Spectrum (disambiguation)

Spectrum (plural spectra) can refer to:

Table of contents

1 Common-noun meanings 1.1 In the physical sciences 1.2 In pure mathematics 2 Proper-noun meanings 3 General description 3.3 Intuitive approach

Common-noun meanings

• The spectrum of activity of a drug
• The political spectrum of opinion
• The bipolar spectrum, in psychology
• A spectre

In the physical sciences

• An emission spectrum or absorption spectrum observed in light
• The energy spectrum of a collection of particles
• The electromagnetic spectrum
  • The optical spectrum or color spectrum, a subset of the electromagnetic spectrum
• The frequency spectrum of a signal
  • The power spectrum of a signal
• The timbre of a musical note

See also cepstrum

In pure mathematics

• The spectrum of an operator
• The spectrum of a matrix
• The spectrum of a ring
• A spectrum (homotopy theory)
• The spectrum of a Boolean algebra is its Stone space.

Proper-noun meanings

• The Wachovia Spectrum, formerly The Spectrum, a sports arena
• IEEE Spectrum, a magazine
• Spectrum (band)
• The ZX Spectrum, a computer
• A fictional army in the television series Captain Scarlet

General description

The noun spectrum (plural: spectra) has a variety of meanings. In most modern usages, there is a unifying theme of a variety of possible cases between extremes at either end. Older usages were not necessarily on that same unifying theme, but nonetheless led to the modern ones through a sequence of events set out below. Some modern usages in mathematics evolved out of that unifying theme but may be difficult to recognize as fitting into it.

). In the 17th century the word spectrum was introduced into optics, referring to the range of colors observed when white light was dispersedd through a prism. Soon the term referred to a plot of light intensity as a function of frequency or wavelength. Max Planck later realized that frequency represents electromagnetic energy:

where E is the energy of a photon, h is Planck's constant, and is the frequency of the light.

The word spectrum then took on the obvious analogous meaning in reference to other sorts of waves, such as sound wave, or other sorts of decomposition into frequency components. Thus a spectrum is a usually 2-dimensional plot, of a compound signal, depicting the components by another measure. Sometimes, the word spectrum refers to the compound signal itself, such as the "spectrum of visible light", a reference to those electromagnetic waves which are visible to the human eye.

Intuitive approach

A spectrum in physics relates to waves: light waves, sound waves, etc. Waves can be represented by graphs that look like waves in the sea and represent some physical quantity as a function of time (or space). However, waves can also be represented as a function of a frequency. A tone without overtones, or monochromatic light can be represented as a sine function of time, but also as a single line as a function of frequency. Sound and light in general can be represented as a superposition of "pure" sine functions with different frequencies and varying intensities. A graph showing the intensities of the "pure" sine functions composing some actual time dependent phenomenon as a function of the frequency is said to show the "spectrum" of that phenomenon.

Essentially, the construction of a spectrum from some time-dependent phenomenon is a purely mathematical construct. However, our eyes and ears also perform the mathematical transformation from the "time domain" to the "frequency domain". Our eyes distinguish colors (light of different frequencies) and our ears distinguish "low" and "high" tones (sounds of different frequencies). Therefore, we are inclined to assume that spectra are real, and to forget that these phenomena always can be represented as a function of time as well.

Spectra are also apparently very realistic, very tangible in communication technology. Parts of the radio spectrum are auctioned and sold like commodities. This is based on the fact, that it is technically fairly easy to filter parts of a spectrum. There is actually just a single ratio signal in the air. However, mathematically it can be seen as a composition of signals of different frequencies. And technologically it is easy to select just the signal related to a certain range of frequencies. That may be related to a radio station. (It should be noted however that such "frequency division multipexing" of radio stations is not the only technology available. There is also "time division multiplexing" where a single channel (from the user perspective) is not related to a confined band of frequencies).

Glass and water transmit light of different frequencies at different speeds. That causes "chromatic aberration" of lenses, and, better known, rainbows. Again, a spectrum appears to be a tangible thing.

The description of a time-dependent phenomenon by a spectrum, i.e. a function of a frequency involves what is called: a transformation from the time domain into the frequency domain. Such transformations are also important from an abstract mathematical perspective. Differentiating (calculating the derivative) of a function of time becomes a multiplication in the time domain. Thus such transformation can be helpful to solve differential equations (which is normally a difficult endeavor).

The transformation of a wave into a superposition of sine waves allegedly was first proposed by Jean Baptiste Joseph Fourier (1768-1830), who studied the propagation of heat waves. Today, the above transformation is still called a "Fourier transform" (which can be performed from numerical data using the "fast Fourier transform" algorithm, invented much later).

For theoretical mathematicians, the choice of the sine function as the basic "composing" function is just one of the possible alternatives. Other sets of "orthogonal" functions may be chosen for an "integral transformations", yielding different kinds of spectra.

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