【Advanced】Channel Coding and Decoding in MATLAB: Implementing Convolutional Codes and the Viterbi Algorithm

# 2.1 Fundamental Concepts of Channel Coding

Channel coding is the process of transforming an information bit sequence into a redundant bit sequence, with the aim of reliably transmitting information over a noisy channel.

### 2.1.1 Channel Model and Channel Capacity

A channel model describes the characteristics of data transmission over a channel, including factors such as noise, fading, and interference. Channel capacity is the maximum information rate that a channel can transmit under a given signal-to-noise ratio, measured in bits per second per Hertz.

### 2.1.2 Coding Gain and Coding Efficiency

Coding gain is the increase in signal-to-noise ratio after channel coding compared to before. Coding efficiency is the ratio of the number of redundant bits to the number of information bits after coding, measuring the redundancy of the coding.

# 2. Fundamentals of Channel Coding Theory

### 2.1 Fundamental Concepts of Channel Coding

#### 2.1.1 Channel Model and Channel Capacity

A channel model descr***mon channel models include:

- **Additive White Gaussian Noise (AWGN) Channel:** Channel noise consists of independent and identically distributed Gaussian noise.
- **Rayleigh Fading Channel:** The signal undergoes multipath effects during transmission, causing random changes in signal amplitude and phase.
- **Multipath Fading Channel:** The signal is affected by multiple paths during transmission, resulting in multiple time delays and phase differences when the signal reaches the receiver.

Channel capacity refers to the maximum reliable information rate that a channel can transmit under given channel model and signal-to-noise ratio (SNR) conditions. Channel capacity is given by Shannon's Theorem:

C = B * log2(1 + SNR)

Where:

- C: Channel Capacity (bits/second)
- B: Channel Bandwidth (Hertz)
- SNR: Signal-to-Noise Ratio

#### 2.1.2 Coding Gain and Coding Efficiency

Coding gain refers to the reduction in bit error rate (BER) that can be achieved by the coded signal under the same SNR conditions compared to the uncoded signal. Coding efficiency measures the ability of a coding scheme to achieve coding gain while adding redundant information.

The relationship between coding gain and coding efficiency is as follows:

Coding Gain = 10 * log10(BER_uncoded / BER_coded)
Coding Efficiency = (Number of Information Bits / Number of Bits after Coding) * 100%

### 2.2 Linear Block Codes

A linear block code is a type of channel coding with a fixed block length and parity bit count.

#### 2.2.1 Generator Matrix and Parity-Check Matrix of Linear Block Codes

The encoding process of a linear block code can be represented by the generator matrix **G**, where each row represents a codeword. The parity-check matrix **H** is the orthogonal matrix of the generator matrix, used for error detection.

**Generator Matrix:**

G = [I | P]

Where:

- I: Identity matrix, representing information bits
- P: Parity-check matrix, generated by the check polynomial

**Parity-Check Matrix:**

H = [P^T | I]

#### 2.2.2 Encoding and Decoding Algorithms for Linear Block Codes

**Encoding Algorithm:**

Codeword = Information Bits * G

**Decoding Algorithm:**

Syndrome = Codeword * H^T
If Syndrome = 0, then the codeword is error-free
Otherwise, use error correction algorithm (such as BCH algorithm) to correct the error

### 2.3 Convolutional Codes

Convolutional codes are a type of channel coding with an infinite block length.

#### 2.3.1 Generator Polynomials and Trellis Diagram for Convolutional Codes

The encoding process of convolutional codes is represented by generator polynomials **g(D)**. A **trellis diagram** is a tool for graphically representing the state transitions and output sequences of convolutional codes.

**Generator Polynomials:**

g(D) = g0 + g1 * D + g2 * D^2 + ... + gn * D^n

Where:

- g0, g1, ..., gn: Coefficients of the generator polynomial

**Trellis Diagram:**

A trellis diagram consists of nodes and branches, where nodes represent the states of the encoder, and branches represent state transitions and output sequences caused by input bits.

#### 2.3.2 Encoding and Viterbi Decoding Algorithms for Convolutional Codes

**Encoding Algorithm:**

Output Sequence = Input Sequence * g(D)

**Viterbi Decoding Algorithm:**

The Viterbi decoding algorithm is a maximum likelihood decoding algorithm that searches the trellis diagram to find the most probable path, thereby estimating the input sequence.

1. Initialize the trellis diagram
2. For each input bit
    - Calculate the metric value for each node
    - Update the trellis diagram, selecting the path with the smallest metric value
3. Output the most probable path, which is the decoded input sequence

# 3. Implementation of Channel Coding in MATLAB

### 3.1 Encoding and Decoding of Linear Block Codes

#### 3.1.1 Using convmtx() and gfdeconv() Functions

**The convmtx() function** is used to generate the generator matrix of linear block codes. The syntax is as follows:

G = convmtx(g, n);

Where:

* `g`: Generator polynomial, a row vector of length `n`.
* `n`: Codeword length.

**The gfdeconv() function** is used to decode linear block codes. The syntax is as follows:

decoded_data = gfdeconv(received_data, h, p);

Where:

* `received_data`: Received codeword, a row vector of length `n`.
* `h`: Check polynomial, a row vector of length `n`.
* `p`: Length of the codeword.

**Example:**

Consider a (7, 4) linear block code, where the generator