Multipath PCA Signal Simulator (Denoising)
Transmitted signal type:
PCA Workflow, Math, and How to Use This Simulator
This simulator models a transmitted signal reaching a receiver through multiple paths, each with random fading and delay. PCA is applied at the receiver to separate the underlying signal components.
Workflow:
- Generate the transmitted signal (sine, noisy sine, or random).
- Simulate multipath propagation with random fading and delay for each path.
- Add AWGN noise based on the chosen SNR.
- Apply PCA to the received signals to compute principal components.
- Use the slider to select the number of top principal components for reconstruction.
- Visualize original, received, eigenvalues, and reconstructed signals.
Math Behind PCA:
\[
\text{Step 1: Mean-Centering: } X_c = X - \bar{X}, \quad \bar{X} = \frac{1}{N} \sum_{i=1}^N X_i
\]
\[
\text{Step 2: Covariance Matrix: } C = \frac{1}{N-1} X_c^T X_c
\]
\[
\text{Step 3: Eigen Decomposition: } C v_j = \lambda_j v_j
\]
\[
\text{Step 4: Projection and Reconstruction: } Y = X_c V, \quad X_\text{recon} = Y V^T + \bar{X}
\]
Eigenvectors \(v_j\) are principal components capturing maximum variance. Eigenvalues \(\lambda_j\) represent the variance explained. By keeping top \(k\) components, we reconstruct signals while reducing noise and separating multipath contributions.
How to Run:
- Select the signal type and parameters (samples, paths, SNR).
- Click "Run Simulation" to generate signals and PCA results.
- Use the slider to change the number of principal components used for reconstruction.
- Observe the effect on the reconstructed signals and variance spectrum.