ddca_ch1 - class1 (1)

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C 1

, 2

C 1

D H . H

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C 1

• •

•

• •

•

•

• •

C 1 ::

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C 1

•

– C , I, , .

• $21 1985 $300 2011

B

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C 1

H

f o c u s

o f t

h i s

c o u r s e

programs

device drivers

instructionsregisters

datapathscontrollers

addersmemories

AND gatesNOT gates

amplifiers

filters

transistorsdiodes

electrons

A

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C 1

•

– – F

–

• D

D A

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C 1

• – 1 0

– 1, E, HIGH, H – 0, FAE, ,

• 1 0 , , , .

• D 1 0

•

D D: B

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C 1

537410

=

1 0 ' s c ol umn

1 0 0 ' s

c ol umn

1 0 0 0

' s c ol umn

1 ' s c o

l umn

11012 =

2 ' s c ol u

mn

4 ' s c ol u

mn

8 ' s c ol u

mn

1 ' s c ol u

mn

• Decimal numbers

• Binary numbers

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C 1

537410

= 5 × 103 + 3 × 102 + 7 × 101 + 4 × 100five

thousands

1 0 ' s c ol umn

1 0 0 ' s

c ol umn

1 0 0 0 ' s c ol umn

threehundreds

seventens

fourones

1 ' s c o

l umn

11012 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 13

10oneeight

2 ' s c ol u

mn

4 ' s c ol u

mn

8 ' s c ol u

mn

onefour

notwo

oneone

1 ' s c ol u

mn

• Decimal numbers

• Binary numbers

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C 1

• 20 =

• 21 =

• 22 =

• 23 =

• 24 =

• 25 =

• 26

=• 27 =

• 28 =

• 29 =

• 210 =

• 211 =

• 212 =

• 213 =

• 214

=• 215 =

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C 1

• 20 = 1

• 21 = 2

• 22 = 4

• 23 = 8

• 24 = 16

• 25 = 32

• 26

= 64• 27 = 128

• Handy to memorize up to 210

• 28 = 256

• 29 = 512

• 210 = 1024

• 211 = 2048

• 212 = 4096

• 213 = 8192

• 214

= 16384• 215 = 32768

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C 1

• 224?

• H 32

?

E

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C 1

• 224?

24 × 220 ≈ 16

• H 32

?22 × 230 ≈ 4

E

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C 1

• Decimal to binary conversion:– Convert 100112 to decimal

• Decimal to binary conversion:– Convert 4710 to binary

C

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C 1

• Decimal to binary conversion:– Convert 100112 to decimal

– 16×1 + 8×0 + 4×0 + 2×1 + 1×1 = 1910

• Decimal to binary conversion:– Convert 4710 to binary

– 32×1 + 16×0 + 8×1 + 4×1 + 2×1 + 1×1 = 1011112

C

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C 1

• – H ?

– ? – E: 3 :

• – H ?

– : – E: 3 :

B

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C 1

• – H ? 10

– ? 0, 10

1 – E: 3 :

• 103 1000 • 0,

• – H ? 2

– : 0, 2 1

– E: 3 :• 23 • 0, 0002 1112

B

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C 1

Hex Digit Decimal Equivalent Binary Equivalent

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

A 10

B 11C 12

D 13

E 14

F 15

H

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C 1

Hex Digit Decimal Equivalent Binary Equivalent

0 0 0000

1 1 0001

2 2 0010

3 3 0011

4 4 0100

5 5 0101

6 6 0110

7 7 0111

8 8 1000

9 9 1001

A 10 1010

B 11 1011

C 12 1100

D 13 1101

E 14 1110

F 15 1111

H

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C 1

• B

• B &

• B

10010110nibble

byte

CEBF9AD7least

significantbyte

mostsignificant

byte

10010110least

significantbit

most

significantbit

B, B,

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C 1

37345168+

8902

carries11

10110011+

1110

11 carries

• Decimal

• Binary

A

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C 1

1001

0101+

10110110+

• Add the following

4-bit binary

numbers


4-bit binary

numbers

B A E

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C 1

1001

0101+1110

1

10110110+

10001

111


4-bit binary

numbers


4-bit binary

numbers

Overflow!

