textbook: digital design, 4 . edition m. morris mano and ...soc.eecs.yuntech.edu.tw/course/digital...

P-1/82 2012/3/14

Textbook: Digital Design, 4th. Edition

M. Morris Mano and Michael D. Ciletti

Prentice-Hall, Inc.

教師 : 蘇慶龍 INSTRUCTOR : CHING-LUNG SU

課程名稱: 數位邏輯設計

E-mail: [email protected]

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Chapter 1

Digital Systems and Binary Numbers

Chapter 1

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Outline of Chapter 1

1.1 Digital Systems

1.2 Binary Numbers

1.3 Number Base Conversions

1.4 Octal and Hexadecimal Numbers

1.5 Complements

1.6 Signed Binary Numbers

1.7 Binary Codes

1.8 Binary Storage and Registers

1.9 Binary Logic

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1.1 Digital Systems

1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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Digital Systems

1. Digital system examples: Digital telephones, digital

TV, DVD, digital cameras (DC), digital videos (DV),

MP3 Player and digital computers.

2. Digital systems: Manipulate discrete data

3. Binary: Numbers are presented by two discrete

values (0 and 1), Binary digit = Bit

4. Group of bits: Binary code

5. Digital systems: A system manipulates discrete

elements of information that is represented

internally binary form.

6. HDL (Hardware description language): A

programming language and suitable for describing

digital circuits in textual form.

1.1 Digital Systems

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1.2 Binary Numbers

1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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Decimal Number

1. Decimal number 7392:

7392=7103+ 3 102+ 9 101+ 2 100

2. Decimal number representation: a5a4a3a2a1a0.a-1a-2a-3

= 105a5+104a4+103a3+102a2+101a1+100a0+10-1a-1+10-2a-2+10-3a-3

1.2 Binary Numbers

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1.2 Binary Numbers

Decimal Number with Binary Number

11010.11=26.75

124+ 123+ 022+ 121+ 020+ 12-1+ 12-2=26.75

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1.2 Binary Numbers

Base-r Number with Binary Number

an rn + an-1 r

n-1 +… + a2 r2 + a1 r

+ a0 + a-1 r

-1 + a-2 r-2 + …+ a-m r

-m

(4021.2)5 = 453+ 052+ 251+ 150+ 25-1 = (511.4)10

(127.4)8 = 182+ 281+ 780+ 48-1 = (87.5)10

(B65F)16 = 11163+ 6162+ 5161+ 15160 = (46687)10

(110101)2 = 32+16+4+1 = (53)10

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1.2 Binary Numbers

Base-r Number System

1. Base-2, Binary

2. Base-8, Octal

3. Base-10, Decimal

4. Base-16, Hexadecimal, (10=A, 11=B, 12=C, D=13, E=14,

F=15)

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1.2 Binary Numbers

Base-2 Power of n

1. n=10, 210=1024, K (kilo)

2. n=20, 220=1048576, M (mega)

3. n=30, 230=130023424, G (giga)

4. n=40, 240=133143986176, T (tera)

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Example: Base-2, Power of n

1. 4K=4210=212

2. 16M=16220=224

3. 4G=4230=232

1.2 Binary Numbers

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Powers of Two

n 2n n 2n n 2n

0 1 8 256 16 65,536

1 2 9 512 17 131,072

2 4 10 1,024 18 262,144

3 8 11 2,048 19 524,288

4 16 12 4,096 20 1,048,576

5 32 13 8,192 21 2,097,152

6 64 14 16,384 22 4,194,304

7 128 15 32,768 23 8,388,608

1.2 Binary Numbers

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Arithmetic Operation - Addition

1.2 Binary Numbers

Augend: 101101

Addend: 100111

Sum: 1010100

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Arithmetic Operation - Subtraction

Minuend: 101101

Subtrahend: 100111

Difference: 000110

1.2 Binary Numbers

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Arithmetic Operation - Multiplication

