Transcript
Page 1: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

CSE20

Lecture 2: Number Systems: BinaryNumbers and Gray Code

CK Cheng

1

Page 2: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

Number Systems

1. Introduction2. Binary Numbers3. Gray code4. Negative Numbers5. Residual Numbers

2

Page 3: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2. Binary Numbers: iClickerWhat is the extent of a binary number

system• A. Coverage of integer and floating point

numbers• B. Mechanism of addition and subtraction

operations• C. Operations of logic functions• D. All of the above• E. None of the above.

3

Page 4: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2. Binary Numbers

1. Definition (radix 2)2. Enumerations (value -> index)3. Addition (logic -> hardware)

4

Page 5: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.1 Definition of Binary Numbers

• Format: An n digit binary number(bn-1, …, b1, b0)2 where bi in {0,1} for 0<= i < n• Value: bn-12n-1+ …+b121+b020

• Non-redundancy: The system is non-redundant, i.e. different binary numbersrepresent different values.

5

Page 6: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.2 Enumeration of Binary Numbers

id b2b1b0

0 0001 0012 0103 0114 1005 1016 1107 111

6

id b1b0

0 001 012 103 11

id b0

0 01 1

1 digit 2 digits 3 digits 4 digits?

id b3b2b1b0

0 00001 00012 00103 0011. .. .. .14 111015 1111

An n digit binary codecovers numbers from0 to 2n-1.

Page 7: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.2 Enumeration of binary numbersiCliker

When we enumerate binary numbers(b3b2b1b0)2 from 0 to 15, the sequence of b3

should be• A. 0101010101010101• B. 0011001100110011• C. 0000111100001111• D. 0000000011111111

7

Page 8: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.3 Addition of Binary Numbers

Given two binary numbers A & B, we derivebinary number S so that the value of S isequal to the sum of the values of A & B,i.e.(an-1…,a1a0)2+(bn-1…b1b0)2=(sn-1…s1s0)2

Caution: Overflow, i.e. the sum is beyondthe range of the representation.

8

Page 9: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.3 Addition: iClickerGiven two binary numbers

A=(an-1…,a1a0)2 and B=(bn-1…b1b0)2

what is the largest possible value of A+B?A.2n+1

B.2n+1-1C.2n+1-2D.None of the above

9

Page 10: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.3 Addition of Binary NumbersEquality of addition

(an-1…,a1a0)2+(bn-1…b1b0)2=(sn-1…s1s0)2

That is to sayan-12n-1+…+a121+a020+bn-12n-1+…+b121+b020

=(an-1+bn-1)2n-1+…+(a1+b1)21+(a0+b0)20

=sn-12n-1+…+s121+s020

10

Page 11: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.3 Addition of Binary Numbers

b2 b1 b0 Value0 0 0 00 0 1 10 1 0 20 1 1 31 0 0 41 0 1 51 1 0 61 1 1 7

8 4 2 10 0 1 10 1 0 1

8 4 2 10 0 1 10 1 1 0

+

+

Examples:

11

Page 12: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.3 Addition of Binary NumbersBiti+1 Biti Biti-1

Carryi+1 Carryi

ai ai-1

bi bi-1

Sumi Sumi-1

12

Formula for Bit i:Carryi+ai+bi= 2xCarryi+1+Sumi

Page 13: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.3 Adding 2 bits in a digit

a b Carry Sum

0 0 0 00 1 0 11 0 0 11 1 1 0

Formula:

a+b=

2xCarry + Sum

13

Page 14: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

2.3 Adding 3 bits in a digitid a b c Carry Sum

0 0 0 0 0 01 0 0 1 0 12 0 1 0 0 13 0 1 1 1 04 1 0 0 0 15 1 0 1 1 06 1 1 0 1 07 1 1 1 1 1

Formula:

a+b+c=

2xCarry + Sum

14

Page 15: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

3. Gray Code

1. Introduction2. Example3. Construction4. Comments

15

Page 16: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

3.1 Gray Code: IntroductionGray: Frank Gray patented the code in1947

A variation of binary code

The code will be used for logic operation(CSE20, CSE140)

Feature: only one bit changes for twoconsecutive numbers

16

Page 17: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

3.2 Gray Code: Example

17

id b2b1b0 g2g1g0

0 000 0001 001 0012 010 0113 011 0104 100 1105 101 1116 110 1017 111 100

id b1b0 g1g0

0 00 001 01 012 10 113 11 10

2 digits 3 digits

Note the differenceof the first and lastrows.

Page 18: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

3.2 Gray Code

18

id b2b1b0 g2g1g0

0 000 0001 001 0012 010 0113 011 0104 100 1105 101 1116 110 1017 111 100

3 digits

id b3b2b1b0 g3g2g1g0

0 0000 00001 0001 00012 0010 00113 0011 00104 0100 01105 0101 01116 0110 01017 0111 01008 1000 ?9 1001

10 101011 101112 110013 110114 111015 1111

Page 19: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

3.2 Gray Code: iClicker

19

A 4-digit Gray code (g3g2g1g0) at id=8 iswritten as (ref: previous page)A. (0101)B. (0110)C. (1100)D. None of the above

Page 20: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

3.3 Gray Code: ConstructionConstruction of n-digit Gray code from n-1digit Gray code

•Copy the n-1 digit Gray code for the top 2n-1

rows. Fill 0 at digit gn-1 in the top rows.

•Reflect and append the n-1 digit code for thebottom 2n-1 rows. Fill 1 at digit gn-1 in thebottom rows.

20

Page 21: Lecture 2: Number Systems: Binary Numbers and Gray …cseweb.ucsd.edu/classes/sp12/cse20-a/notes/lec2.pdf · CSE20 Lecture 2: Number Systems: Binary Numbers and Gray Code CK Cheng

3.4 Gray Code: Comments•There are various codes that satisfy theGray code feature.

•Gray code saves communication powerwhen the signals are continuous in nature,e.g. addresses, analog signals

•Gray code facilitates code checking whenthe signals are supposed to be continuous invalue.•For arithmetic operations, we need toconvert the values.

21


Top Related