Time and frequency representation
Spectrum

The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via Fourier transform (or discrete Fourier transform) of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.

Any signal that can be represented as amplitude that varies with time has a corresponding frequency spectrum. This includes familiar concepts such as visible light (color), musical notes, radio/TV channels, and even the regular rotation of the earth. When these physical phenomena are represented in the form of a frequency spectrum, certain physical descriptions of their internal processes become much simpler. Often, the frequency spectrum clearly shows harmonics, visible as distinct spikes or lines that provide insight into the mechanisms that generate the entire signal.

Spectrum analysis is the technical process of decomposing a whole signal into simpler parts. As described above, many physical processes are best described as a sum of many individual frequency components. Any process that quantifies the various amounts (e.g. amplitudes, powers, intensities, or phases), versus frequency can be called spectrum analysis.

Spectrum analysis can be performed on the entire signal (usually periodic signal). Alternatively, a signal (mainly non periodic or quasi-periodic) can be broken into short segments, called frames, and spectrum analysis may be applied to these individual segments.

The Fourier transform of a function produces a frequency spectrum which contains all of the information about the original signal, but in a different form. This means that the original function can be completely reconstructed (synthesized) by an inverse Fourier transformation.

For perfect reconstruction, the spectrum analyzer must preserve both the amplitude and phase of each frequency component. These two pieces of information can be represented as a 2-dimensional vector, as a complex number, or as magnitude (amplitude) and phase in polar coordinates. A common technique in signal processing is to consider the squared amplitude, or power; in this case the resulting plot is referred to as a power spectrum.

The following table summarizes types of signals and their spectrum.

Signals and their spectrum
Signal | Spectrum
continuous periodic | discrete non-periodic
continuous non-periodic | continuous non-periodic
discrete periodic | discrete periodic
discrete non-periodic | continuous periodic

The following images show basic Fourier transform pairs (only amplitudes are present in the pictures). These can be combined using the Fourier transform theorems below to generate the Fourier transforms of many different functions.

Basic Fourier transforms pairs

The following pictures display two-dimensional signals and their spectrum (magnitude and phase characteristic).

2D rectangular function, magnitude and phase frequency characteristic
2D circular function, magnitude and phase frequency characteristic