fyse410 digital electronics - jyväskylän...

FYSE410 DIGITAL ELECTRONICS

Litterature:

DIGITAL DESIGN

M. Morris Mano

Michael D. Ciletti

[2]

DIGITAL LOGIC CIRCUIT ANALYSIS

& DESIGN

Victor P. Nelson, H. Troy Nagle

J. David Irwin, Bill D. Carroll

[1]

ISBN 0-13-463894-8

LECTURE 1

©Loberg

Digital Logic Design

A. P. Godse

D. A. Godse

[4]

ISBN 0-13-198924-3

Introduction ToDigital Techniques

Dan I. Porat

Arpad Barna

[3]

ISBN 0-471-02924-6

Digital DesignPrinciples and Practices

John F. Wakerly

[5]

ISBN 0-13-186389-4

LECTURE 1

1

NUMBER SYSTEMS AND CODES Number Systems

A number system consists of an ordered set of symbols (digits).

Positional Notation

(((( ))))rm21012n1n aaa.aaaaN −−−−−−−−−−−−−−−−−−−− ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅====

n integer digits m fractional digits[1]

The total number of digits allowed in the number system is called

base or radix (r)

Addition

Subraction

Multiplication

Division ÷÷÷÷××××−−−−++++

1na −−−−

ma−−−−

Most significant digit

Least significant digit

©Loberg

2

NUMBER SYSTEMS AND CODES

(((( ))))21012

10

105103103102101

01.051.0313102100135.123N

−−−−−−−− ××××++++××××++++××××++++××××++++××××====

××××++++××××++++××××++++××××++++××××========

Number Systems

Polynomial Notation

For desimal number : {{{{ }}}}9,8,7,6,5,4,3,2,1,0aand10r i ========

∑∑∑∑−−−−

−−−−========

1n

mi

i

iraN

[1]

©Loberg

3


Commonly Used Number Systems

Desimal number : {{{{ }}}}9,8,7,6,5,4,3,2,1,0aand10r i ========

Number Systems

Binary number : {{{{ }}}}1,0aand2r i ========

Octal number : {{{{ }}}}7,6,5,4,3,2,1,0aand8r i ========

Hexadecimal number : {{{{ }}}}F,E,D,C,B,A,9,8,7,6,5,4,3,2,1,0aand16r i ========

[1]

©Loberg

4


LSBMSB

023272

18128

Byte

bit8Byte1 ====

Unsigned positive numbers

0123456

01

1011

100101110

0123456

0123456

Decimal Binary Octal Hexadecimal

Bit

1or0bitOne

Number Systems

Commonly Used Number Systems

6789

10111213141516

110111

10001001101010111100110111101111

10000

67

101112131415161720

6789ABCDEF

10

First seventeen positive integers[1]

Word

LSBMSB

0272152

1281024

122

409632768

Kb32

bits102432 ××××

bitsK1

1

bit16Byte2Word1 ========

Kb4

For example the size of memory : the number of

one bit memory locations.

©Loberg

5

NUMBER SYSTEMS AND CODES Arithmetic

Binary Arithmetic Addition

0 0

1

0

10

1

1

1

+

Binary addition table

1Carry011

101

110

000

========++++====++++====++++====++++


111010

101111

0010101

111111

+

Carry

Augend

Addend

Sum

Example (((( )))) (((( )))) 10101022 84236110111111101 ====++++====++++

(((( ))))(((( )))) (((( )))) (((( ))))222 110110

111111

====++++++++++++====++++++++

[1]

©Loberg

6


Example

(((( )))) (((( )))) (((( )))) (((( )))) 10101010102222 12817135345010001001101110101101101 ====++++++++++++====++++++++++++

101011

101101

1

The computation is easily performed by adding the numbers in pairs.


Arithmetic


100010

101100

101011

+101011

101101

0100011

1111

+ 100010

101100

011110

1

+

Sum

0111100

0100011

00000001

+

111111

[1]

©Loberg

7


Example(((( )))) (((( )))) (((( )))) (((( ))))2222 010001001101110101101101 ++++++++++++

We can also perform the addition directly to avoid the intermediate steps.


