Number Systems Tutorial
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Questions and Answers

What is the result of 100111 / 11?

1101

Find the representation error for -0.35 using 4-bits two's complement.

0.05

What is the resolution of the code in Method 2 when it is 0.3 and M is 2?

0.15

Find the representation error for -0.25 in fixed point fractional using 5-bits fractional two's complement.

<p>0.1</p> Signup and view all the answers

Which method uses at least 13 levels and requires 4 bits?

<p>Method 3: Direct match</p> Signup and view all the answers

What is the dynamic range for the floating-point code designed with a range of [-15, 20]?

<p>-30, 30</p> Signup and view all the answers

Represent 0.210 in the floating-point code design.

<p>0001101</p> Signup and view all the answers

Calculate the representation error for 0.210 in the floating-point code design.

<p>3.125 x 10^-3</p> Signup and view all the answers

Study Notes

Tutorial 1: Number Systems

  • The lecturer is Dr. (Alex) Yu Gong.
  • The tutorial covers number systems, including binary, decimal, and hexadecimal representations.

Binary Division

  • The tutorial starts with an example of binary division: 10010111 ÷ 11.
  • The solution is shown step-by-step, with the dividend, divisor, and quotient.

Tutorial 2: Two's Complement Codes

  • The tutorial covers three methods to apply two's complement codes.
  • Method 1: Normalize to an integer range.
    • Dynamic range: [-2.4, 1.5]
    • Resolution: 0.3
    • Normalize to [-8, 5]
    • 4-bit two's complement code: 1101
    • Representation error: 0.05
  • Method 2: Fixed-point fractional representation.
    • Resolution: 0.3, choose M = 2
    • Dynamic range: [-2^3, 2^3 - 1] = [-8, 7]
    • 5-bit two's complement code: 11101
    • Representation error: 0.1
  • Method 3: Direct match.
    • Dynamic range: [-2.4, 1.5]
    • Resolution: 0.3
    • At least 13 levels are needed, which require 4 bits.
    • Quantize the values into 16 levels.
    • Representation error: 0.014

Representation of -0.35

  • Method 1: Normalize to an integer range.
    • Representation: 1101
    • Representation error: 0.05
  • Method 2: Fixed-point fractional representation.
    • Representation: 11101
    • Representation error: 0.1
  • Method 3: Direct match.
    • Representation: 1001
    • Representation error: 0.014

Comparison of the Three Methods

  • The three methods are compared in terms of dynamic range, resolution, and overhead.
  • The results are shown in a table.

Tutorial 3: Floating-Point Codes

  • The tutorial covers the design of floating-point codes.
  • The requirements are:
    • Dynamic range: [-15, 20]
    • Relative resolution: 10%
  • The design involves:
    • Sign bit: 1 bit
    • Exponent: 3 bits
    • Fractional part: 4 bits
    • Implied bit: 0
    • Exponent bias: 2

Representation of Numbers

  • The representation of several numbers is shown:
    • 0.2
    • -0.11
    • 0.0051
    • -3.11
  • The representation error is calculated for each number.

Floating-Point Representation

  • The floating-point representation is shown:
    • Sign bit: 1 bit
    • Exponent: 3 bits
    • Fractional part: 4 bits
    • Implied bit: 0
    • Exponent bias: 2
  • The value of the representation is calculated:
    • (−1)^S × 2^(E-2) × (1 + F)
  • The representation of the smallest and largest positive subnormal numbers is shown.

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Description

Solve division problems in number systems, including binary division. Practice exercises and solutions provided by Dr. Alex Yu Gong.

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