takes CONTINUOUS time signal as input and outputs a CONTINUOUS frequency spectrum
is a continuous transformation of a continuous function into its constituent frequencies
cases:
- periodic function → converts its Fourier series in the frequency domain
- non-periodic function → converts it into a continuous frequency domain
can ONLY transform nice well-behaved functions that go to 0 at ±∞; otherwise use Laplace Transform

Fourier Transform - Definition

The Fourier Transform 𝐹{·} of a function 𝑓(𝑥) is defined as:
- $\hat{f} (w) = F {f (x)} (w) = \int_{- \infty}^{\infty} f (x) e^{- i w x} d x$
The Inverse Fourier Transform 𝐹^-1{·} of a function 𝑓ˆ(𝑥) is defined as:
- $f (x) = F^{- 1} {\hat{f} (w)} (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} \hat{f} (w) e^{i w x} d w$

Other