Signal Processing & Time-Series Analysis: From Transient Decay to Spectral Density

A dual approach to analyzing experimental time-series data: modeling transient decays in the time domain and extracting periodic components in the frequency domain.

Python FFT Signal Processing Differential Equations

Project Overview

This project demonstrates a dual approach to analyzing experimental time-series data. I investigated the properties of dynamic systems using both Time-Domain modeling (fitting differential equations to transient signals) and Frequency-Domain analysis (extracting periodic components from noisy datasets).

Key Components

Part 1: Time-Domain Analysis (RLC Circuit)

Objective: Modeled the transient response of a Damped Harmonic Oscillator (RLC circuit).

Method: Applied non-linear curve fitting (scipy.optimize) to voltage data to extract decay constants ($\alpha$) and angular frequencies ($\omega$), verifying the solution to the differential equation $V(t) = V_0 e^{-\alpha t} \cos(\omega t)$.

Application: This mathematical framework is directly analogous to mean-reverting stochastic processes used in financial volatility modeling.

Source Code: Solar Cell FFT Analysis (`p5_hw7.py`)

#!/usr/bin/env python3
#
# Using FFT method

import numpy as np
import matplotlib.pyplot as plt

#part (a)
FS = 920
input_filename = 'solarcell_data.txt'
print('Reading data from %s...' % input_filename)
data = np.loadtxt(input_filename)
print('Successfully read %d data points.' % len(data))
print()

#part (b)
npts = len(data)
data = data-np.mean(data) #removing DC components (average voltage)
ft = np.fft.fft(data, n=16*npts)
ftnorm = np.abs(ft)
ps = ftnorm**2
freqs = np.fft.fftfreq(len(ps), 1.0/FS)
time = np.arange(npts) /FS 
f1, ax1 = plt.subplots()
ax1.plot(time, data)
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('Voltage (V)')
ax1.set_title('Solar Cell Voltage vs Time')
f1.show()

f2, ax2 = plt.subplots()
ax2.plot(freqs, ps)
ax2.set_xlim(0, 200)
ax2.set_xlabel('Frequency (Hz)')
ax2.set_ylabel('Power')
ax2.set_title('Power Spectrum of Solar Cell Signal')
f2.show()

# part (c)
eps_filename = 'power_spectrum_fft.eps'
print('Saving power spectrum to %s...' % eps_filename)
f2.savefig(eps_filename, format='eps')
print('Plot saved successfully.')
print()

real_freq = freqs > 0.5  # Exclude DC and very low frequencies
real_freqs = freqs[real_freq]
real_ps = ps[real_freq]

# Find the peak
max_idx = np.argmax(real_ps)
fundamental_freq = real_freqs[max_idx]
max_power = real_ps[max_idx]

print('Fundamental frequency: %.2f Hz' % fundamental_freq)
print('Power at fundamental frequency: %.2e' % max_power)
print()

input("Press  to exit...\n")

Part 2: Frequency-Domain Analysis (Solar FFT)

Objective: Identified dominant frequency components in high-speed voltage sensor data.

Method: Implemented a Fast Fourier Transform (FFT) algorithm in Python to convert raw time-series data into a Power Spectral Density (PSD) plot, isolating signal from noise.

Application: Spectral analysis is a fundamental technique for identifying cyclical trends and filtering noise in market data.

Detailed Analysis (PDF)

The full technical report, including mathematical derivations and detailed plots, is available below.

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