Fourier Transform Lab (Student Edition): Guided Exercises & Lab Notes

Fourier Transform Lab — Student Edition Workbook: Labs, Examples, and Reports

Introduction

This workbook is a compact, hands-on guide designed for undergraduate and early graduate students to learn the fundamentals and applications of the Fourier transform in a laboratory setting. It balances conceptual explanations, practical experiments, worked examples, and report templates so students can move from theory to practice quickly and confidently.

Learning objectives

• Understand time- and frequency-domain representations of signals.
• Apply continuous and discrete Fourier transforms (FT, DFT, FFT) to real signals.
• Design and run lab experiments that demonstrate spectral analysis, filtering, and aliasing.
• Interpret spectra, estimate signal parameters (frequency, amplitude, phase), and report results clearly.

Required tools and software

• Signals: function generator or recorded wave files (sinusoids, square, sawtooth, chirps).
• Acquisition: oscilloscope or data acquisition (DAQ) device with sampling control.
• Software: Python (NumPy, SciPy, Matplotlib) or MATLAB with FFT functions.
• Optional: audio interface and microphones for acoustic experiments.

Lab 1 — Sinusoids and the Continuous Fourier Transform

Objective: Verify that pure sinusoids produce delta-like frequency components.

Procedure:

1. Generate a 1 kHz sine wave, sample at 48 kHz, record 0.1 s.
2. Compute the DFT using FFT and plot magnitude spectrum (linear and dB).
3. Window the time record (rectangular vs. Hanning) and compare spectral leakage.

Worked example: show code to generate, window, compute FFT, and annotate peaks (include sample output plots).
Expected observations: sharp spectral peaks at ±1 kHz; reduced sidelobes with Hanning window.

Lab 2 — Periodic Waveforms and Harmonic Content

Objective: Analyze harmonic structure of square and sawtooth waves.

Procedure:

1. Generate square and sawtooth waves at 500 Hz.
2. Use sufficiently long record to resolve harmonics; compute and plot spectrum.
3. Measure harmonic amplitudes and compare to theoretical 1/n (square) or 1/n (sawtooth with sign).

Analysis tips: use log-frequency plots to visualize many harmonics; fit amplitudes to expected decay.

Lab 3 — Sampling, Aliasing, and Anti-Aliasing Filters

Objective: Demonstrate sampling effects and necessity of anti-aliasing.

Procedure:

1. Create a signal containing components at 3 kHz and 20 kHz.
2. Sample at 8 kHz and observe aliased 20 kHz component folding into baseband.
3. Repeat with an analog low-pass filter before sampling and show alias suppression.

Report checklist: compute aliased frequencies by folding formula; show before/after spectra.

Lab 4 — Windowing, Resolution, and Zero-Padding

Objective: Trade-offs between spectral resolution and leakage.

Procedure:

1. Create two close sinusoids at 1000 Hz and 1020 Hz.
2. Vary record length and window type; compute FFT with and without zero-padding.
3. Determine minimum record length required to resolve both tones.

Key result: resolution ≈ 1/Tobs; zero-padding interpolates the spectrum but does not increase true resolution.

python
 ffON2NH02oMAcqyoh2UU MQCbz04ET5EljRmK3YpQ CPXAhl7VTkj2dHDyAYAf” data-copycode=“true” role=“button” aria-label=“Copy Code”>Copy CodeCopied
import numpy as np import matplotlib.pyplot as plt 
fs = 48000
t = np.arange(0, 0.1, 1/fs)
x = np.sin(2np.pi1000t)
w = np.hanning(len(x))
X = np.fft.rfft(xw)
f = np.fft.rfftfreq(len(x), 1/fs)

plt.semilogy(f, np.abs(X))
plt.xlabel(‘Frequency (Hz)’)
plt.ylabel(‘Magnitude’)
plt.title(‘Magnitude Spectrum’)
plt.show()