Multiplicand: 1011

Multiplier: 101

1011

0000

1011

Product: 110111

1.2 Binary Numbers

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1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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Decimal to Binary Conversion: Iterative Division

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Example of Decimal to Binary Conversion:

(41)10 to (1011001)2

Integer Reminder Coefficient

Quotient

41/2= 20 + ½ a0=1

20/2= 10 + 0 a1=0

10/2= 5 + 0 a2=0

5/2= 2 + ½ a3=1

2/2= 1 + 0 a4=0

1/2= 0 + ½ a5=1

Answer: (41)10=(a5a4a3a2a1a0)2=(101001)2

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Conveniently Process: (41)10 to (1011001)2

412

202 1

102 0

52 0

22 1

12 0

0 1 Answer: 1 0 1 0 0 1

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Conveniently Process:

Decimal (Base-10) to Octal (Base-8)

1538

198 1

28 3

0 2 Answer: ( 2 3 1 )8

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Fraction Part Conversion : Iterative Multiplication

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Example of Decimal to Binary Conversion:

(0.6875)10 to (0.1011)2

Integer Fraction Coefficient

0.68752= 1 + 0.3750 a-1=1

0.37502= 0 + 0.7500 a-2=0

0.75002= 1 + 0.5000 a-3=1

0.50002= 1 + 0.0000 a-4=1

Answer: (0.6875)10=(0.a-1a-2a-3a-4)2=(0.1011)2


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Conveniently Process: (0.513)10 to Octal

0.513 8= 4.104

0.104 8= 0.832

0.832 8= 6.656

0.656 8= 5.248

0.248 8= 1.984

0.984 8= 7.872

Answer: (0.513)10=(0.406517…)8

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Complete Number Conversion:

1. (41.6875)10 = (101001.1011)2

2. (153.513)10 = (231.406517)8

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1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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Numbers with Different Bases

Decimal Binary Octal Hexadecimal

(base 10) (base 2) (base 8) (base 16)

00 0000 00 0

01 0001 01 1

02 0010 02 2

03 0011 03 3

04 0100 04 4

05 0101 05 5

06 0110 06 6

07 0111 07 7

08 1000 10 8

09 1001 11 9

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F

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Octal and Hexadecimal vs. Binary:

Octal Digit = 23 =3 bits

Hexadecimal Digit = 24 =4 bits


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Number Conversion between Binary to Octal

(10 110 001 101 011 . 111 100 000 110)2 = (26153.7406)8

2 6 1 5 3 . 7 4 0 6

Number Conversion between Binary to Hexadecimal

(10 1100 0110 1011 . 1111 0010)2 = (2C6B.F2)16

2 C 6 B . F 2

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Number Conversion between Octal to Binary

(673.124)8 = (110 111 011 . 001 010 100)2

6 7 3 . 1 2 4

Number Conversion between Hexadecimal to Binary

(306.D)16 = (0011 0000 0110 . 1101)2

3 0 6 . D

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1.5 Complements

1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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1.5 Complements

Complements

Arithmetic Operation : SubtractionLogic Operation : 0 1

Complement

For base-r

r's Complement

r-1's Complement

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1.5 Complements

Diminished Radix (r-1)’s Complements

Given a number N in base-r having n digits,

the (r-1)’s complement of N : (rn-1) - N

If

r =10 r-1=9 :

9’s complements of N = (10n-1) - N

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1.5 Complements

Decimal Example: n = 6 (6 digit) 106-1=999999

The 9’s complement of

546700 is 999999 - 546700 = 453299

012398 is 999999 - 012398 = 987601

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1.5 Complements

For Binary Number (r-1)’s Complement: r=2 r-1=1

The 1’s complement of N is (2n-1)-N, if n = 4

24=(10000)2 then (2n-1)-N = (1111)2-N

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Binary (r-1)’s Example: n = 7 (7 digit)

