Number Systems Tutorial

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.