HisMethod-all

Structured Jupyter notebook that incorporates the steps for Wavelet Transform, Short-Time Fourier Transform (STFT), and Parametric Methods, along with the preprocessing techniques.

```python
# Import necessary libraries
import numpy as np
import matplotlib.pyplot as plt
import pywt
from scipy.signal import stft, detrend, get_window
from statsmodels.tsa.ar_model import AutoReg

# Step 1: Simulate a very low-frequency signal
fs = 100  # Sampling frequency (Hz)
t = np.linspace(0, 1, fs)  # Time vector for 1 second
f_low = 0.1  # Low frequency (Hz)
signal = np.sin(2 * np.pi * f_low * t)

# Plot the original signal
plt.figure(figsize=(12, 4))
plt.plot(t, signal, label='Original Signal')
plt.title('Original Signal')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.legend()
plt.show()

# Step 2: Preprocessing the Signal

# (a) Detrending
detrended_signal = detrend(signal)

# (b) Windowing
window = get_window('hamming', fs)
windowed_signal = detrended_signal * window

# (c) Downsampling
downsampling_factor = 2
downsampled_signal = signal[::downsampling_factor]
downsampled_t = t[::downsampling_factor]

# Plot the preprocessed signal
plt.figure(figsize=(12, 4))
plt.plot(t, signal, label='Original Signal')
plt.plot(t, detrended_signal, label='Detrended Signal')
plt.plot(t, windowed_signal, label='Windowed Signal')
plt.plot(downsampled_t, downsampled_signal, label='Downsampled Signal', linestyle='--')
plt.title('Preprocessed Signals')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.legend()
plt.show()

# Step 3: Wavelet Transform

# Continuous Wavelet Transform (CWT)
widths = np.arange(1, 128)
cwt_matrix, freqs = pywt.cwt(signal, widths, 'mexh', 1.0 / fs)

# Plot the CWT result
plt.figure(figsize=(12, 6))
plt.imshow(np.abs(cwt_matrix), extent=[0, 1, 1, 128], cmap='PRGn', aspect='auto', vmax=abs(cwt_matrix).max(), vmin=-abs(cwt_matrix).max())
plt.title('Wavelet Transform (CWT)')
plt.xlabel('Time (s)')
plt.ylabel('Scale')
plt.colorbar(label='Magnitude')
plt.show()

# Step 4: Short-Time Fourier Transform (STFT)

# Compute STFT
f, t_stft, Zxx = stft(signal, fs, nperseg=20)

# Plot STFT result
plt.figure(figsize=(12, 6))
plt.pcolormesh(t_stft, f, np.abs(Zxx), shading='gouraud')
plt.title('Short-Time Fourier Transform (STFT)')
plt.xlabel('Time (s)')
plt.ylabel('Frequency (Hz)')
plt.colorbar(label='Magnitude')
plt.show()

# Step 5: Parametric Methods (Autoregressive Model)

# Fit AR model to the detrended signal
model = AutoReg(detrended_signal, lags=5)
model_fit = model.fit()
ar_psd = model_fit.predict(start=0, end=len(detrended_signal)-1)

# Plot AR model results
plt.figure(figsize=(12, 4))
plt.plot(t, detrended_signal, label='Detrended Signal')
plt.plot(t, ar_psd, label='AR Model PSD', linestyle='--')
plt.title('Autoregressive Model PSD')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.legend()
plt.show()
```

### Explanation of the Notebook:

1. **Signal Simulation**: Creates a very low-frequency sinusoidal signal.
2. **Preprocessing the Signal**:
   - **Detrending**: Removes trends in the data.
   - **Windowing**: Applies a Hamming window to the signal.
   - **Downsampling**: Reduces the sampling rate to focus on low-frequency components.
3. **Wavelet Transform**:
   - Uses the Continuous Wavelet Transform (CWT) to analyze the signal.
4. **Short-Time Fourier Transform (STFT)**:
   - Computes the STFT to get the time-frequency representation.
5. **Parametric Methods (Autoregressive Model)**:
   - Fits an Autoregressive (AR) model to the signal and estimates the Power Spectral Density (PSD).

This notebook provides a comprehensive analysis of very low-frequency signals using various signal processing techniques and ensures the signal is preprocessed appropriately to enhance the quality of the spectral analysis.