Lab: Fourier Analysis & Audio Fingerprinting

Learning Objectives

Understand the transition from the Time Domain to the Frequency Domain.
Implement the Short-Time Fourier Transform (STFT) to visualize music.
Identify "Spectral Peaks" as a method of data compression for fingerprinting.

1. The Mathematical Foundation

The Fourier Transform allows us to see which frequencies are present in a signal. However, music changes over time. To capture this, we use the STFT, which applies the Fourier Transform to small, overlapping windows of time.

$X(m, \omega) = \sum_{n=-\infty}^{\infty} x[n] w[n - m] e^{-j \omega n}$
(Where $w[n]$ is a window function like Hamming or Hann)

2. Lab Procedures

Step 1: Visualizing the Waveform vs. the Spectrum

Load a 5-second clip of a song. Notice how the raw waveform (Amplitude vs. Time) tells us nothing about the melody or pitch.

import numpy as np
import matplotlib.pyplot as plt
from scipy.io import wavfile

fs, data = wavfile.read('sample_audio.wav')
plt.plot(data[:fs*5])
plt.title("Time Domain: Signal Amplitude")

Step 2: Generating the Spectrogram

Use the Discrete Fourier Transform (DFT) repeatedly across the signal to create a Spectrogram. This is the "Heat Map" of music.

Task: Experiment with window sizes (NFFT).
Observation: How does a larger window affect your frequency resolution vs. your time resolution?

plt.specgram(data, Fs=fs, NFFT=2048, noverlap=1024)
plt.ylabel('Frequency [Hz]')
plt.xlabel('Time [sec]')

Step 3: Peak Extraction (The Constellation Map)

Shazam doesn't store the whole spectrogram—it’s too much data. It only stores the strongest frequency peaks. These peaks are robust against background noise.

Exercise:

Find local maxima in the spectrogram.
Filter out frequencies below 300Hz (bass) and above 5000Hz (harmonics) to focus on the "information-rich" zone.
Plot these points on a coordinate plane (Time vs. Frequency). This is your Constellation Map.

Step 4: Combinatorial Hashing

Pick an "Anchor Point" in your constellation map. Pair it with several "Target Points" in a window ahead of it. Create a hash for each pair:

Hash = [Freq_Anchor | Freq_Target | Delta_Time]

This hash is unique to the song's structure and remains the same even if the song starts playing at a different time.

Lab: Deconstructing Audio with Fourier Transforms