14.1 Error Control in Digital Communication Systems
3 min read•august 7, 2024
Error control in digital communication systems is crucial for ensuring reliable . This topic covers various techniques like and , which add or request retransmissions to combat errors.
Advanced codes like convolutional codes, , and LDPC codes are explored. These powerful codes, along with performance metrics like BER and SNR, help achieve near-optimal communication rates in noisy channels.
Error Control Techniques
Forward Error Correction (FEC) and Automatic Repeat Request (ARQ)
Forward Error Correction (FEC) adds redundancy to transmitted data enabling the receiver to detect and correct errors without retransmission
FEC techniques include block codes (Reed-Solomon codes) and convolutional codes
Automatic Repeat Request (ARQ) detects errors at the receiver and requests retransmission of erroneous data
ARQ protocols include Stop-and-Wait ARQ, Go-Back-N ARQ, and Selective Repeat ARQ
Stop-and-Wait ARQ sends one frame at a time and waits for an acknowledgment (ACK) before sending the next frame
Go-Back-N ARQ sends multiple frames and retransmits all frames starting from the erroneous one upon receiving a negative acknowledgment (NACK)
Hybrid ARQ and Interleaving
combines FEC and ARQ to achieve better error control performance
In Type I Hybrid ARQ, the receiver attempts to correct errors using FEC and requests retransmission if correction fails
Type II Hybrid ARQ sends additional redundancy information in retransmissions to assist error correction
rearranges the order of transmitted bits to spread out burst errors
Interleaving improves the performance of FEC by making errors appear more random and easier to correct
Examples of interleaving include block interleaving and convolutional interleaving
Error Correction Codes
Convolutional Codes and Turbo Codes
Convolutional codes generate redundancy by combining the current input bit with previous input bits using shift registers and modulo-2 adders
The encoding process of convolutional codes can be represented by a trellis diagram
Decoding of convolutional codes is performed using the Viterbi algorithm, which finds the most likely transmitted sequence based on the received sequence
Turbo codes are a class of high-performance error correction codes that achieve near-Shannon limit performance
Turbo codes consist of two or more convolutional encoders connected in parallel with an interleaver between them
The decoding of turbo codes involves iterative exchange of soft information between component decoders
Low-Density Parity-Check (LDPC) Codes
Low-Density Parity-Check (LDPC) codes are linear block codes with sparse parity-check matrices
LDPC codes can be represented by a Tanner graph, where nodes represent variable nodes (bits) and check nodes (parity-check equations)
Decoding of LDPC codes is performed using message-passing algorithms such as the sum-product algorithm or the min-sum algorithm
LDPC codes have excellent error correction performance and are widely used in modern communication systems (5G, Wi-Fi, satellite communications)
Examples of LDPC codes include regular LDPC codes and irregular LDPC codes
Performance Metrics
Bit Error Rate (BER) and Signal-to-Noise Ratio (SNR)
(BER) is the ratio of the number of bit errors to the total number of transmitted bits
BER is a key performance metric for evaluating the effectiveness of error control techniques
(SNR) is the ratio of the signal power to the noise power in a communication system
SNR is often expressed in decibels (dB) and indicates the quality of the communication channel
Higher SNR values generally correspond to lower BER and better error correction performance
The relationship between BER and SNR depends on the modulation scheme and error control technique used
Channel Capacity
is the maximum rate at which information can be reliably transmitted over a communication channel
The Shannon-Hartley theorem gives the channel capacity C as C=Blog2(1+NS), where B is the channel bandwidth, S is the signal power, and N is the noise power
Error correction codes aim to achieve reliable communication at rates close to the channel capacity
The gap between the achievable rate of a practical error correction code and the channel capacity is called the Shannon limit
Modern error correction codes (turbo codes, LDPC codes) can operate within a fraction of a decibel from the Shannon limit
Key Terms to Review (26)
Automatic Repeat Request: Automatic Repeat Request (ARQ) is an error control method used in digital communication systems that ensures reliable data transmission by automatically retransmitting erroneous or lost data packets. This technique enhances the integrity of data communication by allowing the receiver to request the sender to resend any corrupted or missing packets, thereby maintaining a robust connection in the presence of errors.
