Definition
The Fourier transform decomposes a function or signal into its constituent frequencies. For a function f(t), the transform is with the inverse recovering the original via integration against the complex exponentials.
Why It Matters
Any sufficiently nice signal — sound, light, electrical voltage, images, quantum wave functions — can be exactly represented as a sum (or integral) of pure sinusoids. Operations that are difficult or nonlocal in the time domain (differentiation, convolution, filtering) become simple multiplications in the frequency domain. This is the mathematical basis of spectrum analysis, image compression, quantum mechanics (momentum representation), and much of modern communications.
Core Concepts
- Frequency Domain Representation: A signal is equivalently described by how much of each frequency it contains (amplitude and phase spectrum).
- How to read: “The function f hat of omega equals the integral of f of t times e to the negative i omega t with respect to t.”
- Meaning: Each complex value \hat f(ω) encodes the contribution of the pure tone at angular frequency ω.
- Linearity and Superposition: The transform of a sum is the sum of the transforms; this is why linear systems are so powerful when analyzed in frequency space.
- Convolution Theorem: Convolution in time ↔ multiplication in frequency (and vice versa). Filtering becomes pointwise multiplication of spectra.
- Parseval/Plancherel: Energy (or power) is preserved between domains: the integral of |f(t)|² equals (scaled) integral of | \hat f(ω) |².
- Uncertainty Principle (informal): A function and its Fourier transform cannot both be arbitrarily concentrated; spread in time implies spread in frequency.