🔢Coding Theory Unit 10 – Convolutional Codes and Trellis Decoding

Key Concepts and Terminology

• Convolutional codes are a type of error-correcting code used in digital communication systems to protect data from noise and interference during transmission
• Trellis diagrams visually represent the encoding process of convolutional codes, showing the possible state transitions and outputs
• Constraint length ($K$) defines the number of memory elements in the encoder, influencing the complexity and error-correcting capability of the code
• Code rate ($R$) is the ratio of input bits to output bits, determining the redundancy and efficiency of the code
  • Common code rates include 1/2, 1/3, and 2/3
• Free distance ($d_{free}$) measures the minimum Hamming distance between any two distinct codewords, indicating the error-correcting strength of the code
• Catastrophic codes are convolutional codes that can cause an infinite number of output errors for a finite number of input errors and should be avoided
• Puncturing removes selected bits from the encoded sequence to increase the code rate without changing the encoder structure

Fundamentals of Convolutional 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 is described by a generator matrix $G(D)$, which consists of generator polynomials that define the connections between the shift registers and adders
• The memory of the encoder is determined by the constraint length, which is the maximum number of previous input bits that influence the current output
• Convolutional codes are classified as systematic or non-systematic, depending on whether the input bits appear directly in the output sequence or not
• The Viterbi algorithm is the most widely used decoding method for convolutional codes, which finds the most likely transmitted sequence based on the received sequence
• Convolutional codes have a continuous encoding process, where the encoder state depends on the previous input bits, unlike block codes that encode fixed-size blocks independently

Encoding Process and Structures

• The encoding process of convolutional codes involves passing the input sequence through a linear finite-state shift register
• The shift register consists of $K-1$ memory elements, where each element stores one input bit
• The output sequence is generated by combining the current input bit and the contents of the memory elements using modulo-2 adders
  • The connections between the shift register and adders are defined by the generator polynomials
• The encoder can be represented by a state diagram, which shows the possible states and the corresponding output for each input bit
• The state of the encoder is determined by the contents of the memory elements, and the number of states is $2^{K-1}$
• Recursive Systematic Convolutional (RSC) codes are a special type of convolutional code where the input bits are included in the output sequence, and the encoder has a feedback structure
• Puncturing is a technique used to increase the code rate by removing selected bits from the encoded sequence according to a puncturing pattern

Trellis Representation

• Trellis diagrams are a graphical representation of the encoding process, showing the possible state transitions and outputs over time
• The trellis consists of nodes, which represent the encoder states, and branches, which represent the state transitions and the corresponding output bits
• Each branch is labeled with the input bit and the output bits associated with that transition
• The trellis structure depends on the constraint length and the generator polynomials of the convolutional code
• The trellis diagram helps visualize the decoding process, as the decoder searches for the most likely path through the trellis based on the received sequence
• The Viterbi algorithm uses the trellis representation to perform maximum-likelihood decoding, by calculating the path metrics and selecting the path with the minimum distance from the received sequence
• The trellis can be terminated by appending $K-1$ zeros to the input sequence, ensuring that the encoder returns to the all-zero state

Decoding Algorithms

• The Viterbi algorithm is the most widely used decoding method for convolutional codes, which performs maximum-likelihood decoding on the trellis
• The algorithm calculates the path metrics for each state at each time step, which represent the cumulative distance between the received sequence and the possible transmitted sequences
• The path metrics are updated recursively using the branch metrics, which measure the distance between the received bits and the expected output bits for each branch
• The algorithm keeps track of the surviving paths at each state, which are the paths with the minimum path metric leading to that state
• At each time step, the algorithm discards the less likely paths and retains only the surviving paths, reducing the complexity of the decoding process
• The traceback operation is performed to determine the most likely transmitted sequence, by following the surviving paths backward through the trellis
• The decoding complexity of the Viterbi algorithm grows exponentially with the constraint length, making it impractical for codes with large memory
• Other decoding algorithms for convolutional codes include the Fano algorithm and the stack algorithm, which offer trade-offs between performance and complexity

Performance Analysis

• The performance of convolutional codes is typically evaluated using the bit error rate (BER) or the frame error rate (FER) as a function of the signal-to-noise ratio (SNR)
• The free distance of the code determines the asymptotic error-correcting capability, as it represents the minimum number of bit errors that can be corrected
• The error bounds, such as the union bound, provide upper limits on the error probability of the code based on the weight distribution of the codewords
• The weight distribution can be calculated using the transfer function of the convolutional code, which represents the input-output relationship of the encoder
• The performance of convolutional codes can be improved by using techniques such as puncturing, which increases the code rate without changing the encoder structure
• Soft-decision decoding, which uses the reliability information of the received bits, can provide better performance than hard-decision decoding
• The performance of convolutional codes is often compared to that of other error-correcting codes, such as block codes and turbo codes, under similar channel conditions

Applications and Real-World Examples

• Convolutional codes are widely used in digital communication systems, such as wireless networks, satellite communications, and deep-space communications
• In mobile communication standards, such as GSM and CDMA, convolutional codes are employed to protect the voice and data transmissions from channel errors
• Deep-space missions, such as the Voyager and Cassini-Huygens missions, rely on convolutional codes to ensure reliable communication between the spacecraft and Earth
• Convolutional codes are used in conjunction with other error-correcting techniques, such as Reed-Solomon codes and interleaving, to provide multi-layer protection against different types of errors
• In digital video broadcasting (DVB) systems, convolutional codes are applied to the video and audio streams to maintain the quality of the transmitted content
• Convolutional codes are also used in data storage systems, such as magnetic recording and flash memories, to improve the reliability and integrity of the stored data

Advanced Topics and Future Directions

• Turbo codes, which consist of two or more convolutional codes connected in parallel with interleavers, offer near-optimal performance and have become a popular choice for modern communication systems
• Low-density parity-check (LDPC) codes, which are based on sparse parity-check matrices, have emerged as an alternative to convolutional codes due to their excellent performance and parallelizable decoding algorithms
• Polar codes, invented by Erdal Arıkan, are a new class of error-correcting codes that achieve the capacity of binary-input symmetric memoryless channels and have gained significant attention in recent years
• Space-time convolutional codes are designed for multi-antenna systems, exploiting the spatial diversity to improve the reliability and throughput of wireless communications
• Quantum convolutional codes are being developed to protect quantum information from errors and decoherence, enabling reliable quantum communication and computation
• The design of convolutional codes for specific applications, such as ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC), is an active area of research
• The integration of convolutional codes with other technologies, such as compressed sensing and machine learning, is being explored to develop more efficient and adaptive error-correcting schemes