Arithmetic


(((( )))) (((( )))) (((( )))) (((( )))) (((( ))))222 100101011111111 ====++++====++++++++++++====++++++++++++100010

101100

101011

101101

+

10110101010

00000001

(((( ))))210000010 ====++++++++++++++++(((( )))) (((( )))) (((( )))) (((( )))) (((( ))))222 10001010111101111 ====++++++++====++++++++++++====++++++++++++++++

(((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( ))))2222 1001010010110010110 ====++++====++++++++++++++++====++++++++++++++++

(((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( ))))2222 1001010101010101010 ====++++====++++++++++++++++====++++++++++++++++

(((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( ))))2222 1001010001110001110 ====++++====++++++++++++++++====++++++++++++++++

[1]

©Loberg

8


Subtraction


Example(((( )))) (((( )))) 10101022 542377101111001101 ====−−−−====−−−−

0124 356

1 10 Borrows

Column

Arithmetic

Binary Arithmetic

1110or1ofborrowawith110

000

011

101

====−−−−====−−−−====−−−−====−−−−====−−−−

1011001

11101

0

0

1

1000

10

10 10

11 Difference

Subtrahend

Minuend

Borrows

[1]

©Loberg

9


Multiplication


0 0

1

0

1

1

0

0

x

Example

(((( )))) (((( )))) 10101022 2301023101010111 ====++++====××××

0101

11101

x 0101

11101

x

Arithmetic

Binary Arithmetic

[1]

1 10

Binary Multiplication table

0101

00000

x

11101

0000011101

000 1 1111

0101x

111011

11101000 1 1111

1 1

We don't need to list an all-zero partial

product for a multiplier bit of 0.

(((( )))) 1010 230243264128 ====++++++++++++++++

©Loberg

10


Division


Example : Divide

(((( )))) 102 1191110111 ==== (((( )))) 102 91001 ====by

Example : Divide

(((( )))) 102 651000001 ==== (((( )))) 102 131101 ====by

1011

DivisorQuotient 13====

1011 1000001

1010 105====

Arithmetic

Binary Arithmetic

01

1101

1001 11101111011

1001

1001

10011101

Dividend

Reminder 2====0

1

00001

1011 10000010000

1011

1011

0110000000011000

100====

[1]

©Loberg

11


Octal Arithmetic Addition


Example (((( )))) (((( ))))88 75204163 ++++

10

0123

8

216313866415124

838681844163

====××××++++××××++++××××++++××××====××××++++××××++++××××++++××××====

0123

8 808285877520 ××××++++××××++++××××++++××××====

Arithmetic

Octal addition table 10392010826455127 ====××××++++××××++++××××++++××××====

Carries

Augend

Addend

Sum

3614

1+ 7 5 2 0

3 7 0 3

1 1

[1]

©Loberg

12


Subtraction

Example (((( )))) (((( ))))88 51736204 −−−−

10

0123

8

320414806425126

848082866204

====××××++++××××++++××××++++××××====××××++++××××++++××××++++××××====

Borrows

Minuend

Subtrahend

4026+ 5 1 7 3

101

Arithmetic

Octal Arithmetic


10

0123

8

268313876415125

838781855173

====××××++++××××++++××××++++××××====××××++++××××++++××××++++××××====

Subtrahend

Difference

+ 5 1 7 3

1 0 1 1

[1]

©Loberg

13


Multiplication

10

0123

8

216717866415124

878681844167

====××××++++××××++++××××++++××××====××××++++××××++++××××++++××××====

10

0123

8

134713806455122

838085822503

====××××++++××××++++××××++++××××====××××++++××××++++××××++++××××====

Example (((( )))) (((( ))))88 25034167 ××××7614

+

51 5

32 0

15

402

4

54

0

0 0 02 3

x

13 50530 51 6

1

Multiplicand

Multiplier

Partial products

Product

Arithmetic

Octal Arithmetic


[1]

Octal multiplication table

10134713806455122 ====××××++++××××++++××××++++××××====

10

0234567

8 2918949858480858081838113105045 ====××××++++××××++++××××++++××××++++××××++++××××++++××××++++××××====

