Lecture 11: Carom Maps and Simplifying Functions

Introduction

• Continuation of carom maps discussion
• Learn to use carom maps to simplify functions
• Provide a set of guidelines or "recipe"

Guidelines for Simplifying Functions

1. Identify Rectangular Groupings of One Cells

   • Groupings must contain a number of one cells that is a power of two
   • Start with largest groupings, then proceed to smaller ones
   • Never include a grouping entirely contained in another
   • Grouping = Prime implicant

2. Identify Essential One Cells

   • Essential one cell: contained in only one prime implicant

3. Identify Essential Prime Implicants

   • Essential Prime Implicant: contains an essential one cell

4. Find All Minimal Sums

   • Each minimal sum must contain all essential prime implicants
   • Add any other prime implicants as needed

Example Problem 5.3A

• Objective: Find all minimal sums of products for function F1 of ABC = m0 + m2 + m5 + m6
• Approach:
  • Write function in algebraic notation
  • Use caral map for compact representation

Steps in Example

• Step 1: Identify Prime Implicants
  • No groupings of 8 or 4 one cells
  • Groupings of 2 one cells:
    • Horizontal and wrapped edges
• Step 2: Name Prime Implicants
  • Look for constant variables
  • Names: BC', A'C', A'B'C
• Step 3: Identify Essential One Cells
  • Essential ones found in the map
• Step 4: Minimal Sums
  • Include all essential prime implicants
  • Verify by checking accounted cells

Practical Application

• Simplification leads to efficient hardware design
• Reduces number of gates and inputs needed

Example Problem 5.3B

• Objective: Find all minimal sums for F2 of DEF = sum of m0, m1, m2, m4

Steps in Example

• Step 1: Prime Implicants
  • 8 cells total, no groupings of 8 or 4
  • Groupings of 2 one cells identified
• Step 2: Name Prime Implicants
  • Names: D'F', E'F', D'E'
• Step 3: Essential One Cells
  • Check each one cell against prime implicants
• Step 4: Minimal Sums
  • Include all essential prime implicants
  • Verify coverage of all cells

Example Problem 5.3C

• Objective: Simplify F3 of RST = RT' + RP'S' + RP'S

Steps in Example

• Step 1: Prime Implicants
  • Use caral map
  • Groupings of 4 one cells identified
• Step 2: Name Prime Implicants
  • Constant variable analysis

Conclusion

• Carom maps are a powerful tool for function simplification
• Importance in both theoretical understanding and practical hardware design
• Practice with examples to master the technique