The Fourier transform is a fundamental mathematical tool that allows decomposition of any signal into a sum of sinusoids of different frequencies. The FFT (Fast Fourier Transform) is the fast algorithm that allows smartphones to perform this analysis in real time.
Discover FizziQ
How to measure it in class
With the FizziQ app, FFT is used behind the scenes by several instruments: frequency meter, tuner, spectrogram.
Steps:
- Open FizziQ and select the “Spectrogram” or “Spectrum” instrument
- Produce a pure sound (tuning fork, whistle): a single frequency appears
- Produce a complex sound (voice, instrument): multiple frequencies appear
- Observe that natural sounds are composed of a fundamental frequency and harmonics
- Use the synthesizer to create pure sounds and verify their spectrum
Scientific activities on this topic
- Analyze the spectrum of a tuning fork vs that of a guitar: https://www.fizziq.org/en/activities/the-tone-of-an-instrument/
- Understanding harmonics of an instrument
- Study acoustic beats: https://www.fizziq.org/en/activities/acoustic-beats/
Learn more
Fourier’s theorem:
Joseph Fourier demonstrated in 1822 that any periodic function can be written as a sum (potentially infinite) of sinusoids. This decomposition reveals the “frequency content” of the signal.
Physical interpretation:
A musical sound is composed of:
- A fundamental frequency f0 that determines the perceived pitch
- Harmonics (multiples of f0) whose relative amplitudes determine the timbre
A tuning fork produces an almost pure sound (mainly f0). A violin or voice contains many harmonics.
FFT in the smartphone:
The FFT algorithm (Cooley-Tukey, 1965) reduces computation time from N squared to N times log(N) operations. This is what allows real-time sound analysis. FizziQ typically analyzes 1024 or 2048 samples at a time.
Resolution and trade-offs:
The more samples analyzed, the better the frequency resolution, but the more temporal resolution is lost. This is Heisenberg’s uncertainty principle applied to signal processing.
Formula
The Fourier transform of a signal x(t) is:
X(f) = integral of x(t) times e to the power of (-2 pi i f t) dt
For a discrete signal of N samples:
X(k) = sum of x(n) times e to the power of (-2 pi i k n / N) for n = 0 to N-1
where:
- X(k): complex amplitude at frequency k times (f_s / N)
- f_s: sampling frequency
- N: number of samples
Application examples
- The FizziQ frequency meter uses FFT to find the dominant frequency
- Audio equalizers modify frequency components identified by FFT
- MP3 compression uses a variant of FFT to eliminate inaudible frequencies
- Medical imaging (MRI) uses the Fourier transform to reconstruct images
FAQ
Q: Why is it called “fast”? A: The direct algorithm requires N squared operations. The Cooley-Tukey FFT requires only N times log(N). For 1024 samples: 1 million vs 10,000 operations!
Q: Can the original sound be reconstructed from the FFT? A: Yes, thanks to the inverse transform. Information is not lost, just represented differently.
Q: Why are low frequencies less well resolved? A: To distinguish two close frequencies, the signal must be observed long enough. Low periods being long, they require more observation time.
Related concepts
Frequency - Sound Spectrum - Spectrogram - Harmonics - Sampling - Signal Analysis