lecturer 吳安宇教授access.ee.ntu.edu.tw/course/logic_design_95first... · graduate institute...
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ACCESS IC LAB
Graduate Institute of Electronics Engineering, NTU
CH1 Number Systems and ConversionCH1 Number Systems and Conversion
Lecturer:吳安宇 教授
Date:2006/09/22
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.2
OutlineOutlineDigital Systems and Switching CircuitsNumber Systems and ConversionBinary ArithmeticRepresentation of Negative NumbersAddition of 2’s Complement NumbersAddition of 1’s Complement NumbersBinary Codes
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.3
PurposePurposeDesign a switching network (Logic Function)
:Binary number{ }1,0∈iX { }1,0∈iZ
Input OutputSwitching
Circuits
……
……
X1X2
Xm
Z1Z2
Zm
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.4
OutlineOutlineDigital Systems and Switching CircuitsNumber Systems and ConversionBinary ArithmeticRepresentation of Negative NumbersAddition of 2’s Complement NumbersAddition of 1’s Complement NumbersBinary Codes
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.5
Number System and ConversionNumber System and ConversionDecimal (Base 10) Number (separated by decimal point):
953.7810(210.-1-2)
= 9*102 + 5*101 + 3*100 + 7*10-1 +8*10-2
Binary (Base 2) Number (separated by binary point):
1011.112(3210.-1-2)
= 1*23 + 0*22 + 1*21 + 1*20 + 1*2-1 + 1*2-2
= 8 + 0 +2 +1+1/2 +1/4= (11.75)10
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.6
Generalized Representation of an Generalized Representation of an Positive integerPositive integer NN with with Base (Radix) RBase (Radix) R::
Name Decimal Binary Octal Hexadecimal
Radix 10 2 8 16
Digits 0,1,2,3,4,5,6,7,8,9 0,1 0,1,2,3,
4,5,6,7
0,1,2,3,4,5,6,7,8,9,A,B,
C,D,E,F
Firstseventeenpositiveintegers
012345678910111213141516
0110111001011101111000100110101011110011011110111110000
01234567101112131415161720
0123456789ABCDEF10
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.7
Generalized Representation of an Positive Generalized Representation of an Positive integer N with Base (Radix) Rinteger N with Base (Radix) R::
N= (a4 a3 a2 a1 a0 . a-1 a-2 a-3)R= a4*R4 + a3*R3 + a2*R2 + a1*R1 + a0*R0
+ a-1*R-1 + a-2*R-2 + a-3*R-3
EX:R=8 , Digits = { 0,1,2,3,4,5,6,7 }(147.3)8 = 1*82 + 4*81 + 7*80 + 3*8-1
= (103.375)10
EX:R=16, Digits = { 0,1,2,...,A,B,C,D,E,F }(A2F)16 = 10*162 + 2*161 + F*160
= (2607)10
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.8
Integer Conversion Using Integer Conversion Using ““Division MethodDivision Method”” (1/2)(1/2)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.9
Integer Conversion Using Integer Conversion Using ““Division MethodDivision Method”” (2/2)(2/2)
EX:Convert (53)10 to Binary no.
MSB
(53)10 = (110101)2
MSB
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.10
Conversion of A Decimal Fraction Using Conversion of A Decimal Fraction Using ““Successive MultiplicationSuccessive Multiplication”” (1/3)(1/3)
Fi:Fraction Number
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.11
Conversion of A Decimal Fraction Using Conversion of A Decimal Fraction Using ““Successive MultiplicationSuccessive Multiplication”” (2/3)(2/3)
EX:Convert (0.625)10 to Binary number
(0.625)10 = (0.101)2
MSB MSB
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.12
EX:Convert (0.7)10 to Binary number
Repeated Process 0.8, 1.6, 1.2, 0.4, ….
(0.7)10 = 0.1 0110 0110 0110 …… (Base2)
No exact conversion !!!
