Equations · Calculus and analysis
Fourier transform
A signal in time rewritten as the frequencies it contains.
F(ω) = ∫₋∞^∞ f(t) e^(−iωt) dt
Aethel says
F of omega equals the integral from negative infinity to infinity of f of t, times e to the power of negative i omega t, d t
What it says
The transform compares the signal f(t) with a pure oscillation at each angular frequency ω. The size of F(ω) says how much of that frequency the signal contains, and its angle says the phase.
It turns hard problems in time, such as filtering or solving differential equations, into simpler ones in frequency.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| F(ω) | F of omega | the transform: the strength and phase at angular frequency ω |
| f(t) | f of t | the signal, as a function of time |
| e^(−iωt) | e to the power of negative i omega t | a pure oscillation at frequency ω |
Where you meet it
Signal processing, physics, image compression and probability.
Questions
How do you say Fourier transform out loud?
F(ω) = ∫₋∞^∞ f(t) e^(−iωt) dt is read: “F of omega equals the integral from negative infinity to infinity of f of t, times e to the power of negative i omega t, d t.” That is exactly what Aethel says when it meets the formula in a book.
What does F(ω) mean in Fourier transform?
F(ω) (said “F of omega”) is the transform: the strength and phase at angular frequency ω.
What does f(t) mean in Fourier transform?
f(t) (said “f of t”) is the signal, as a function of time.
Can I listen to the equations in my own textbook like this?
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