Frequency spectrum

A frequency spectrum describes the frequency components of a function. A Fourier series supplies discrete Fourier coefficients; the Fourier transform of an integrable function supplies a continuous, generally complex-valued description. An amplitude plot shows magnitudes and omits wave phases, so it need not determine the original function. In the example with three Fourier harmonics above, the one-sided amplitude plot has heights $1$, $1/2$, and $1/4$. The corresponding complex Fourier coefficients at positive and negative frequencies each have half those heights: $\cos (2\pi \nu t)=(e^{2\pi i\nu t}+e^{-2\pi i\nu t})/2$.

A spectrogram displays how a local frequency spectrum changes over time, usually using colour for squared Fourier transform magnitude. Longer analysis windows give finer frequency spacing while averaging over a longer time interval.