lecture 2: number systems: binary numbers and gray...
TRANSCRIPT
CSE20
Lecture 2: Number Systems: BinaryNumbers and Gray Code
CK Cheng
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Number Systems
1. Introduction2. Binary Numbers3. Gray code4. Negative Numbers5. Residual Numbers
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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.
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2. Binary Numbers
1. Definition (radix 2)2. Enumerations (value -> index)3. Addition (logic -> hardware)
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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.
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2.2 Enumeration of Binary Numbers
id b2b1b0
0 0001 0012 0103 0114 1005 1016 1107 111
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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.
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
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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.
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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
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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
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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:
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2.3 Addition of Binary NumbersBiti+1 Biti Biti-1
Carryi+1 Carryi
ai ai-1
bi bi-1
Sumi Sumi-1
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Formula for Bit i:Carryi+ai+bi= 2xCarryi+1+Sumi
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
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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
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3. Gray Code
1. Introduction2. Example3. Construction4. Comments
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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
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3.2 Gray Code: Example
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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.
3.2 Gray Code
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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
3.2 Gray Code: iClicker
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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
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.
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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.
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