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Digital Logic & Design Lecture 03

Digital Logic & Design Lecture 03. Recap. Number System Conversion Sum-of-Weights for converting to decimal Repeated division for converting from decimal Binary Arithmetic Similar to Decimal Arithmetic Multiplying by a constant by shifting left Dividing by a constant by shifting right.

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Digital Logic & Design Lecture 03

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  1. Digital Logic & Design Lecture 03

  2. Recap • Number System Conversion • Sum-of-Weights for converting to decimal • Repeated division for converting from decimal • Binary Arithmetic • Similar to Decimal Arithmetic • Multiplying by a constant by shifting left • Dividing by a constant by shifting right

  3. Recap • Representing Numbers • Unsigned • Signed Magnitude • 2’s Complement

  4. 2’s Complement form • 1’s complement form • 2’s complement form Binary number 01101 (13) 1’s complement 10010 + 1 2’s complement 10011 (-13)

  5. Addition and Subtraction with 2’s Complement 0101 +5 0101 +5 0010 +21110 -2 0111 +7 10011 +3 1011 -5 1011 -5 1110 -20010 +2 11001 -7 1101 -3

  6. Addition and Subtraction2’s complement vs. Signed 2’s Complement Signed Binary 0101 +5 0101 +5 0010 +20010 +2 0111 +7 0111 +7 1011 -5 1101 -5 1110 -21010 -2 11001 -7 10111 -7

  7. Addition and Subtraction2’complement vs. Signed 0101 +5 0101 +5 1110 -21010 -2 10011 +3 1111 +3 1011 -5 1101 -5 0010 +20010 +2 1101 -3 1111 -3

  8. Range of Numbers • Unsigned • Positive Numbers Only (0 to 7) • 3-bit • Signed Magnitude • Positive & Negative Numbers (-7 to 7) • 4-bit • 2’s Complement • Positive & Negative Numbers (-8 to 7) • 4-bit

  9. Range & Overflow • 2 + 7 using 3-bit unsigned binary? • 9 overflow? • 2 – 7? • Can not represent -7 in unsigned binary

  10. Range and Overflow 1011 -5 1101 -3 11000 -8 1011 -5 0101 +5 1100 -40100 +4 10111 -9 1001 +9

  11. Range of Binary Numbers • Processors can handle 64-bit unsigned binary values. • Maximum unsigned decimal number is 18.446 x 1018 • How to represent larger numbers? • How to represent very small numbers? • How to represent numbers with integer part and fraction part?

  12. Floating Point Representation38 • 32-bit Floating Point Representation • ANSI/IEEE Standard 754 defines • 32-bit Single-Precision Floating Point • Sign Bit 1 • Exponent Bits 8 • Mantissa Bits 23 • 64-bit Double-Precision Floating Point

  13. Floating Point Format • 15 digit decimal number format • Sign digit 1 • Exponent digits 2 • Mantissa digits 12 • 6918.3125 = 6.9183125 x 103 • Magnitude 69183125 • Exponent 3

  14. Floating Point Format • Normalized form 0.69183125 x 104 • Magnitude 69183125 • Exponent 4 • Max Number 0.999,999,999,999 x 1099 • No negative exponent

  15. Floating Point Format • Option I • Increase exponent field to 3 digits • 1099 to10-99 • Option II • Biased 50 Exponent • add 50 1049→ 99 10-50→ 0

  16. Floating Point Format • Zero ? • Infinity (∞) ?

  17. Floating Point Format • Allow 1048→ 98 10-49→ 1 • Exponent 99 → ∞ • Exponent 0 → 0 • Decrease Bias to 49 • 1049→ 98 10-48→ 1 • Exponent 99 → ∞ • Exponent 0 → 0

  18. Single-Precision F.P. format • Representing 6918.3125 • 6918.3125 = 1101100000110.01012 • Normalized form • 1.1011000001100101 x 212 • S = 0 • E = 10001011 (127 + 12 = 139) • M = 10,110,000,011,001,010,000,000 • Hidden 1

  19. Floating Point Numbers +1.101 x 25 S=0 Exponent=10000100 Mantissa 101 0000 0000 0000 0000 0000 -1.01011 x 2-126 S=1 Exponent=00000001 Mantissa 010 1100 0000 0000 0000 0000 0 S=0 Exponent=00000000 Mantissa 000 0000 0000 0000 0000 0000 ∞ S=0 Exponent=11111111 Mantissa 000 0000 0000 0000 0000 0000

  20. Hexadecimal Number System • Base 16 • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • Representing Binary in compact form • 11011000001102 = 1B06 H