Conversion of A Decimal Fraction Using Conversion of A Decimal Fraction Using ““Successive MultiplicationSuccessive Multiplication”” (3/3)(3/3)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.13
Conversion (1/2)Conversion (1/2)EX:Convert (231.3)4 to Base 7 number(231.3)4 = 2*42 + 3*41 + 1*40 + 3*4-1 = (45.75)10
Integer Fraction
(45.75)10= (63.51 51 51 ……)2
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.14
Conversion (2/2)Conversion (2/2)Conversion from Binary (Base 2)
to Octal (Base 8)to Hexadecimal (Base 16)
(11010111110.0111 )2 = (3276.34)8
(11010111110.0011)2 = (6BE.3)16
3 2 7 6 3 4(補0)
6 B E 3Binary point ( R=2 )
Hexadecimal point
decimal point (R=10)
00
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.15
OutlineOutlineDigital Systems and Switching CircuitsNumber Systems and ConversionBinary ArithmeticRepresentation of Negative NumbersAddition of 2’s Complement NumbersAddition of 1’s Complement NumbersBinary Codes
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.16
Binary Arithmetic Binary Arithmetic ---- AdditionAdditionAddition
0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 10 (sum 0 & carry 1)
EX:1111 (Carry)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.17
Binary Arithmetic Binary Arithmetic ---- SubtractionSubtractionSubtraction
0 – 0 = 01 – 0 = 11 – 1 = 00 – 1 = 1 (with borrow 1 from next column)
EX:
1111 (Borrow)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.18
Binary Arithmetic Binary Arithmetic ---- MultiplcationMultiplcationMultiplication
0 * 0 = 00 * 1 = 01 * 0 = 01 * 1 = 1
EX: (13)10
(11)10
copy of multiplicand if “1”
multiplicandmultiplier
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.19
Binary Arithmetic Binary Arithmetic ---- DivisionDivisionDivision
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.20
OutlineOutlineDigital Systems and Switching CircuitsNumber Systems and ConversionBinary ArithmeticRepresentation of Negative NumbersAddition of 2’s Complement NumbersAddition of 1’s Complement NumbersBinary Codes
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.21
Signed Number RepresentationSigned Number Representation
S Magnitude
signed bit
bit bit bit bit bit bit bit bit
A computer “word”, wordlength n = 8 bits(other popular n = 16 bits, 32 bits)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.22
Singed Magnitude NumbersSinged Magnitude NumbersN = (an-1 an-2 ……a1 a0)r
N = ( s, an-2 ……a1 a0)2sm
EX:
N = -(13)10 = -(0,1101)2 = (1,1101)2sm
±
s = 0 if N 0
s = 1 if N 0≥≤
Signed magnitude
(2sm = Binary Singed Magnitude)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.23
Radix ComplementRadix ComplementDefinition: The “radix complement [N]” of a number (N)r is defined as:
N* (notation in textbook) = [N]r = rn – (N)r
where n is the number of digits (wordlength) in (N)r
The largest positive number (positive full scale) = rn-1 – 1The most negative number (negative full scale) = - rn-1
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.24
22’’s Complements ComplementN* = [N]2 = 2n – (N)2
EX:2’s complement of (N)2 = (01100101)2
[N]2 = [01100101]2= 28 – (01100101)2= (100000000)2 – (01100101)2= (10011011)2
EX:show that (N)2 + [N]2 = 0011001011001101100000000
(carry)
+1 [N]2 = - (N)2
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.25
22’’s Complements ComplementEX:check (N)2 = [ [N]2 ]2 (by yourself)
EX:2’s complement of (N)2 = (10110)2 for n=8
[N]2 = 28 – (10110)2= (100000000)2 – (00010110)2= (11101010)2
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.26
22’’s Complements ComplementConvert (N)2 to [N]2:
Method 1
N = 0 1 1 0 0 1 0 1
[N]2 = 1 0 0 1 1 0 1 1
N = 1 1 0 1 0 1 0 0
[N]2 = 0 0 1 0 1 1 0 0
First nonzero digit
First nonzero digit
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.27
22’’s Complements ComplementConvert (N)2 to [N]2:
Method 2
N = 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 complement the bits
1 add 11 0 0 1 1 0 1 1
⎩⎨⎧
→→
→1001
,kk aa Flip then Add 1
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.28
22’’s Complements ComplementSigned Decimal Sign Magnitude
BinaryTwo’s Complement
SystemOne’s Complement
System
+15+14+13+12+11+10+9+8+7+6+5+4+3+2+10
0,11110,11100,11010,11000,10110,10100,10010,10000,01110,01100,01010,01000,00110,00100,00010,0000
(1,0000)
0,11110,11100,11010,11000,10110,10100,10010,10000,01110,01100,01010,01000,00110,00100,00010,0000
0,11110,11100,11010,11000,10110,10100,10010,10000,01110,01100,01010,01000,00110,00100,00010,0000
(1,1111)
for n = 5
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.29
