1 / 75

EENG 2710 Chapter 1

EENG 2710 Chapter 1. Number Systems and Codes. Chapter 1 Homework. 1.1c, 1.2c, 1.3c, 1.4e, 1.5e, 1.6c, 1.7e, 1.8a, 1.9a, 1.10b, 1.13a. Number Systems. Binary Number System. Uses two digits, 0 and 1. Represents any number using the positional notation. Positional Notation.

shayna
Télécharger la présentation

EENG 2710 Chapter 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. EENG 2710 Chapter 1 Number Systems and Codes

  2. Chapter 1 Homework 1.1c, 1.2c, 1.3c, 1.4e, 1.5e, 1.6c, 1.7e, 1.8a, 1.9a, 1.10b, 1.13a

  3. Number Systems

  4. Binary Number System • Uses two digits, 0 and 1. • Represents any number using the positional notation.

  5. Positional Notation • The value of a digit depends on its placement within a number. • In base 10, the positional values are (starting to the left of the decimal) –1 (100), 10 (101), 100 (102), 1000 (103), etc. • In base 2, the positional values are1 (20), 2 (21), 4 (22), 8 (23), etc.

  6. Binary Weights

  7. Fractional Binary Weights

  8. Bit • Shorthand for binary digit, a logic 0 or 1. • The most significant bit (MSB) is the leftmost bit of a binary number. • The least significant bit (LSB) is the rightmost bit of a binary number.

  9. Binary Inputs • Digital circuits operate by accepting logic levels (0,1) at their input(s). • The corresponding output(s) logic level will change (0,1).

  10. Binary Inputs

  11. 4-Input Digital Circuit

  12. Base Conversions Methods • Series substitution method • Sum powers of 2 • Radix method • Repeated Division • Repeated Multiplication

  13. Series Substitution Method(Binary to Decimal)

  14. Sum Powers of 2(Decimal to Binary) • Step 1: • Determine the largest power of 2 less than or equal to the number to be converted. • Place a 1 in that positional location. • Convert 5710 to binary • 6410 5710  3210

  15. Sum Powers of 2 • Step 2: • Subtract the number found in Step 1 from the number to be converted. • 57 – 32 = 25 • For the new number, determine if the next lowest power of 2 is less than or equal to that number. • 25 – 16 = 9

  16. Sum Powers of 2 • Step 3: • If the new power of two from Step 2 is larger, place a 0 in that positional location. • If the new value is less than or equal, place a 1 in that positional location.

  17. Sum Powers of 2 • Step 4: • Repeat Steps 2 and 3 until there is nothing left to subtract. • All remaining bits are set to 0. 5710 = 1110012

  18. Radix Method (Repeated Division by 2) 4610 = 1011102

  19. Fractional Binary Numbers • Radix point: • The generalized decimal point. The dividing line between positive and negative powers for positional multipliers. • Binary point: • The radix point for binary numbers.

  20. Fractional Binary Values • The value immediately to the right of the binary point is 2–1 = 0.5. • The next value to the right is 2–2 = 0.25. • The next value to the right is 2–4 = 0.125, and so on.

  21. Series Substitution Method(Binary Fraction to Decimal Fraction)

  22. Radix Method for 0.210 to Binary (Repeated Multiplication by 2) • Step 1:Multiply the decimal fraction by 2. • Step 2: Integer part is 0 or 1 left of decimal point. 0.2 x 2 = 0.4 Integer part = 0 0.4 x 2 = 0.8 Integer part = 0 0.8 x 2 = 1.6 Integer part = 1 0.6 x 2 = 1.2 Integer part = 1 0.2 x 2 = 0.4 Integer part = 0 (stop 0011repeats) Read integer parts from top to bottom Therefore, 0.210 = 0.0011 0011 0011

  23. Hexadecimal Numbers • Base 16 number system. • Primarily used as a shorthand form of binary numbers.

  24. Counting in Hexadecimal • Values range from 0 to F with the letters A to F used to represent the values 10 to 15 respectively. • Positional multipliers are powers of 16:160 = 1, 161 = 16, 162 = 256, etc.

