【Advanced】Signal Compressed Sensing and Reconstruction in MATLAB

# 1. Fundamental Principles of Signal Compressed Sensing

Signal compressed sensing is a revolutionary technology that allows the recovery of high-fidelity signals from severely under-sampled signals. The fundamental principle is that most natural signals are sparse or compressible, ***

***pressed sensing exploits this property by under-sampling signals using random measurement matrices, thereby obtaining far fewer measurements than the signal length. These measurements contain important information about the signal, and an optimized problem can be solved to recover the sparse signal from these measurements.

# 2. Signal Compressed Sensing Algorithms

**2.1 Greedy Algorithms**

Greedy algorithms are iterative algorithms that choose the optimal local solution at each step until the global optimal solution is found. In signal compressed sensing, greedy algorithms are used to iteratively select basis functions to approximate the sparse signal.

**2.1.1 Orthogonal Matching Pursuit (OMP)**

OMP is a greedy algorithm that approximates the sparse signal by iteratively selecting the basis function that most closely matches the residual signal. The algorithm steps are as follows:

Input: Sparse signal x, basis function dictionary D
Output: Sparse representation s

Initialization: Residual r = x
Initialization: Support set S = empty set
while length of S is less than k:
     Select the basis function d that best matches r
     Update support set S = S ∪ {d}
     Update residual r = r - d * <r, d> / <d, d>
end

*Parameters:*

* `x`: Sparse signal
* `D`: Basis function dictionary
* `k`: Sparsity (size of support set)

*Code Logic Analysis:*

The OMP algorithm starts with an empty support set and adds the most matching basis function one by one. At each step, the algorithm calculates the inner product of the residual signal with all basis functions in the dictionary and selects the basis function with the largest inner product. Then, the algorithm updates the support set and residual signal and repeats this process until the support set reaches the predefined sparsity.

**2.1.2 Compressed Sensing Matching Pursuit (CoSaMP)**

CoSaMP is an improved greedy algorithm that increases efficiency by selecting multiple basis functions simultaneously. The algorithm steps are as follows:

Input: Sparse signal x, basis function dictionary D
Output: Sparse representation s

Initialization: Residual r = x
Initialization: Support set S = empty set
while length of S is less than k:
     Select the 2k most matching basis functions
     Update support set S = S ∪ {2k basis functions}
     Update residual r = r - D_S * <r, D_S> / <D_S, D_S>
end

*Parameters:*

* `x`: Sparse signal
* `D`: Basis function dictionary
* `k`: Sparsity (size of support set)

*Code Logic Analysis:*

The CoSaMP algorithm is similar to the OMP algorithm, but it selects 2k most matching basis functions at each step instead of one basis function. This increases the efficiency of the algorithm, but it may lead to redundant basis functions in the support set.

**2.2 Convex Optimization Algorithms**

Convex optimization algorithms are used to solve convex optimization problems. In signal compressed sensing, convex optimization algorithms are used to find the sparse representation that minimizes the error between the reconstructed signal and the original signal.

**2.2.1 Basis Pursuit (BP)**

BP is a convex optimization algorithm that finds the sparse representation by solving the following optimization problem:

min ||x - Dx||_2^2 + lambda ||x||_1

*Parameters:*

* `x`: Sparse representation
* `D`: Basis function dictionary
* `lambda`: Regularization parameter

*Code Logic Analysis:*

The BP algorithm uses the L1 norm as a regularization term to promote sparsity. The algorithm iteratively solves the optimization problem to find the sparse representation until convergence.

**2.2.2 Regularized Least Squares (RLS)**

RLS is a convex optimization algorithm that finds the sparse representation by solving the following optimization problem:

min ||x - Dx||_2^2 + lambda ||x||_2^2

*Parameters:*

* `x`: Sparse representation
* `D`: Basis function dictionary
* `lambda`: Regularization parameter

*Code Logic Analysis:*

The RLS algorithm uses the L2 norm as a regularization term to promote smooth sparse representation. The algorithm iteratively solves the optimization problem to find the sparse representation until convergence.

# 3. Signal Reconstruction Algorithms

### 3.1 Basis Pursuit Algorithm

The basis pursuit algorithm is a greedy algorithm that reconstructs signals by iteratively selecting the most relevant basis vectors. At each step, the algorithm selects the basis vector most related to the residual signal and adds it to the reconstructed signal.

#### 3.1.1 Orthogonal Matching Pursuit (OMP)

The OMP algorithm is a basis pursuit algorithm that selects the most relevant basis vectors by orthogonalizing the residual signal. The algorithm starts with a zero vector and iteratively adds the mos