©Loberg

14


Division

Example : Divide (((( )))) 108 21634163 ==== (((( )))) 108 2125 ====by

52 3614

741Dividend

Quotient012

8 878481147 ××××++++××××++++××××====

Arithmetic

Octal Arithmetic


[1]

641

852 8361452

421322

000322

Reminder

Dividend

10

8

1031784641

878481147

====××××++++××××++++××××====××××++++××××++++××××====

©Loberg

15


Hexadecimal Arithmetic Addition

Example (((( )))) (((( ))))1616 0D7158A2 ++++

Arithmetic


[1]

Carries

Augend

Addend

Sum

85A2+7 1 D09C 2 8

1

Hexadecimal addition table

©Loberg

16


Borrows

Minuend

Subtrahend

B1F9- 4 A 3 6

11E


Subtraction

Example (((( )))) (((( ))))1616 36A4B1F9 ++++

Arithmetic

Hexadecimal Arithmetic

Difference

-5 4 E5

[1]

©Loberg

17


Multiplication

A2C51 07 Dx

Multiplicand

Multiplier

Example (((( )))) (((( ))))1616 0D71A2C5 ××××

Arithmetic



Hexadecimal Multiplication table

+

1

2

07 D

C5E0

2

2

A4

9

0 0 0

A

F8 0C658 22 6

2

Partial

products

Product

2

[1]

©Loberg

18


Division

C31

16E3 16ACF72

15A

C62

631

Dividend

Quotient

Example : Divide (((( ))))16FCA27 (((( ))))16E3by

Arithmetic



[1]

C31631A6

C2E3

Reminder

©Loberg

19

NUMBER SYSTEMS AND CODES Base Conversion

We have numbers NA and NBAssumption: {{{{ }}}},...3,2,1k,AB k ========

Group the digits of N in groups of k digits in both directions from the radix point

and then replace each group with the equivalent digit in base B.

82ABand2A 33 ================Examples : Binary to Octal

0110110011011011 →→→→ 1331011011 →→→→

Conversion :BA NN →→→→BA <<<<

[1]

01101100110110112 →→→→

81 83 8382 1331011011 →→→→

85

100011101.011011001101011100.1011011 2 →→→→81 83 83 83 84

82 534.133101011100.1011011 ====

©Loberg

20


Assumption: {{{{ }}}},...3,2,1k,AB k ========

Replace each base B digit in N with equivalent k digits in base A.

44 2A16Band2A ================Example : Hexadecimal to Binary 4k ====

BA NN ←←←←

Base Conversion

Conversion :

BA <<<<

216 110001100001.11111010C16.AF →→→→ Replace each base 16 digit in N 16 withequivalent 4 digits in base 2 .

4k ====

33 2A8Band2A ================Example : Octal to Binary 3k ====

28 100101101000.1111010100554.257 ==== Replace each base 8 digit in N 8 withequivalent 3 digits in base 2 .

[1]

©Loberg

21


Assumption:34 2Aand2B ========Hexadecimal to Octal

000001011011.10101111110001100001.11111010C16.AF ====→→→→

Example : Convert16C16.AF to base 8

Base Conversion

BA NN ←←←←Conversion :

[1]

2216 000001011011.10101111110001100001.11111010C16.AF ====→→→→A C1 6F

8

2

0554.257

100101101000.111101010

====

©Loberg

22


Conversion : Radix Divide MethodBA NN →→→→

For integers

(((( )))) 0

0

1n

1nAI Bb...BbN ++++++++==== −−−−−−−−

Integer in base AThe bi 's represent the digits of (NI)B in base A

Example : Convert10234 to base 8

Base Conversion

[1]

(((( ))))(((( ))))

0

0

1

2n

1n

0

0

1

1

1n

1n

A

AI

bBb...Bb

B

BbBb...Bb

B

N

++++++++++++====

++++++++====

−−−−−−−−

−−−−−−−−

Quotient Q1 Reminder R0

Quotient Q2Reminder R1

Quotient Q3 Reminder R2

Stop when Quotient is zero

(((( ))))A1 BQ====

(((( ))))A2 BQ====

102341029

108167472

0b2 ====

1029103

10824

1b5 ====

103100

1080

2b3 ====

810 352234 ====

©Loberg

23


Example : Convert10234 to base 16

102341014

101616

1014100

10160

Base Conversion

Conversion : Radix Divide Method

For integers

[1]

102341016167464

(((( )))) 01610 bA10 ========

101410160

(((( )))) 116 bE14 ========

1610 EA234 ====

©Loberg

24


For fractoins

(((( )))) m

m

2

2

1

1AF Bb...BbBbN −−−−−−−−

−−−−−−−−

−−−−−−−− ++++++++====

The fraction can be written in series form.