  25. Hexadecimal vs. Decimal Numbers

  26. Counting In Hexadecimal 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 10,11,12,13,14,15,16,17,18,19,1A,1B,1C,1D,1E,1F 20,21,22,23,24,25,26,27,28,29,2A,2B,2C,2D,2E,2F 30,31,32,33,34,35,36,37,38,39,3A,3B,3C,3D,3E,3F

  27. Decimal-to-Hexadecimal Conversion(Repeated division by 16) 3158110 = 7B5DH

  28. Hex-Decimal-Binary Table

  29. Conversion Between Hexadecimal and Binary • Each hexadecimal digit represents 4 binary bits. F D 6 9 1111 1101 0110 1001 FD69H = 11111101011010012

  30. Signed/Unsigned Binary Numbers • Signed Binary Number: • A binary number of fixed length whose sign (+/–) is represented by one bit (usually MSB) and its magnitude by the remaining bits. • Unsigned Binary Number: • A binary number of fixed length whose sign is not specified by a bit. All bits are magnitude and the sign is assumed +.

  31. Unsigned Binary Arithmetic • Sum: • Result of an Addition Operation of two (or more) binary numbers (operands). • Carry: • A digit (or bit) that is carried over to the next most significant bit during an n-Bit addition operation. • The carry bit is a 1 if the result was too large to be expressed in n bits.

  32. Basic Rules (Unsigned)

  33. Basic Subtraction • Basic Subtraction of x = a–b, with a = minuend, b = subtrahend, and x = difference or result. • Requires a Borrow Bit if a < b. • There are other forms of subtraction such as 2’s Complement Addition used by microprocessors (such as in a PC).

  34. Binary Subtraction with Borrow Examples

  35. Signed Binary Numbers • Sign Bit: • A bit (usually the MSB) that indicates whether a number is positive (= 0) or negative (= 1). • Magnitude Bits: • The bits of a signed binary number that tell how large it is in value.

  36. Signed Binary Numbers • True-Magnitude Form: • A form of signed binary whose magnitude bits are the TRUE binary form (not complements). • 1’s Complement: • A form of signed binary in which negative numbers are created by complementing all bits. • 2’s Complement: • A form of signed binary in which the negative numbers are created by complementing all the bits and adding a 1 (1’s Complement + 1).

  37. True-Magnitude Form • 5-Bit Numbers Negative Sign (S = 1) • +25 = 011001 (Note sign bit (MSB) Sign = 0) • –25 = 111001 (Same as +25 with sign = 1) • +12 = 001100 • –12 = 101100

  38. 1’s Complement Form • 8-Bit 1’s Complement Negative (S = 1) • +57 = 00111001 • –57 = 11000110 (All Bits Inverted) • +72 = 01001000 • –72 = 10110111

  39. 2’s Complement Form 57 = 0011 1001 -57 = 1100 0110 + 1 1100 0111

  40. Signed Binary Addition (8-Bit) Signed Addition Positive (S = 0) +30 = 00011110 +75 = 01001011 105 01101001

  41. Subtraction with 1’s Complement • Add the 1’s Complement and then Carry. • Uses an End around carry addition method. (80 – 65)

  42. 2’s Complement Subtraction • Add 2’s Complement to Minuend. (80 – 65) Discord Carry Bit From Results

  43. 2’s Complement Subtraction 20010 – 510 = 19510 (Use 16 bit word) 20010 = 00000000110010002 510 = 00000000000001012 -510 = 11111111111110102 = 1’s complement + 1 11111111111110112 00000000110010002 + 11111111111110112 100000000110000112 = 00000000110000112 = 19510

  44. Negative Results • If the True-Magnitude Form is used for subtraction, the results are incorrect. • If the result is from 1’s or 2’s Complement and the result is negative (S = 1), the magnitude is found by taking the complement of the result.

  45. Negative Result Example Thus, = -1510

  46. More Binary Addition

  47. More Binary Subtraction

  48. Binary Multiplication

  49. Binary Division

  50. Range of Signed Numbers • Range of Positive Numbers is 0 to 2n– 1 for a number with n magnitude bits. • Range of Negative Numbers is –1 to –2n for a number with n magnitude bits. • 8-Bit Example: 8-Bit Number Range is –2n x  +2n – 1 or –128 to +127

More Related