Bit error: A bit error occurs when a single bit of data is altered during transmission or storage, resulting in the loss of accuracy in the information being communicated. This alteration can happen due to various factors, including noise, interference, or hardware malfunctions. Understanding bit errors is crucial for implementing effective error control strategies in digital communication systems, as they can significantly impact the reliability and integrity of data transmission.
Bit error rate: Bit error rate (BER) is a metric that quantifies the number of bit errors in a digital transmission system, expressed as a ratio of the number of erroneous bits to the total number of transmitted bits. This measurement is critical for assessing the performance and reliability of communication systems, particularly in the presence of noise and interference. A lower BER indicates a more reliable system and is essential in designing effective error correction techniques.
Block Code: A block code is a type of error-correcting code that divides the data into fixed-size blocks and encodes each block separately to ensure reliable transmission. This method is essential in digital communication systems, where errors can occur during data transmission due to noise or interference. By using block codes, systems can detect and correct errors efficiently, improving overall data integrity and communication reliability.
Burst error: A burst error is a type of data corruption that affects a sequence of bits in a digital transmission. This error occurs when multiple adjacent bits are altered or corrupted during transmission, often due to factors like noise, interference, or signal degradation. Understanding burst errors is crucial for developing effective error control methods in digital communication systems, as they can significantly impact the integrity of the transmitted data.
Channel Capacity: Channel capacity is the maximum rate at which information can be transmitted over a communication channel without error. It reflects the inherent limits of a channel’s ability to transmit data, influenced by factors like noise, bandwidth, and signal strength. Understanding channel capacity is crucial for designing efficient communication systems that minimize errors and maximize data throughput.
Convolutional code: A convolutional code is a type of error-correcting code used in digital communication that encodes data by passing it through a series of shift registers and applying a set of mathematical operations. This coding technique helps ensure reliable data transmission by adding redundancy to the data stream, which can be used to detect and correct errors that occur during transmission. Convolutional codes are particularly important in improving the performance of communication systems, especially when combined with algorithms for decoding like the Viterbi Algorithm.
Cyclic Redundancy Check: A cyclic redundancy check (CRC) is an error-detecting code used to identify accidental changes to raw data in digital communication systems. It uses polynomial division to compute a checksum that represents the data, allowing the receiver to verify if the data has been altered during transmission. This method is highly efficient and effective for detecting common types of errors that occur in data storage and transmission.
Data transmission: Data transmission refers to the process of sending and receiving digital information over a communication medium, such as wires, optical fibers, or airwaves. This process is fundamental in digital communication systems, where the integrity and accuracy of the transmitted data are crucial. Various coding techniques are employed to ensure that data can be sent efficiently and accurately, protecting against errors that can occur during transmission.
Error Correction: Error correction is the process of detecting and correcting errors that occur during data transmission or storage. This method ensures the integrity and reliability of data by enabling systems to identify mistakes and recover the original information through various techniques.
Error detection: Error detection is the process of identifying errors in transmitted or stored data to ensure the integrity and accuracy of information. It plays a crucial role in various systems by allowing the detection of discrepancies between the sent and received data, which can be essential for maintaining reliable communication and storage.
Forward error correction: Forward error correction (FEC) is a technique used in digital communication systems that allows the receiver to detect and correct errors without needing to request retransmission of the original data. By adding redundant data to the original message before transmission, FEC ensures that even if some bits are altered during transit, the intended message can still be reconstructed accurately. This method is particularly valuable in scenarios where retransmission is costly or impractical, such as in satellite communications or streaming media.
Gilbert-Varshamov Bound: The Gilbert-Varshamov bound provides a crucial limit on the maximum number of codewords in a binary code of a certain length and minimum distance, indicating the capacity of error-correcting codes. This bound shows that, for a given length and minimum distance, it is possible to construct codes that approach this bound, thereby informing the design and assessment of error-correcting capabilities in digital communication systems.