27-1=(1111111)2

The 1’s (r-1)’s complement of

1.5 Complements

1111111 - 1011000 = 01001111111111 - 0101101 = 1010010

Toggle Each Digit : 01

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1.5 Complements

Diminished Radix r’s Complements

Given a number N in base-r having n digits,

the r’s complement of N :

rn-N, for N 0

0, for N 0

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1.5 Complements

Comparison of r’s and (r-1)’s complement

r’s complement: (r-1)’s complement:

rn- N

= [(rn-1) – N]+1

= (r-1)’s complement +1

(rn-1) – N

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1.5 Complements

Example of r’s complement

10’s (r’s) complement of decimal

2389 7610 (9’s complement) +1= 7611

012398 987601 +1= 987602

246700 753299 +1=753300

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1.5 Complements

Example of r’s complement

2’s (r’s) complement of binary

1101100 0010011 +1= 0010100

0110111 1001000 +1= 1001001

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1.5 Complements

Subtraction with Complements

1. Add the minuend, M, to the r’s complement of

subtrahend, N. This performs M+(rn-N)=M-N+rn

2. If M N, the sum will produce an end carry, rn,

which can be discarded; what is left is the result M-

N.

3. If M < N, the sum dose not produce an end carry

and is equal to rn-(N-M), which is the r’s

complement of (N-M). To obtain the answer in a

familiar form, take the r’s complement of the sum

and place a negative sign in front.

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1.5 Complements

Example of Subtraction with Complements:

Using 10’s (r’s) complement, subtract 72532-3250

M= 7 2 5 3 2

10's complement of N= + 9 6 7 5 0

Sum = 1 6 9 2 8 2

Discard and carry 105= -1 0 0 0 0 0

Answer = 6 9 2 8 2

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1.5 Complements


Using 10’s (r’s) complement, subtract 3250-72532

M= 0 3 2 5 0

10's complement of N= + 2 7 4 6 8

Sum = 3 0 7 1 8

No carry !!

Answer : - (10's complement of 30718)

= - 6 9 2 8 2

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1.5 Complements


Using 2’s (r’s) complement, perform X-Y and Y-X

X= 1010100

2's complement of Y= + 0111101

Sum = 10010001

Discard end carry 27 = - 10000000

Answer X-Y = 0010001

X=1010100, Y=1000011

(a) X-Y :

Y= 1000011

2's complement of X= + 0101100

Sum = 1101111

No carry 27, Answer= -(2'sc of 1101111)

Answer = - 0010001

(a) Y-X :

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1.5 Complements


Using 1’s (1’s) complement, perform X-Y and Y-X

X= 1010100

1's complement of Y= + 0111100

Sum = 10010000

End-around carry 27 = + 1

Answer X-Y = 0010001

X=1010100, Y=1000011

(a) X-Y :

Y= 1000011

1's complement of X= + 0101011

Sum = 1101110

No carry 27, Answer= -(1'sc of 1101110)

Answer = - 0010001

(a) Y-X :

1'sc+1=2'sc

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1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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Signed Binary Numbers

an, an-1, an-2, …, a0

Sign Bit Magnitude Bits

(0=+, 1=-)

Example

01001

9 (unsigned binary)

9 (signed binary)

11001

25 (unsigned binary)

-9 (signed binary)

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Presentation formats of value -9

-9

Signed-magnitude 10001001

Signed-1'sc 11110110

Signed-2'sc 11110111


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Presentation formats for 4-bit binary value


Decimal 2’sc 1’sc Signed-Magnitude

+7 0111 0111 0111

+6 0110 0110 0110

+5 0101 0101 0101

…

+1 0001 0001 0001

+0 0000 0000 0000

-0 ------ 1111 1000

-1 1111 1110 1001

-2 1110 1101 1010

…

-6 1010 1001 1110

-7 1001 1000 1111

-8 1000 ------ ------

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Features of Number Systems


Signed-Magnitude System: For Ordinary Arithmetic

1'sc: For Logical Operation

2'sc: For Computer Arithmetic Operation

(without end-around carry)

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Arithmetic Addition of Signed-Magnitude System


1. The same signed number:

Adder the two magnitudes and give the sum common

sign.