22’’s Complements ComplementSigned Decimal Sign Magnitude
BinaryTwo’s Complement
SystemOne’s Complement
System
-1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16
1,00011,00101,00111,01001,01011,01101,01111,10001,10011,10101,10111,11001,11011,11101,1111--------
1,11111,11101,11011,11001,10111,10101,10011,10001,01111,01101,01011,01001,00111,00101,00011,0000
1,11101,11011,11001,10111,10101,10011,10001,01111,01101,01011,01001,00111,00101,00011,0000--------
for n = 5
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.30
22’’s Complements ComplementEX:2’s complement of –(13)10 for n = 8
(13)10 = (1011)2 = (00001101)2-(00001101)2 = [00001101]2 = (11110011)2
EX:(n=8) Determine the decimal no. of N=(1,111,1010)2
1111010 (?) (-6)0000110 (6)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.31
Radix Complement ArithmeticRadix Complement ArithmeticEX:compute (9)10 + (5)10 for 5-bit 2’s complement
0 1 0 0 1 (+9)+ 0 0 1 0 1 (+5)
0 1 1 1 0 (+14)EX:compute (12)10 + (7)10
0 1 1 0 0 (+12)+ 0 0 1 1 1 (+7)
1 0 0 1 1 (-13)EX:compute (12)10 – (5)10 = (12) + (-5)
0 1 1 0 0 (+12)+ 1 1 0 1 1 (2’s complement of (5)2)1 0 0 1 1 1 (+7)
discard the carry
Add two positive no. and obtain a negative no.(overflow occurs!)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.32
Radix Complement ArithmeticRadix Complement ArithmeticEX:(-9) – (5) = (-9) + (-5)
9 = 0 1 0 0 1 -9 = 1 0 1 1 15 = 0 0 1 0 1 -5 = 1 1 0 1 1
1 0 1 1 1 (-9)+ 1 1 0 1 1 (-5)1 1 0 0 1 0 (-14)
discard (why?)EX:(-12) – (5) = (-12) + (-5)
1 0 1 0 0 (-12)+ 1 1 0 1 1 (-5)1 0 1 1 1 1 (+15) (overflow occurs)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.33
11’’s & 2s & 2’’s Complements Complement2’s complement is the main streamCheck SIGN for the overflow!
(+) + (+) (-)
(-) + (-) (+)overflow!!
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台灣大學 吳安宇 教授 pp.34
Overflow ConditionOverflow Condition
A B A+B A-B
+ + - ˇ x
+ - x - ˇ
- + x +ˇ
- - + ˇ x
ˇ:overflowx :no overflow
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.35
Diminished Radix ComplementDiminished Radix Complement1’s complement
EX: 1 0 1 1 0 1 0 0 (N)2
0 1 0 0 1 0 1 1 1’s complement of (N)2
⎩⎨⎧
→→
→1001
,kk aa
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.36
Addition of 1Addition of 1’’s Complement Numberss Complement Numbers“End-around carry” :
Instead of discarding the last carry (as in 2’s complement), it is added to the n-bit sum in the position furthest to the right.
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.37
Addition of 1Addition of 1’’s Complement Numberss Complement NumbersAddition of positive & negative numbers(a)
+5 0101-6 1001-1 1110 (correct)
(b)-5 1010+6 0110+1 1 0000
1 (end-around carry)0001 (correct, no overflow)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.38
Addition of 1Addition of 1’’s Complement Numberss Complement NumbersAdding two negative numbers(a)
-3 1100-4 1011-7 1 0111
1 (end-around carry)1000 (-7) (correct, no overflow)
(b) -5 1010-6 1001
-11 1 00111 (end-around carry)
0100 (wrong, overflow!!)
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.39
Addition of 1Addition of 1’’s Complement Numberss Complement NumbersEX:Addition for a word-length of 8(a) (-11) + (-20) in 1’s complement
+11 = 00001011 (-11) = 11110100+20 = 00010100 (-20) = 11101011
(1)11011111 1
(+31) 00011111 11100000 (-31)
(b) (-8) + (+19) in 2’s complement11111000 (-8)00010011 (+19)
(1)00001011 (+11)discard the last carry
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.40
OutlineOutlineDigital Systems and Switching CircuitsNumber Systems and ConversionBinary ArithmeticRepresentation of Negative NumbersAddition of 2’s Complement NumbersAddition of 1’s Complement NumbersBinary Codes
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.41
Binary CodesBinary CodesBCD (Binary Coded Decimal) codesEX: 1 9 8 9
0001 1001 1000 1001
0: 00001: 00012: 00103: 00114: 0100
5: 01016: 01107: 01118: 10009: 1001
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台灣大學 吳安宇 教授 pp.42
Binary CodesBinary CodesASCII codeskeyboard computer
Character Binary Code Hexadecimal Code
Digital
1000100110100111001111101001111010011000011101100
4469676974616C
Encode the word Digital in ASCII code, representing each character by two hexadecimal digits
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Graduate Institute of Electronics Engineering, NTU
台灣大學 吳安宇 教授 pp.43
ASCII Code (Table 1ASCII Code (Table 1--3 on p.22)3 on p.22)
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台灣大學 吳安宇 教授 pp.44
Binary Codes for Decimal DigitsBinary Codes for Decimal Digits