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Signal Analysis Algorithms and Formula Reference

This document provides the mathematical formulas and algorithmic steps used for identifying frequency and amplitude from 1D array signal datasets.

1. Amplitude Statistics

Peak-to-Peak Amplitude

The total range of the signal from its lowest to highest points. Useful for understanding the maximum swing. $$A_{pk-pk} = \max(x) - \min(x)$$

RMS (Root Mean Square) Amplitude

A measure of the signal's effective power, representing the square root of the arithmetic mean of the squares of the values. $$A_{rms} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2}$$

Standard Deviation ($\sigma$)

Quantifies the amount of variation or dispersion of the signal values around the mean. $$\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}$$

2. Frequency Identification (FFT)

Discrete Fourier Transform (DFT)

The algorithm transforms the signal from the time domain to the frequency domain to identify periodic components. $$X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-j \frac{2\pi}{N} kn}$$

Dominant Frequency Identification

The frequency associated with each FFT index $k$ is calculated as: $$f_k = \frac{k \cdot F_s}{N}$$ where:

  • $F_s$: Sampling rate (e.g., 256 Hz)
  • $N$: Total number of samples (e.g., 256)
  • $k$: Bin index (frequency index)

Algorithm Steps:

  1. Compute the FFT of the 1D signal array.
  2. Calculate the magnitude of the FFT output ($|X[k]|$).
  3. Focus on positive frequencies (indices $1$ to $N/2$).
  4. Identify the index $k_{max}$ with the largest magnitude.
  5. Convert $k_{max}$ to Hz using the formula above to get the Dominant Frequency.

3. Waveform Generation Formulas

The following formulas are used in src/dataset.py to generate the 1D signal arrays for training and testing. Signals are typically generated over the interval $t \in [0, 1]$ with $n=256$ samples.

Periodic Waveforms

  • Sine Wave: $x(t) = A \sin(2\pi f t + \phi)$
  • Cosine Wave: $x(t) = A \cos(2\pi f t + \phi)$
  • Square Wave: $x(t) = A \cdot \operatorname{sgn}(\sin(2\pi f t + \phi))$
  • Sawtooth Wave: $x(t) = A \cdot (2(\frac{t}{T} - \lfloor \frac{t}{T} + \frac{1}{2} \rfloor))$
  • Triangle Wave: $x(t) = A \cdot |2(\frac{t}{T} - \lfloor \frac{t}{T} + \frac{1}{2} \rfloor)|$

Noise and Stochastic Signals

  • White Noise: $x(t) = A \cdot \mathcal{N}(0, 1)$ (Gaussian distribution)
  • Pink Noise: $x(t) = A \cdot \operatorname{normalize}(\sum \mathcal{N}(0, 1))$ (1/$f$ approximation)
  • Brownian Noise: $x(t) = A \cdot \operatorname{normalize}(\sum \sum \mathcal{N}(0, 1))$ (1/$f^2$ approximation)

Mathematical Funstions

  • Exponential Decay: $x(t) = A \cdot e^{-rt}$ (where $r$ is the decay rate)
  • Logarithmic Decay: $x(t) = A \cdot (1 - \operatorname{normalize}(\ln(t)))$
  • Hyperbolic Tangent: $x(t) = A \cdot \tanh(f \sin(t))$
  • Sigmoid Wave: $x(t) = A \cdot \left(\frac{2}{1+e^{-f\sin(t)}} - 1\right)$

4. General Statistics

Mean ($\mu$)

The arithmetic average of all signal values, also known as the DC level or offset. $$\mu = \frac{1}{n} \sum_{i=1}^{n} x_i$$

Median

The middle value of the signal when sorted in ascending order. $$\operatorname{Median}(x) = \begin{cases} x_{(n+1)/2} & \text{if } n \text{ is odd} \ \frac{x_{n/2} + x_{n/2 + 1}}{2} & \text{if } n \text{ is even} \end{cases}$$

Variance ($\sigma^2$)

The average of the squared differences from the Mean. $$\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$$

Skewness

Measures the asymmetry of the probability distribution of the signal values about its mean. $$\operatorname{Skewness} = \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^3}{\left(\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2\right)^{3/2}}$$

Kurtosis

Measures the "tailedness" of the probability distribution of the signal values. $$\operatorname{Kurtosis} = \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^4}{\left(\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2\right)^2}$$

5. Model Comparison Framework: Technical Deep-Dive

To identify signal parameters (like frequency or amplitude) optimally, this project compares two distinct modeling approaches: Manual Feature Engineering and Automated Feature Learning (CNN).

A. Feature Extraction (Classical ML Workflow)

Classical models like XGBoost or Random Forest cannot process raw waveform arrays effectively. We transform the 256-point signal into a compact feature vector:

1. Time-Domain Statistical Features

  • RMS (Root Mean Square): Calculates the effective power. For periodic signals, this is highly correlated with amplitude.
  • Peak-to-Peak: Measures the maximum swing, direct proxy for amplitude in noise-free signals.
  • Skewness: Measures the asymmetry of the signal's distribution (e.g., distinguishing a sine wave from a lopsided pulse).
  • Kurtosis: Measures the "peakedness" or tail behavior, identifying noise spikes vs. smooth oscillations.

2. Frequency-Domain (FFT) Features

FFT shifts the signal from Time (Amplitude vs. Time) to Frequency (Magnitude vs. Hz).

  • Dominant Frequency: The system identifies the index $k$ with the highest magnitude $|X[k]|$ and converts it to Hz. This is the primary feature for frequency regression.
  • Spectral Energy: The total power in the frequency domain, used to differentiate between high-energy signals and low-amplitude noise.

B. 1D CNN Architecture (Deep Learning Workflow)

The 1D Convolutional Neural Network learns to extract features automatically using sliding filters.

  • Conv1D Layers: Small kernels (e.g., size 7) slide across the signal to detect "Local Motifs" like edges, curves, or periodic segments.
  • Max Pooling: Reduces the signal length by half after each convolution to isolate the most important activation points and improve computational efficiency.
  • Hierarchical Features: Earlier layers detect simple slopes; later layers detect complex periodic patterns.
  • Global Adaptive Pooling: Collapses the final 128 feature maps into a single vector, making the model robust to "Stationarity" (it doesn't matter when a sine cycle starts, just that it exists).

C. Evaluation & Ranking Algorithm

The framework uses a rigorous statistical cycle to rank models:

  1. Standardization: Both features and raw signals are normalized to $\mu=0, \sigma=1$ to ensure gradient stability.
  2. 5-Fold Cross-Validation: Data is split into 5 "folds." Models are trained on 4 and tested on 1, rotating until every sample has been used for testing. This prevents Overfitting.
  3. Metrics Ranking:
    • $R^2$ Score: The primary ranking metric. Indicates what percentage of the variance is captured by the model (Goal: 1.0).
    • MAE & RMSE: Used to quantify the "error magnitude" in physical units, with RMSE specifically punishing large "outlier" misses.