2. Different signed number:

Subtract the smaller magnitude form the larger and

give the result the sign of the larger magnitude.

(+25)+(-37)=-(37-25)=-12

Additional 2 Steps: Compare Magnitude and Signed bit !!

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Four 2’sc Cases of Arithmetic Operations


+6 00000110

+13 00001101

+19 00010011

-6 11111010

+13 00001101

+7 00000111

+6 00000110

-13 11110011

-7 11111001

-6 11111010

-13 11110011

-19 11101101

Discard

n bits

n+1 bits

Signed-bitExtension

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Four 2’sc Cases of Arithmetic Subtraction


(A)-(+B)=(A)+(-B)

(A)-(-B)=(A)+(+B)

2'sc of -B

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1.7 Binary Codes

1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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1.7 Binary Codes

Binary Codes

n-bit binary code can distinct 2n discrete values !!

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1.7 Binary Codes

BCD (Binary Coded Decimal)

Decimal Symbol BCD Digit

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

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1.7 Binary Codes

Feature of BCD

k-digit decimal requires 4k-bits in BCD !!

Example of BCD

(185)10=(0001 1000 0101)BCD=(10111001)2

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1.7 Binary Codes

BCD Addition

k-digit decimal requires 4k-bits in BCD !!

Example of BCD Addition

4 0100

+5 +0101

9 1001

4 0100

+8 +1000

12 1100>9

Carry out

1100

+0110

1 0010

8 1000

+9 +1001

17 10001>9

Carry out

10001

+0110

1 0111

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1.7 Binary Codes

Why add 0110 into result if result >9 ?

Decimal Binary BCD

0 0000 0000

1 0001 0001

2 0010 0010

3 0011 0011

4 0100 0100

5 0101 0101

6 0110 0110

7 0111 0111

8 1000 1000

9 1001 1001

10 1010 +6 1 0000

11 1011 +6 1 0001

12 1100 +6 1 0010

13 1101 +6 1 0011

14 1110 +6 1 0100

15 1111 +6 1 0101

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1.7 Binary Codes

Example of BCD Addition: 184+576 in BCD

BCD carry 1 1

0001 1000 0100 184

+0101 0111 0110 +576

Binary Sum 0111 10000 1010

Add 6 0110 0110

BCD Sum 0111 0110 0000 760

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1.7 Binary Codes

Decimal Arithmetic

(+375) + (-240) = +135

0 375

+ 9 760

0 135

Negative

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1.7 Binary Codes

Other Decimal Codes

Decimal BCD 2421 Excess-3 8 4-2-1

digit 8421

0 0000 0000 0011 0 0 0 0

1 0001 0001 0100 0 1 1 1

2 0010 0010 0101 0 1 1 0

3 0011 0011 0110 0 1 0 1

4 0100 0100 0111 0 1 0 0

5 0101 1011 1000 1 0 1 1

6 0110 1100 1001 1 0 1 0

7 0111 1101 1010 1 0 0 1

8 1000 1110 1011 1 0 0 0

9 1001 1111 1100 1 1 1 1

1010 0101 0000 0 0 0 1

1011 0110 0001 0 0 1 0

Unused Bits 1100 0111 0010 0 0 1 1

1101 1000 1101 1 1 0 0

1110 1001 1110 1 1 0 1

1111 1010 1111 1 1 1 0

Self-complementing Code

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1.7 Binary Codes

Example of Self-complementing Code:

9’s complement of (395)10 in Exceed-3 code

(395)10

= 0110 1100 1000 (in Exceed-3)

= 1001 0011 0111 (9'sc 604)

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1.7 Binary Codes

Gray Code Decimal Binary Gray

Code

0 0000 0000

1 0001 0001

2 0010 0011

3 0011 0010

4 0100 0110

5 0101 0111

6 0110 0101

7 0111 0100

8 1000 1100

9 1001 1101

10 1010 1111

11 1011 1110

12 1100 1010

13 1101 1011

14 1110 1001

15 1111 1000

Binary: 0 bn ...

b3 b

2 b

1 b

0

Gray Code: gn ...

g3 g

2 g

1 g

0

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1.7 Binary Codes

Feature of Gray Code

Only one bit in the code group changes from one number

to the next !

