Guess the Song with Fourier Music Decomposition!

Table of Contents

• Introduction
• Step 1: Downloading the Song
• Step 2: Understanding WAV Files
• Step 3: The Fourier Transform
• Step 3.1: Decomposing the Wave into Sine Waves
• Step 4: Visualizing the Frequency Spectrum
• Step 5: Discrete Fourier Transform
• Step 6: Filtering the Data
• Step 7: Reconstructing the Original Song
• Conclusion

Step 1: Downloading the Song

🎵 Exploring the First Step: Downloading the Song

To create a song guessing Game using the concept of the Fourier transform, the first step is to download the WAV file for a song. A WAV file is a 1D array of values that represent the displacement of a speaker over time. By capturing the details of the song in this format, we can analyze and manipulate its sound using the Fourier transform.

Step 2: Understanding WAV Files

🔊 Unraveling WAV Files: A Closer Look

WAV files contain audio data represented as a continuous function of time. When you hear sound, it's because pressure waves created by a speaker are reaching your ear. WAV files store these pressure waves as numerical values between -1 and 1. These values correspond to the displacement of the speaker from its rest position at different points in time. Understanding the structure of WAV files is essential for analyzing and transforming audio data.

Step 3: The Fourier Transform

🎚️ Diving into the Fourier Transform

The Fourier transform is an essential mathematical tool for analyzing periodic waves. By performing the Fourier transform on a wave, it is decomposed into a series of infinite sine waves. Each of these sine waves has a specific frequency and amplitude. In the context of the song guessing game, the Fourier transform allows us to break down a song into its underlying sine waves.

Step 3.1: Decomposing the Wave into Sine Waves

🌊 Dissecting the Wave: Decomposing into Sine Waves

When we perform a Fourier transform on a wave, we can decompose it into a series of sine waves with different frequencies. In our example, we created a wave using just three sine waves with frequencies of 0.2, 2, and 3 Hertz. Each wave contributes to the overall Shape of the original wave, with the amplitude of each sine wave determining its influence. Visualizing the decomposition reveals the impact of each individual sine wave.

Step 4: Visualizing the Frequency Spectrum

📊 Unmasking the Frequency Spectrum

The frequency spectrum summarizes the amplitude and frequency of each sine wave Present in a wave. By plotting the amplitude versus the frequency for all the individual sine waves, we can observe the frequency spectrum. In our example, with only three non-zero frequencies, the spectrum appears as a bar Chart. By understanding the frequency spectrum, we gain insights into the composition of the original song.

Step 5: Discrete Fourier Transform

🔢 The Discrete Fourier Transform: A Finite Perspective

In practice, audio data is represented by a discrete set of points rather than a continuous waveform. To analyze this type of data, we use the Discrete Fourier Transform (DFT). The DFT decomposes the audio data into a finite number of sine waves, with the number of waves dictated by the number of points in the original sample. This transforms the continuous frequency spectrum into a bar chart-like representation.

Step 6: Filtering the Data

🔍 Filtering the Data: Selecting Key Frequencies

To make the song guessing game interesting, we need to filter the frequency spectrum and select only a certain percentage of frequencies for each round. By sorting the frequencies by their amplitudes and setting the amplitudes of unwanted frequencies to zero, we gradually recreate the original wave. This filtering process adds batches of the most impactful frequencies, making the wave progressively Resemble the original song.

Step 7: Reconstructing the Original Song

🎶 Piecing It Together: The Final Reveal

After filtering and selecting the desired frequencies, we can reconstruct the original song. Using the inverse Fourier transform, we combine all the individual sine waves back together. By adding the frequencies in batches, each round of the game brings the song closer to its original form. The song gradually sounds more like the real song until its final version is revealed.

Conclusion

🔮 Wrapping Up the Journey

In this Tutorial, we embarked on a fascinating journey through the world of the Fourier transform and its application to a song guessing game. We learned how to download a song, understand WAV files, decompose waves into sine waves, Visualize the frequency spectrum, perform the Discrete Fourier Transform, filter the data, and reconstruct the original song. By diving into the complexities of sound and Music, we explored the beauty of mathematics in action.