The bi 's represent the digits of (NF)B in base AFraction in base A

Base Conversion

Conversion : Radix Multiply Method

[1]

Fraction in base A

(((( )))) (((( ))))(((( ))))1m

m

1

21

m

m

2

2

1

1AAFA

Bb...Bbb

Bb...BbBbBNB

−−−−−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

−−−−−−−−

−−−−−−−−

++++++++====

++++++++××××====××××

©Loberg

25


Example : Convert101285.0 to base 8

100280.0

108

102240.0

101285.0

108

100280.1×××× ××××

102240.0

108

107920.1××××

107920.0

108

103360.6××××

103360.0

108

106880.2××××

106880.0

108

105040.5××××

105040.0

108

100320.4××××

100320.0

108

102560.0××××

Base Conversion


For fractoins

[1]

1b−−−− 2b−−−− 3b−−−− 4b−−−− 5b−−−− 6b−−−− 7b−−−− 8b−−−−

(((( ))))810 ...10162540.01285.0 ====

©Loberg

26


Example : Convert10828125.0 to base 2

10

10

312500.02625000.0

656250.02312500.1

828125.02656250.1

××××====××××====××××====MSD

Base Conversion


For fractoins

10

10

10

10

500000.02000000.1

250000.02500000.0

625000.02250000.1

312500.02625000.0

××××====××××====××××====××××====

LSD

210 110101.0828125.0 ====

[1]

©Loberg

27

NUMBER SYSTEMS AND CODES Signed Number Representation

Sign and Magnitude

(((( ))))rsmm101n aa.aasN −−−−−−−−−−−− ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅====

negativeNif1rs

positiveNif0swhere

====−−−−============

(((( )))) (((( )))) (((( )))) sm2210 1101,1110113N ====−−−−====−−−−====

(((( )))) (((( )))) sm1010 13,913N ====−−−−====

Sign = r-1

Examples :

[1]

©Loberg

28


[[[[ ]]]] (((( ))))rnr NrN −−−−====

Radix Complement [[[[ ]]]] rN

(((( ))))rNindigitsofnumbern ====

Largest positive number is 1r 1n −−−−−−−−

Most negative number is 1nr −−−−−−−−5n ====

Complements are used in digital computers to simplify the

subtraction operation. (simpler and less expensive circuits)

In general, for base-r system : ( r's complement )

Signed Number Representation

Most negative number is r−−−−

[1], [2]

[[[[ ]]]] [[[[ ]]]] (((( ))))(((( )))) (((( ))))(((( ))))

2

22

2

8

22

10011011

01100101100000000

01100101201100101N

====−−−−====

−−−−========

Example of Two's Complement :

"Invert bits and add one"

Two's Complement : [[[[ ]]]] (((( ))))2n2 N2N −−−−====2r ====

©Loberg

29


Diminished Radix Complement

[[[[ ]]]] (((( )))) 1NrN r

n1r −−−−−−−−====−−−− (((( ))))rNindigitsofnumbern ====

Ones's Complement :

[[[[ ]]]] (((( )))) 1NrN n −−−−−−−−====2r ====

In general, diminished radix complement [N]r-1 of number (N)r is: ( (r-1)'s complement )


"Invert bits"

Example of One's Complement :

[[[[ ]]]] [[[[ ]]]] (((( ))))(((( )))) (((( )))) (((( )))) (((( ))))

2222

2

8

1212

100110100000000101100101100000000

101100101201100101N

====−−−−−−−−====

−−−−−−−−======== −−−−−−−−

[1]

