The Fourier transform represents a nonperiodic continuous-time signal by a continuous range of frequency components.

For a suitable signal $x(t)$, define

$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-i\omega t},dt.$$

The inverse transform reconstructs the signal:

$$x(t)=\frac1{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{i\omega t},d\omega.$$

Time domain and frequency domain

$x(t)$ describes how the signal changes with time. $X(\omega)$ describes the amplitude and phase associated with each angular frequency.

These are not two different signals; they are two representations of the same signal.

Bandwidth

A signal is band-limited when its Fourier transform is zero outside a bounded frequency interval. If

$$X(\omega)=0\qquad\text{for }|\omega|>\omega_B,$$

then $\omega_B$ is a highest angular frequency present in the idealized signal.

Operations become simpler

Differentiation in time corresponds to multiplication in frequency:

$$\frac{dx}{dt}\longleftrightarrow i\omega X(\omega).$$

Convolution in time corresponds to multiplication in frequency. These relationships make Fourier transforms central in signal processing and linear systems.

The idea of a finite bandwidth is also what allows the sampling theorem to state when a continuous-time signal can be reconstructed from discrete samples.