Error Detection & Correction Virtual Laboratory

Theoretical Background

1. Error Detection vs. Correction

• Detection: Identifies that errors have occurred (Parity, CRC)
• Correction: Identifies and fixes errors (Hamming, Convolutional)

Redundancy Ratio = (n-k)/k × 100%

2. Types of Errors

Single Bit Error: Only one bit in a data unit is changed
Burst Error: Two or more consecutive bits are changed
Random Error: Bits changed randomly throughout the data

3. Coding Techniques Overview

Block Codes

k-bit message → n-bit codeword. Memoryless encoding.

Examples: Hamming, Reed-Solomon, BCH

Convolutional Codes

Uses memory (shift registers). Current output depends on past inputs.

Used in: 3G, 4G, WiFi, Deep Space

Cyclic Codes

Subset of linear block codes with cyclic property.

Examples: CRC, BCH codes

4. Key Parameters

(n,k)

Code Parameters
n=block length, k=message

dmin

Minimum Hamming Distance

Error Correction Capability
t = ⌊(dmin-1)/2⌋

R=k/n

Code Rate

Interactive Simulations

Single & Double Parity Check

Detect single-bit errors using even/odd parity

Detection Only Simple

Data Input (7 bits)

Parity Type:

2D Parity Matrix (Optional)

2D parity can detect and correct single-bit errors by checking row and column parity simultaneously.

Hamming (7,4) Code

Corrects single-bit errors, detects double-bit errors

Correction dmin = 3

4-Bit Message

Generator Matrix G

Parity Check Equations:

p1 = d1 ⊕ d2 ⊕ d4
p2 = d1 ⊕ d3 ⊕ d4
p3 = d2 ⊕ d3 ⊕ d4

Syndrome Calculation:

s = r × H^T

Non-zero syndrome indicates error position

Cyclic Redundancy Check (CRC)

Powerful error detection using polynomial division

Detection Burst Error

Data Bits (Binary):

Generator Polynomial:

Long Division Visualization

Click "Calculate CRC" to see division steps...

CRC Properties:

• Detects all single-bit errors
• Detects all double-bit errors (if polynomial has ≥3 terms)
• Detects all odd number of errors
• Detects all burst errors ≤ degree of polynomial

Laboratory Procedure

Experiment 1: Parity Check Codes

Set the 7-bit data word using the toggle switches (0 or 1)
Select Even or Odd parity scheme
Click "Calculate Parity Bit" to generate the 8-bit codeword
Observe how the parity bit is calculated (XOR of all data bits for even parity)
Click "Inject Random Error" to simulate channel noise
Click "Verify at Receiver" to check if the error is detected
Repeat with different error patterns (single, double, triple errors)
Record in which cases the error goes undetected

Experiment 2: Hamming (7,4) Code

Input a 4-bit message (m3, m5, m6, m7 positions)
Click "Encode" to generate the 7-bit Hamming codeword
Identify the parity bits (p1, p2, p4) and their positions
Verify the parity equations manually:
- p1 covers positions 1,3,5,7
- p2 covers positions 2,3,6,7
- p4 covers positions 4,5,6,7
Click on any bit in the channel to flip it (inject single error)
Click "Detect & Correct" to observe syndrome calculation
Note the syndrome value and how it identifies the error position
Try injecting two errors and observe that correction fails (detects but miscorrects)

Experiment 3: CRC Implementation

Enter a binary data sequence (e.g., 110101)
Select a generator polynomial (start with CRC-3)
Observe the polynomial representation (e.g., x³ + x + 1)
Click "Calculate CRC" to see the modulo-2 division process
Note the remainder (CRC bits) appended to the data
Inject errors and verify detection capability
Test with burst errors of length > degree of polynomial
Compare different CRC standards (CRC-8, CRC-16, CRC-32)

Experiment 4: Convolutional Coding

Set up a 4-bit input sequence
Observe the 3-stage shift register (constraint length K=3)
Click "Encode" to generate the output using generators g1=[111], g2=[101]
For each input bit, note the state transitions
Click "Show Trellis Diagram" to visualize the state machine
Trace the path through the trellis for your input sequence
Calculate the free distance (minimum weight of non-zero codeword)

Laboratory Report Guidelines

Report Structure

Your laboratory report should follow this format:

1. Title Page

Experiment Title: Error Detection and Correction Techniques
Course Code and Name
Student Name, ID, Date
Instructor Name

2. Objectives

Understand the difference between error detection and correction
Implement and analyze parity check codes
Study Hamming code construction and decoding
Analyze CRC polynomial division and error detection capabilities
Understand convolutional encoding and trellis diagrams

3. Theory (Summary)

Briefly explain (max 1 page each):

Hamming distance and its relation to error control capability
Parity check matrix and generator matrix
Systematic vs. non-systematic codes
Hard decision vs. soft decision decoding

4. Experimental Results

Include for each experiment:

Tabulated input data, codewords, and received words
Screenshots of simulations (if applicable)
Calculated vs. simulated syndrome values
Error patterns tested and detection/correction results

Required Tables:

Table 1: Parity Check - Error detection rate vs. number of errors
Table 2: Hamming Code - All 16 possible 4-bit messages and their codewords
Table 3: CRC - Remainder values for different data patterns
Table 4: Convolutional - State transition table

5. Analysis and Discussion

Compare the overhead (redundancy) of different schemes
Analyze undetected error probability for each method
Discuss trade-offs between complexity and performance
Why can't Hamming code correct double-bit errors?
Applications: Where is each technique used in real systems?

6. Conclusion

Summarize key findings and state whether objectives were achieved.

7. References

Cite textbooks (e.g., Proakis, Sklar, Haykin) and IEEE papers.

Grading Rubric (100 points)

Theory Understanding: 20 pts

Experimental Data: 25 pts

Analysis Quality: 30 pts

Presentation/Format: 15 pts

Conclusions: 10 pts

Error Detection & Correction Laboratory