[[[[ ]]]] (((( )))) 1NrN 2

n12 −−−−−−−−====−−−−2r ====

©Loberg

30

(((( )))) (((( )))) (((( )))) [[[[ ]]]] (((( )))) (((( )))) n

rrrrrr rNMNMNM ++++−−−−====++++====−−−−


Subtraction with Radix Complement

(((( )))) (((( ))))rr NM ≥≥≥≥If The sum will generate an end carry nr Discard the end carry The result is M-N

(((( )))) (((( ))))rr NM ≤≤≤≤If The sum does not produce an end carry The result is [[[[ ]]]] rMN −−−−

( r's complement of (N-M) )


( r's complement of (N-M) )

Examples of 10's Complement :

(((( )))) (((( )))) 325072532NM 1010 −−−−====−−−−

10's complement of N is : 96750

725329675016928210000069282

+

-Discard end carry 105

(((( )))) (((( )))) 725323250NM 1010 −−−−====−−−−

10's complement of N is : 27468

032502746830718

+

10's complement of

30718 is : [[[[ ]]]] 6928230718 10 −−−−====−−−−

©Loberg

31

NUMBER SYSTEMS AND CODES Computer Codes

Numeric Codes Fixed-point Numbers

Excess or Biased Representations

32Excess −−−−

1n−−−− 2n−−−− 3n−−−− 01⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

1n−−−−2n−−−−3n−−−−0 1 ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

Sign bit

Implied binary

point

Implied binary point

Magnitude

representation

Fixed-point integer

Fixed-point fraction

[1]

32Excess −−−−

Excess-K representation of a code C is C+K.

Excess-2n numbers are two's complement numbers

with the sign bit reversed.

EXCESS-8 code

©Loberg

32


Floating-point Numbers

ErMN ××××====exponent

mantissa

========

E

Mwhere

Computer Codes

Numeric Codes

(((( ))))rsmm1nM aa.SM −−−−−−−− ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅==== numberpositive0SM ====

Mantissa M is often coded

in sign magnitude, usually

as a fraction.

[1]

©Loberg

33


Exponent E is most often coded in excess-K two's complement. Bias K is added to the 2's complement

integer value of the exponent.

For binary floating-point numbers, K is usually selected to be1e2 −−−− where e is the number of

bits in the exponent.

(((( ))))(((( )))) 1EE

1EE rrMrMN

−−−−

++++

××××××××====××××====××××÷÷÷÷====××××====

Computer Codes

Numeric Codes Floating-point Numbers

(((( )))) 1EE rrMrMN −−−−××××××××====××××====

(((( ))))(((( ))))(((( ))))(((( )))) 6

2

5

2

4

2

2

20011010101.0

2011010101.0

211010101.0

0101.1101M

××××====

××××====

××××====

++++====

Example :

[1]

©Loberg

34


Character and Other Codes Binary Coded Decimal (BCD)

0: 00001: 00012: 00103: 0011

The BCD code is weighted code: Each bit position in

the code has a fixed weight associated with it.

(8421-code)

Computer Codes

[1]

3: 00114: 01005: 01016: 01107: 01118: 10009: 1001

BDC Codes

Example

(((( )))) (((( ))))BCD10 00000101011110019750N ========

©Loberg

35


Gray Code (unit distance code)

Computer Codes

Character and Other Codes

[1]

©Loberg

36


ASCII

(American Standard Code for Information Interchange)

Computer Codes

Character and Other Codes

[1]

©Loberg

37


Error Detection - and Correction Codes Parity Codes

Even Parity P=1 Information bits : the number of 1 is odd

P Information bits

Parity Bit

Computer Codes

[1]

Odd Parity

P=0 Information bits : the number of 1 is even

P=0 Information bits : the number of 1 is odd

P=1 Information bits : the number of 1 is even

[2]

©Loberg

38


Classification of Binary Codes

Weighted Non-weighted Reflective Sequential Alphanumeric Error Detecting

and Correcting

Binary

BCD

Excess-3 Gray 2421 ASCII5211 Excess-3 8421

Excess-3 Hollerith

EBCDIC

Parity Hamming

[4]

Odd

Computer Codes

[3]

Even

©Loberg

39

The End

40

The End

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