  21. Counting in Hexadecimal

  22. Counting in Hexadecimal

  23. Binary-Hexadecimal Conversion • Binary to Hexadecimal Conversion • 11010110101110010110 • 1101 0110 1011 1001 0110 • D 6 B 9 6 • Hexadecimal to Binary Conversion • FD13 • 1111 1101 0001 0011

  24. Decimal-Hexadecimal Conversion • Decimal to Hexadecimal Conversion • Indirect Method • Decimal →Binary → Hexadecimal • Repeated Division by 16

  25. Decimal-Hexadecimal Conversion • Hexadecimal to Decimal Conversion • Indirect Method • Hexadecimal →Binary → Decimal • Sum-of-Weights

  26. Hexadecimal Addition & Subtraction • Hexadecimal Addition • Carry generated • Hexadecimal Subtraction • Borrow weight 16

  27. Repeated Division by 16

  28. Sum-of-Weights CA02 (C x 163) + (A x 162) + (0 x 161) + (2 x 160) (12 x 163) + (10 x 162) + (0 x 161) + (2 x 160) (12 x 4096) + (10 x 256) + (0 x 16) + (2 x 1) 49152 + 2560 + 0 + 2 51714

  29. Hexadecimal Addition Carry 1 2AC6 6+5=11d Bh + 92B5 C+B=23d 17h BD7B A+2+1=13d Dh 2+9=11d Bh

  30. Hexadecimal Subtraction Borrow 111 92B5 21-6=15d Fh - 2AC6 26-C=14d Eh 67EF 17-A=7d 7h 8-2=6d 6h

  31. Octal Number System • Base 8 • 0, 1, 2, 3, 4, 5, 6, 7 • Representing Binary in compact form • 11011000001102 = 154068

  32. Counting in Octal

  33. Counting in Octal

  34. Binary-Octal Conversion • Binary to Octal Conversion • 11010110101110010110 • 011 010 110 101 110 010 110 • 3 2 6 5 6 2 6 • Octal to Binary Conversion • 1726 • 001 111 010 110

  35. Decimal-Octal Conversion • Decimal to Octal Conversion • Indirect Method • Decimal →Binary → Octal • Repeated Division by 8

  36. Decimal-Octal Conversion • Octal to Decimal Conversion • Indirect Method • Octal →Binary → Decimal • Sum-of-Weights

  37. Octal Addition & Subtraction • Octal Addition • Carry generated • Octal Subtraction • Borrow weight 8

  38. Repeated Division by 8

  39. Sum-of-Weights 4033 (4 x 83) + (0 x 82) + (3 x 81) + (3 x 80) (4 x 512) + (0 x 64) + (3 x 8) + (3 x 1) 2048 + 0 + 24 + 3 2075

  40. Octal Addition Carry 1 7602 2+1=3d 3O + 5771 0+7=7d 7O 15573 6+7=13d 15O 1+7+5=13d 15O

  41. Octal Subtraction Borrow 11 7602 2-1=1d 1O - 5771 8-7=1d 1O 1611 13-7=6d 6O 6-5=1d 1O

  42. Alternate Representations • BCD Code • BCD Addition • Gray Code

  43. Alternate Representations • BCD (Binary Coded Decimal) Code

  44. BCD Addition • Multi-digit BCD numbers can be added together 23 0010 0011 45 0100 0101 68 0110 1000 23 0010 0011 48 0100 1000 71 0110 1011 • 1011 is illegal BCD number

  45. BCD Addition • Add a 0110 (6) to an invalid BCD number • Carry added to the most significant BCD digit 23 0010 0011 48 0100 1000 71 0110 1011 0110 0111 0001

  46. Gray Code • Binary Code more than 1 bit change • Electromechanical applications of digital systems restrict bit change to 1 • Shaft encoders • Braking Systems • Un-Weighted Code

  47. Gray Code

  48. Gray Code Application

  49. Alphanumeric Code • Numbers, Characters, Symbols • ASCII 7-bit Code • American Standard Code for Information Interchange • 10 Numbers (0-9) • 26 Lower Case Characters (a-z) • 26 Upper Case Characters (A-Z) • Punctuation and Symbols • 32 Control Characters

  50. ASCII Code • Numbers 0 to 9 • ASCII 0110000 (30h) to 0111001 (39h) • Alphabets a to z • ASCII 1100001 (61h) to 1111010 (7Ah) • Alphabets A to Z • ASCII 1000001 (41h) to 1011010 (5Ah) • Control Characters • ASCII 0000000 (0h) to 0011111 (1Fh)

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