[Practical Exercise] Bit Error Rate for BPSK Modulation with LDPC Codes under AWGN Channel in MATLAB

# 1. Introduction to AWGN Channel and BPSK Modulation

The AWGN channel (Additive White Gaussian Noise channel) is a common channel model that assumes the addition of Gaussian white noise to the channel with constant power spectral density. BPSK (Binary Phase Shift Keying) modulation is a digital modulation technique that maps binary data onto two carriers with different phases.

In an AWGN channel, the BPSK modulated signal can be represented as:

s(t) = A * cos(2πf_c * t + θ)

Where A is the signal amplitude, f_c is the carrier frequency, and θ is the phase, θ∈{0, π}. When the input data is 0, θ=0; when the input data is 1, θ=π.

# 2. Theoretical Foundations of LDPC Codes

### 2.1 Structure and Characteristics of LDPC Codes

#### 2.1.1 Sparse Parity-Check Matrix

LDPC codes (Low-Density Parity-Check codes) are a type of linear block codes with a parity-check matrix that is sparse, meaning that the number of non-zero elements in the matrix is much less than the total number of elements in the matrix. A sparse parity-check matrix can be represented as:

H = [
     ***
     ***
     ***
]

Here, 1 represents a non-zero element, and 0 represents a zero element. The sparsity of the parity-check matrix gives LDPC codes good coding efficiency and decoding performance.

#### 2.1.2 Low-Density Parity-Check

The parity-check rules of LDPC codes differ from traditional parity-checks. In LDPC codes, each information bit is assigned one or more check bits. The value of the check bits is calculated through linear combinations of the information bit values using the parity-check matrix. For instance, for the above parity-check matrix, the calculation formulas for the check bits c1, c2, and c3 corresponding to the information bits x1, x2, and x3 are:

c1 = x1 + x4 + x5
c2 = x2 + x3 + x6
c3 = x3 + x4 + x7

The low-density parity-check rules of LDPC codes result in code words with high error correction capability.

### 2.2 Decoding Algorithms for LDPC Codes

There are two main types of decoding algorithms for LDPC codes: BP algorithm and MPA algorithm.

#### 2.2.1 BP Algorithm

The BP algorithm (Belief Propagation algorithm) is an iterative decoding algorithm. Its basic principle is to estimate the probability of information bits through the mutual exchange of information between information bits and check bits. The BP algorithm process is as follows:

1. Initialize the probability distribution of information bits and check bits.
2. Iteratively update the probability distribution of information bits and check bits.
3. Repeat step 2 until the convergence condition is met.

The BP algorithm has high decoding efficiency, but its performance can be affected by factors such as signal-to-noise ratio and code length.

#### 2.2.2 MPA Algorithm

The MPA algorithm (Maximum A Posteriori Probability algorithm) is a soft-dec