Hamming Code: Hamming Code is a method of error detection and correction that can identify and correct single-bit errors in transmitted data. It achieves this by adding redundancy through parity bits, allowing the receiver to determine which bit may have been corrupted during transmission, making it essential in various coding techniques used to ensure reliable data communication and storage.
Hybrid ARQ: Hybrid Automatic Repeat reQuest (ARQ) is an error control method that combines error detection with retransmission of erroneous data and forward error correction. This approach optimizes the efficiency of data transmission by using both techniques to manage and correct errors in digital communication systems, ensuring more reliable delivery of information.
Information entropy: Information entropy is a measure of the uncertainty or randomness associated with a set of possible outcomes, quantifying the average amount of information produced by a stochastic source of data. It provides insight into how much information is contained in a message and plays a crucial role in error control by indicating the limits of how much information can be reliably transmitted over a communication channel.
Interleaving: Interleaving is a technique used in coding theory to rearrange the order of data symbols to improve the performance of error correction codes. By spreading out errors across multiple code words, it enhances the reliability of data transmission, making it easier to recover the original message despite the presence of noise or interference. This method is particularly effective in mitigating burst errors, which are common in communication systems.
Low-density parity-check codes: Low-density parity-check (LDPC) codes are a class of error-correcting codes that use sparse parity-check matrices to detect and correct errors in data transmission. They are particularly effective due to their ability to approach the Shannon limit of channel capacity, providing efficient error correction in digital communication systems. LDPC codes utilize an iterative decoding process that leverages the structure of their sparse matrices, which is crucial for improving performance in noisy environments.
Parity Check: A parity check is a method used to detect errors in binary data by adding an extra bit, known as a parity bit, to ensure that the total number of 1s in the data is even or odd. This simple error detection scheme allows systems to identify whether data has been altered during transmission or storage. Parity checks are fundamental in various error detection and correction processes, making them essential for maintaining data integrity in digital communication systems.
Redundancy: Redundancy in coding theory refers to the intentional inclusion of extra bits in a message to ensure that errors can be detected and corrected. This additional information provides a safety net that helps maintain the integrity of data during transmission or storage, enhancing the reliability of communication systems.
Reed-Solomon Code: Reed-Solomon codes are a type of error-correcting code that can detect and correct multiple symbol errors in data transmission and storage. These codes work by encoding data into larger blocks of symbols, allowing for the recovery of the original information even when a certain number of symbols are corrupted. This makes them particularly valuable in applications such as digital communication and data storage systems, where reliability is crucial.
Shannon's Theorem: Shannon's Theorem, formulated by Claude Shannon, defines the maximum data transmission rate over a noisy communication channel without error, known as the channel capacity. This theorem highlights the critical balance between data rate, bandwidth, and noise, showing how efficient coding techniques can approach this theoretical limit. Understanding this concept is essential for various coding techniques that aim to minimize errors and optimize data transfer in digital communication systems.
Signal-to-Noise Ratio: Signal-to-noise ratio (SNR) is a measure that compares the level of a desired signal to the level of background noise. A higher SNR indicates that the signal is clearer and more distinguishable from the noise, which is crucial for effective communication, data integrity, and overall performance in various systems.
Storage media: Storage media refers to the physical devices or materials used to store digital data and information. This can include various forms such as hard drives, solid-state drives, optical discs, and flash drives. Understanding storage media is crucial for ensuring data integrity and reliability, especially when considering the potential for errors during data retrieval and transmission in digital communication systems.
Throughput: Throughput refers to the rate at which data is successfully transmitted over a communication channel in a given amount of time. This key performance indicator reflects the efficiency of a system, affecting both error detection and correction capabilities and overall system performance. Higher throughput can lead to better error correction mechanisms and effective digital communication, ultimately enhancing the performance of coding strategies and information transmission.
Turbo Codes: Turbo codes are a class of error correction codes that use two or more convolutional codes in parallel, combined with an interleaver, to achieve near Shannon limit performance on communication channels. They revolutionized coding theory by enabling significant improvements in error correction capabilities, making them widely used in modern digital communication systems.