring 2009 Martha Kim

Enabling example: Car control

• Inputs:

• IG (ignition switch)

• LS (light switch)

• RS (radio switch)

• WS (power window switch)

8

• Outputs

• L (lights)

• R (radio)

• W (power windows)

What are the enabling relationships?

What is the truth table for this system?

Decoding

CSEE 3827, Spring 2009 Martha Kim

Decoder (1:2)

10

Converts n-bit input to m-bit output, where n <= m <= 2n

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

3-17

A D0 D1

0 1 01 0 1

(a) (b)

D1 AA

D0 A

CSEE 3827, Spring 2009 Martha Kim

Decoder (2:4)

11

n

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

3-18

A1

0011

A0

0101

D0

1000

D1

0100

D2

0010

D3

0001

(a)

D0 A1 A0

D1 A1 A0

D2 A1 A0

D3 A1 A0

(b)

A1

A0

CSEE 3827, Spring 2009 Martha Kim

Decoder (3:8)

12

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

3-19

3-to-8 Line decoder

1-to-2-Line decoders

4 2-input ANDs 8 2-input ANDs

2-to-4-Linedecoder

D0A0

A1

A2

D1

D2

D3

D4

D5

D6

D7

Hierarchical design: use small decoders to build bigger decoder

CSEE 3827, Spring 2009 Martha Kim

Decoder-based circuits

• If decoders produce minterms...

13

3:8decoder

AB

C

‘A’B’C‘A’BC‘AB’C‘ABCA’B’CA’BCAB’CABC

Encoding

CSEE 3827, Spring 2009 Martha Kim

Encoders

15

Inverse of a decoder: converts m-bit input to n-bit output, where n <= m <= 2n

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

T 3-7

TABLE 3-7Truth Table for Octal-to-Binary Encoder

Inputs Outputs

D7 D6 D5 D4 D3 D2 D1 D0 A2 A1 A0

00000001

00000010

00000100

00001000

00010000

00100000

01000000

10000000

00001111

00110011

01010101

CSEE 3827, Spring 2009 Martha Kim

Encoders

16

Inverse of a decoder: converts m-bit input to n-bit output, where n <= m <= 2n

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

T 3-7

TABLE 3-7Truth Table for Octal-to-Binary Encoder

Inputs Outputs

D7 D6 D5 D4 D3 D2 D1 D0 A2 A1 A0

00000001

00000010

00000100

00001000

00010000

00100000

01000000

10000000

00001111

00110011

01010101

CSEE 3827, Spring 2009 Martha Kim

Priority Encoder

17

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

T 3-8

Inputs Outputs

D3 D2 D1 D0 A1 A0 V

00001

0001X

001XX

01XXX

X0011

X0101

01111

TABLE 3-8Truth Table of Priority Encoder

Code conversion

CSEE 3827, Spring 2009 Martha Kim

General code conversion

19

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

3-3

(a) Segment designation

a

bf

e cg

d(b) Numeric designation for display

Multiplexers

CSEE 3827, Spring 2009 Martha Kim

Multiplexers

• Combinational circuit that selects binary information from one of many input lines and directs it to one output line

21

(2 inputs)n

(n selection bits)

(1 output)

CSEE 3827, Spring 2009 Martha Kim

Truth table for a 4:1 mux

22

i0i1i2i3

s1 s0

o

CSEE 3827, Spring 2009 Martha Kim

Internal mux organization

23

© 2008 Pearson Education, Inc.M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS, 4e

3-26

S1Decoder

S0

Y

S1Decoder

S0

Y

S1Decoder

4 2 AND-ORS0

Y

I2

I3

I1

I0

CSEE 3827, Spring 2009 Martha Kim

Multibit multiplexing

24