🔍 FFT Signal Reconstruction and Analysis

MATLAB tool for signal reconstruction using the Fast Fourier Transform (FFT). Processes time-domain data from a .txt file, decomposes it into frequency components, reconstructs the signal using a selectable number of harmonics, and quantifies accuracy via Mean Squared Error (MSE).

---

📊 Results

Signal reconstruction — original vs FFT approximation

Original signal (blue) vs FFT approximation (red).
Visual overlap confirms high-fidelity reconstruction across the full time window.

Mean Squared Error

MSE = 7.3581 × 10⁻³⁰ — effectively zero numerical error,
confirming that the FFT reconstruction is virtually indistinguishable from the original signal.

---

🧠 Key Features

• Automated Data Handling — detects and transposes data columns from .txt inputs automatically
• FFT Implementation — converts time-domain signals to the frequency domain using MATLAB's fft
• Selectable harmonics — user defines N at runtime; more harmonics = higher fidelity
• Error Analysis — quantifies reconstruction accuracy with MSE
• Coefficient Table — outputs Fourier coefficients (aₙ and bₙ) in structured format
• Visualization — side-by-side comparison plot + squared error plot

---

⚙️ Mathematical Background

Any periodic signal x(t) can be approximated by a Fourier Series:

$$ x(t) \approx \frac{a_0}{2} + \sum_{n=1}^{N} \left[ a_n \cos(2\pi n f_0 t) + b_n \sin(2\pi n f_0 t) \right] $$

Where:

• N = number of harmonics selected by the user
• aₙ, bₙ = Fourier coefficients derived from the FFT output

Reconstruction accuracy is validated by the Mean Squared Error (MSE):

$$ MSE = \frac{1}{M} \sum_{i=1}^{M} \left( x_{\text{original}}[i] - x_{\text{reconstructed}}[i] \right)^2 $$

For curve1.txt with N harmonics, the result was MSE = 7.3581 × 10⁻³⁰, confirming near-perfect reconstruction.

---

🧩 Folder Structure

Signal-Analyzer/
├── Signal Samples/
│   ├── curve1.txt           # Primary test signal
│   └── noisy_data.txt       # Optional noisy signal
├── images/
│   ├── Reconstruction.jpg   # Comparison plot: original vs FFT
│   └── MSE.jpg              # Squared error over time
├── analog_2_continuos.m     # Main MATLAB processing script
└── README.md

---

🚀 How to Use

Prerequisites

• MATLAB R2020a or newer
• Input signal file (.txt) placed inside Signal Samples/

Input File Format

Two whitespace-separated columns:

0.000  0.125
0.001  0.150
0.002  0.100

Column	Description
1	Time (t)
2	Amplitude (A)

Execution

git clone https://github.com/VoIkmer/Signal_Analyzer.git

1. Open MATLAB and navigate to the project folder
2. Run:

   analog_2_continuos

3. Select your .txt file when prompted
4. Enter the number of harmonics N

---

📈 Output & Analysis

Output	Description
Reconstruction plot	Original (black) vs FFT approximation (red dashed)
MSE plot	Squared error at each time step
Coefficient table	aₙ and bₙ values for each harmonic
MSE value	Printed in the plot title and command window

Note on Gibbs phenomenon: near signal discontinuities, increasing N may introduce overshoot. This is a known theoretical property of Fourier reconstruction, not a script error.

---

🧑‍💻 Author

Carlos Eduardo
Electrical Engineering Student — UFBA

• Email: cguimaraesbarbosa03@gmail.com
• GitHub: VoIkmer
• LinkedIn: carl0sedu

---

📚 License

MIT License — free to use, modify, and distribute.