/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 One way of detecting errors is t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 15

One way of detecting errors is to transmit data as a block of \(n\) rows of \(k\) bits per row and add parity bits to each row and each column. The bit in the lower-right corner is a parity bit that checks its row and its column. Will this scheme detect all single errors? Double errors? Triple errors? Show that this scheme cannot detect some four-bit errors.

Short Answer

Expert verified
This scheme detects all single, most double, and some triple errors, but cannot detect certain four-bit errors positioned in specific rows and columns.

Step by step solution

01

Understanding Parity Bits

Parity bits are added to check for errors in data. For each row and each column, there is a parity bit that makes the total number of 1s even (even parity) or odd (odd parity). The parity bit in the lower-right corner ensures the entire row and column have the correct parity.
02

Detecting Single Errors

When a single bit is in error (either 0 is flipped to 1 or vice versa), it changes the parity of its row and its column, causing both the row and column parity bits to indicate an error. Thus, a single error will be detected as it alters two parity bits.
03

Detecting Double Errors

When two bits are in error within one row or one column, the parity of that row or column will remain correct due to a second error canceling the first. However, if the errors occur in different rows and columns, the parity bits will detect the error due to the unchanged lower-right parity bit.
04

Detecting Triple Errors

For triple errors, one scenario is that two errors are in the same row or column and one error is in another; this affects one row or column parity bit and the corner parity bit, thus detected. A different arrangement may involve three rows and columns, making errors detectable as at least one parity check is violated.
05

Failure to Detect Certain Four-Bit Errors

The scheme will fail with four-bit errors if all errors occur such that two errors are in the same row and column; this won't change the row or column parity and will also cancel out at the lower-right parity bit. For example, if errors are precisely positioned so each row and column cancels the parity imbalance with upper row and column parity unchanged, the error is not detected.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Parity Bits
Parity bits play a crucial role in error detection. They are additional bits added to data, either making the count of ones (1s) in a set even or odd, depending on whether even or odd parity is being used.
Each row and column in a dataset will have its own parity bit, plus one extra in the lower-right corner to verify both row and column parities together.
This setup helps detect if any error has occurred during data transmission. If the calculated parity does not match the expected parity, an error is flagged.
Detecting Single Errors
Single errors are quite effectively detected using parity bits. When a single data bit is erroneous, it changes the parity, flipping the calculated result for both the row and the column.
This discrepancy is picked up immediately by the parity bits, highlighting that an error has occurred.
  • Flipping a single 0 to 1 or 1 to 0 inverts the parity check outcome.
  • Both the row's and column's parity checks will reveal this inconsistency.
  • The error's location is identified by which row and column parity bits are incorrect.
Challenges with Double Errors
Detecting double errors can be tricky when both errors reside within the same row or column. Here, one error cancels the effect of the other, maintaining the parity and masking the error.
However, if two errors stretch across different rows and columns, the unchanged lower-right corner parity bit helps in sensing the discrepancy, leading to successful detection.
  • Double errors in one row/column cancel out each other's effect.
  • Errors in separate rows and columns do trigger a parity failure.
  • The corner parity bit is crucial in spotting errors across multiple rows and columns.
Limitations with Four-Bit Errors
The four-bit error detection presents a significant limitation for this scheme. When four errors are positioned such that two errors are found in the same row and the corresponding other two in the same column, everything balances out.
This creates an illusion of as if no error occurred. The row, column, and corner parity bits remain unaltered, failing to raise a flag for errors.
  • Two errors in one row and two corresponding in the same column will go undetected.
  • The parity bits do not change, leading to undetected errors.
  • Data integrity cannot be assured in such four-bit error scenarios.
These cases underscore the necessity of more advanced error-detecting mechanisms for better reliability.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Imagine a sliding window protocol using so many bits for sequence numbers that wraparound never occurs. What relations must hold among the four window edges and the window size, which is constant and the same for both the sender and the receiver?

A bit string, 0111101111101111110 , needs to be transmitted at the data link layer. What is the string actually transmitted after bit stuffing?

A bit stream 10011101 is transmitted using the standard CRC method described in the text. The generator polynomial is \(x^{3}+1\). Show the actual bit string transmitted. Suppose that the third bit from the left is inverted during transmission. Show that this error is detected at the receiver's end. Give an example of bit errors in the bit string transmitted that will not be detected by the receiver.

In the discussion of ARQ protocol in Section 3.3.3, a scenario was outlined that resulted in the receiver accepting two copies of the same frame due to a loss of acknowledgement frame. Is it possible that a receiver may accept multiple copies of the same frame when none of the frames (message or acknowledgement) are lost?

Consider a protocol that uses piggybacking, a sending window size of 4, and 400 -bit frames. This protocol is used to transfer data over a \(200 \mathrm{kbps}\) channel with a \(4 \mathrm{msec}\) one-way propagation delay. Unfortunately, the receiver has no data to send back. It needs to send its acknowledgements in separate frames. What is the maximum amount of time the receiver can wait before sending, such that the bandwidth efficiency does not drop below \(50 \%\) ?
See all solutions

Recommended explanations on Computer Science Textbooks

Cloud Services

Read Explanation

Blockchain Technology

Read Explanation

Data Structures

Read Explanation

Cybersecurity in Computer Science

Read Explanation

Fintech

Read Explanation

Big Data

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.