For Example 7 8

Gray Code: 0100 1100

Binary Code: 0111 1000

Only one bitchange !!

4 bit change

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1.7 Binary Codes

ASCII Code (American Standard Code for Information Interchange)

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1.7 Binary Codes

Extended ASCII Code

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1.7 Binary Codes

Error-Detecting Code:

Communication and computation will cause error

Even Parity Odd Parity

ASCII A=1000001 01000001 11000001

ASCII T=1010100 11010100 01010100

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1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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Register

1. A register is a group of binary cells.

2. A register with n cells can store n-bit information.

3. A register with 16 cells can be in one of 216 possible

states, i.e. number 0~ 216-1 .

4. A 16-bit example: 1100 0011 1100 1001

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Register Transfer

1. A register transfer operation is a basic operation in

digital systems.

2. It transfer binary information from one set of

registers to another set of registers.

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Transfer of Information with Register

keybord

J

O

H

N

Keyboard

ControllerStrike

8-bitRegister

Input Unit

8-bitRegister

8-bitRegister

8-bitRegister

8-bitRegister

Processor Unit

Processor Register

01001010 01001111 11001000 11001110

J O H N

Memory Unit

Memory Register

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Transfer of

Information with

Register

Digital logic

circuits for

binary addition

0100100011

0001000010

0011100001

R1

R2

R3

Processor Unit

0000000000

0011100001

0001000010

Memory Unit

Operand 1

Operand 2

Sum

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1.9 Binary Logic

1.1 Digital Systems

1.2 Binary Numbers



1.5 Complements


1.7 Binary Codes


1.9 Binary Logic

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Binary Logic

1. 0/1

2. True/False

3. Yes/No

1.9 Binary Logic

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Definition of Binary Logic

1.9 Binary Logic

Binary Logic: 1. Arithmetic operation

2. Logical operation

Basic Logical Operation: AND / OR/ NOT

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Binary Logic

1.9 Binary Logic

AND

NOT

OR

x y=z

or

xy=z

x y=z

x'=z

or

x=z

x AND y is equal to z

x OR y is equal to z

NOT x is equal to z

Why?

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Truth Table for AND/OR/NOT

1.9 Binary Logic

x y x y

0 0 0

0 1 0

1 0 0

1 1 1

AND

x y x+y

0 0 0

0 1 1

1 0 1

1 1 1

OR

x x’

0 1

1 0

NOT

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Example of a Digital System

1.9 Binary Logic

0

1

3

4

Range for Logic 0

Range for Logic 1

Transition occursbetween these limits

Volts

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Symbol for Digital Logic Circuit

1.9 Binary Logic

x

yz=x y

x

yz=x+y

x x'

Two Input

AND Gate

Two Input

OR Gate

NOT Gate or

Inverter

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Input-Output Signal for Logic Gates

1.9 Binary Logic

x 0 1 1 0 0

y 0 0 1 1 0

AND: x · y 0 0 1 0 0

OR : x + y 0 1 1 1 0

NOT: x’ 1 0 0 1 1

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Gates with Multiple Inputs

1.9 Binary Logic

A

B F=A B C

G=A+B+C

3 Input

AND Gate

3 Input

OR Gate

C

A

B

C

textbook: digital design, 4 . edition m. morris mano and ...soc.eecs.yuntech.edu.tw/course/digital...

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