B A E

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C 1

1001

0101+1110

1

10110110+

10001

111

• Substract the

following 4-bit

binary numbers

• Substract the

following 4-bit

binary numbers

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C 1

• Sign/Magnitude Numbers

• Two’s Complement Numbers

B

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C 1

• 1 sign bit, N -1 magnitude bits

• Sign bit is the most significant (left-most) bit

– Positive number: sign bit = 0

– Negative number: sign bit = 1

• Example, 4-bit sign/mag representations of ± 6:+6 =

- 6 =

• Range of an N -bit sign/magnitude number:

{ }

1

1 2 2 1 0

2

0

: , , , ,

( 1) 2n

N N

n

a i

i

i

a a a a a

A a−

− −

−

=

= − ∑

L

/

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C 1

• 1 sign bit, N -1 magnitude bits

• Sign bit is the most significant (left-most) bit

– Positive number: sign bit = 0

– Negative number: sign bit = 1

• Example, 4-bit sign/mag representations of ± 6:+6 = 0110

- 6 = 1110

• Range of an N -bit sign/magnitude number:[-(2N-1-1), 2N-1-1]

{ }

1

1 2 2 1 0

2

0

: , , , ,

( 1) 2n

N N

n

a i

i

i

a a a a a

A a−

− −

−

=

= − ∑

L

/

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C 1

• Problems:

– Addition doesn’t work, for example -6 + 6:

1110

+ 0110

10100 (wrong!)

– Two representations of 0 (± 0):

1000

0000

/

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C 1

• Don’t have same problems as sign/magnitude

numbers:– Addition works

– Single representation for 0

C

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C 1

( )

21

1

02 2

n

n i

n i

i A a a

−

−

−

=

= − +∑

• MSB has value of -2 N -1

• Most positive 4-bit number:

• Most negative 4-bit number:

• The most significant bit still indicates the sign

(1 = negative, 0 = positive)• Range of an N -bit two’s comp number:

C

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C 1

( )

21

1

02 2

n

n i

n i

i A a a

−

−

−

=

= − +∑

• Msb has value of -2 N -1

• Most positive 4-bit number: 0111

• Most negative 4-bit number: 1000

• The most significant bit still indicates the sign

(1 = negative, 0 = positive)• Range of an N -bit two’s comp number:

[-(2N-1), 2N-1-1]

C

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C 1

• Flip the sign of a two’s complement number

• Method:

1. Invert the bits

2. Add 1

• Example: Flip the sign of 310 = 00112

C

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C 1

• Flip the sign of a two’s complement number

• Method:

1. Invert the bits

2. Add 1

• Example: Flip the sign of 310 = 001121. 1100

2. + 1

1101 = -310

C

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C 1

• Take the two’s complement of 610 = 01102

• What is the decimal value of 10012?

C E

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C 1

• Take the two’s complement of 610 = 011021. 1001

2. + 1

10102 = -610

• What is the decimal value of the two’s

complement number 10012?

1. 01102. + 1

01112 = 710, so 10012 = -710

C E

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C 1

+01101010

+11100011

• Add 6 + (-6) using two’s complement

numbers

• Add -2 + 3 using two’s complement numbers

C A

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C 1

+01101010

10000

111

+11100011

10001

111

• Add 6 + (-6) using two’s complement

numbers

• Add -2 + 3 using two’s complement numbers

C A

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C 1

• Sign bit copied to msb’s

• Number value is same

• Example 1:

– 4-bit representation of 3 = 0011

– 8-bit sign-extended value: 00000011

• Example 2:

– 4-bit representation of -5 = 1011

– 8-bit sign-extended value: 11111011

E

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C 1

• Zeros copied to msb’s

• Value changes for negative numbers

• Example 1:

– 4-bit value = 00112 = 310

– 8-bit zero-extended value: 00000011 = 310

• Example 2:

– 4-bit value = 1011 = -510

– 8-bit zero-extended value: 00001011 = 1110

E

C

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C 1

-8

1000 1001

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111 Two's Complement

10001001101010111100110111101111

00000001 0010 0011 0100 0101 0110 0111

1000 1001 1010 1011 1100 1101 1110 11110000 0001 0010 0011 0100 0101 0110 0111

Sign/Magnitude

Unsigned

Number System Range

Unsigned [0, 2 N -1]

Sign/Magnitude [-(2

N -1

-1), 2

N -1

-1]Two’s Complement [-2 N -1, 2 N -1-1]

For example, 4-bit representation:

C

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E: 4 ( )

Value Sign-and-Magnitude

1sComp.

2sComp.

+7 0111 0111 0111

+6 0110 0110 0110+5 0101 0101 0101+4 0100 0100 0100+3 0011 0011 0011+2 0010 0010 0010

+1 0001 0001 0001+0 0000 0000 0000

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E: 4 ( )

Value Sign-and-Magnitude

1sComp.

2sComp.

-0 1000 1111 --1 1001 1110 1111-2 1010 1101 1110-3 1011 1100 1101-4 1100 1011 1100-5 1101 1010 1011

-6 1110 1001 1010-7 1111 1000 1001-8 - - 1000

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E:1. For 2’s complement binary numbers, the range of

values for 5-bit numbers is

a. 0 to 31 b. -8 to +7 c. -8 to +8

d. -15 to +15 e. -16 to +15

2. In a 6-bit 2’s complement binary number system,what is the decimal value represented by (100100)2s?

a. -4 b. 36 c. -36 d. -27 e. -28

2

ddca_ch1 - class1 (1)

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