tags: ece-402
Fourier Transform
A Fourier Transform is a mathematical tool for frequency decomposition, used to decompose a signal into its pure, single-frequency components.
- from MUS 409: a mathematical operation mapping a continuous signal to amplitude/phase data corresponding to spectrum
- i.e. conversion of time domain to frequency domain
In practice, Fourier transforms are often used with [inverse Fourier transforms]: reproducing a signal (intensity over time) from a Fourier Transform (intensity in terms of frequency)
The Fourier transform of an intensity vs. time graph, usually $g(t)$, is a new function $\hat{g}(t)$, which:
- doesn't have time as an input, but instead takes a frequency (the "winding" frequency)
- outputs a complex number, some point in the 2D plane, that corresponds to the strength of a given frequency in the original signal
$$ \hat{g}(f) = \int^{t^2}_{t_1}g(t)e^{-2 \pi i f t} dt $$
Key concepts for understanding this function:
- exponentials correspond to rotation (Euler's number)
- multiplying the exponential by the original input $g(t)$ means drawing a wound up version of that graph around the origin
- an integral of a complex valued function can be interpreted in terms of a center-of-mass idea
Sound
Fourier transforms are used in [spectral analysis] to decompose and isolate [frequencies] from [sound].
Example (MUS 409)
- A DFT of a digital recording of Stravinsky's Rite of Spring (or some other really long piece)
- Treat the entirety of the piece as one period of a complex vibration
- DFT derives amplitude/phas einformation for the fundamental, 2nd harmonic, 3rd harmonic, etc. through N
- note that the "fundamental"i n this case is vastly below our range of hearing - around 0.00056 Hz if the piece is 30 minutes long
- when summed together, frequency components perfectly reconstruct the original time domain signal
- FFT tells us the sinusoidal components we need to reconstruct this piece of music in its entirety, but not enough information to tell where components of certain sinusoidal segments of a piece are located
- To do this, we have to take STFTs