MATLAB Curve Denoising: Removing Impurities and Extracting Useful Signals

# 1. Overview of MATLAB Curve Denoising

MATLAB curve denoising is a technique that utilizes MATLAB software to remove noise from data curves. Noise refers to unnecessary interference superimposed on useful signals, which can affect the accuracy and readability of the signal. MATLAB curve denoising can effectively remove noise from data curves by applying various algorithms, thereby improving signal quality.

Curve denoising has a wide range of applications in scientific research, engineering applications, and data analysis. For example, in the biomedical field, curve denoising can be used to remove noise from electrocardiograms and electroencephalograms to improve the accuracy of diagnosis. In industrial data analysis, curve denoising can be used to remove noise from sensor data to enhance the reliability and credibility of the data.

# 2. Theoretical Basis of MATLAB Curve Denoising

### 2.1 Noise Models and Denoising Principles

#### 2.1.1 Types and Characteristics of Noise

Noise is an unwanted signal superimposed on a useful signal that can affect the quality and reliability of the signal. According to the statistical characteristics of the noise, it can be classified into several types:

- **Additive White Noise:** Constant amplitude, flat spectrum, and uncorrelated with the signal.
- **Additive Gaussian Noise:** Amplitude follows a normal distribution, and is uncorrelated with the signal.
- **Multiplicative Noise:** Amplitude is proportional to the signal, and the spectrum is correlated with the signal.
- **Impulse Noise:** Large amplitude, short duration, and randomly occurring.

#### 2.1.2 Classification of Denoising Algorithms

Denoising algorithms can be classified into two categories based on their processing methods:

- **Time-Domain Denoising:** Directly processes the time-domain data of the signal, such as moving average filtering, median filtering, etc.
- **Frequency-Domain Denoising:** Transforms the signal into the frequency domain, filters the noise, and then transforms it back to the time domain, such as Fourier transform, filter design, etc.

### 2.2 Time-Domain and Frequency-Domain Denoising Methods

#### 2.2.1 Time-Domain Denoising: Moving Average and Median Filtering

- **Moving Average Filtering:** Averages each point of the signal to eliminate random noise.
- **Median Filtering:** Sorts each point of the signal and takes the middle value as the output to eliminate impulse noise.

#### 2.2.2 Frequency-Domain Denoising: Fourier Transform and Filter Design

- **Fourier Transform:** Transforms the signal from the time domain to the frequency domain, separating noise from the signal.
- **Filter Design:** Designs a filter to filter out the noise frequency band and then transforms the signal back to the time domain.

**Code Example:**

% Fourier Transform Denoising
Fs = 1000;
t = 0:1/Fs:1;
signal = sin(2*pi*100*t) + randn(size(t));

% Compute Fourier Transform
X = fft(signal);

% Design Filter
order = 10;
cutoff_freq = 200;
[b, a] = butter(order, cutoff_freq/(Fs/2));

% Filtering
Y = filter(b, a, X);

% Inverse Fourier Transform
denoised_signal = ifft(Y);

% Plotting Comparison
figure;
plot(t, signal, 'r');
hold on;
plot(t, denoised_signal, 'b');
legend('Original Signal', 'Denoised Signal');
title('Fourier Transform Denoising');

**Logical Analysis:**

- The Fourier Transform decomposes the signal into sine and cosine components.
- Filter design filters out the noise components based on the cutoff frequency.
- The Inverse Fourier Transform converts the filtered frequency domain signal back into the time domain signal.

**Parameter Explanation:**

- `Fs`: Sampling Frequency
- `order`: Filter Order
- `cutoff_freq`: Cutoff Frequency

# 3.1 Selection of Denoising Algorithms and Parameter Settings

#### 3.1.1 Selection of Denoising Algorithms f***

***mon types of noise include:

- **Additive Gaussian White Noise:** Random noise with constant power spectral density, usually treated with time-domain or frequency-domain filtering methods.
- **Multiplicative Noise:** Noise whose amplitude changes over time, usually treated with wavelet transforms or deep learning methods.
- **Impulse Noise:** Noise in