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In the physical sciences, the term spectrum was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersed through a prism. OpenStax Astronomy, "Spectroscopy in Astronomy". OpenStax CNX. September 29, 2016 The word "spectrum" to describe a band of colors that has been produced, by refraction or diffraction, from a beam of light first appears on p. 3076. Soon the term referred to a plot of light intensity or power as a function of frequency or wavelength, also known as a spectral density plot.
Later it expanded to apply to other , such as and that could also be measured as a function of frequency (e.g., noise spectrum, sea wave spectrum). It has also been expanded to more abstract "", whose power spectrum can be analyzed and processed. The term now applies to any signal that can be measured or decomposed along a continuous variable, such as energy in electron spectroscopy or mass-to-charge ratio in mass spectrometry. Spectrum is also used to refer to a graphical representation of the signal as a function of the dependent variable.
Light from many different sources contains various colors, each with its own brightness or intensity. A rainbow, or prism, sends these component colors in different directions, making them individually visible at different angles. A graph of the intensity plotted against the frequency (showing the brightness of each color) is the frequency spectrum of the light. When all the visible frequencies are present equally, the perceived color of the light is white, and the spectrum is a flat line. Therefore, flat-line spectra in general are often referred to as white, whether they represent light or another type of wave phenomenon (sound, for example, or vibration in a structure).
In radio and telecommunications, the frequency spectrum can be shared among many different broadcasters. The radio spectrum is the part of the electromagnetic spectrum corresponding to frequencies lower below 300 GHz, which corresponds to wavelengths longer than about 1 mm. The microwave spectrum corresponds to frequencies between 300 MHz (0.3 hertz) and 300 GHz and wavelengths between one meter and one millimeter.Pozar, David M. (1993). Microwave Engineering Addison–Wesley Publishing Company. .Sorrentino, R. and Bianchi, Giovanni (2010) Microwave and RF Engineering , John Wiley & Sons, p. 4, . Each broadcast radio and TV station transmits a wave on an assigned frequency range, called a channel. When many broadcasters are present, the radio spectrum consists of the sum of all the individual channels, each carrying separate information, spread across a wide frequency spectrum. Any particular radio receiver will detect a single function of amplitude (voltage) vs. time. The radio then uses a tuned circuit or tuner to select a single channel or frequency band and modulation or decode the information from that broadcaster. If we made a graph of the strength of each channel vs. the frequency of the tuner, it would be the frequency spectrum of the antenna signal.
In astronomical spectroscopy, the strength, shape, and position of absorption and emission lines, as well as the overall spectral energy distribution of the continuum, reveal many properties of astronomical objects. Stellar classification is the categorisation of based on their characteristic electromagnetic spectra. The spectral flux density is used to represent the spectrum of a light-source, such as a star.
In radiometry and colorimetry (or color science more generally), the spectral power distribution (SPD) of a light source is a measure of the power contributed by each frequency or color in a light source. The light spectrum is usually measured at points (often 31) along the visible spectrum, in wavelength space instead of frequency space, which makes it not strictly a spectral density. Some spectrophotometers can measure increments as fine as one to two and even higher resolution devices with resolutions less than 0.5 nm have been reported. the values are used to calculate other specifications and then plotted to show the spectral attributes of the source. This can be helpful in analyzing the color characteristics of a particular source.
A source of sound can have many different frequencies mixed. A musical tone's timbre is characterized by its harmonic spectrum. Sound in our environment that we refer to as noise includes many different frequencies. When a sound signal contains a mixture of all audible frequencies, distributed equally over the audio spectrum, it is called white noise.
The spectrum analyzer is an instrument which can be used to convert the sound of the musical note into a visual display of the constituent frequencies. This visual display is referred to as an acoustic spectrogram. Software based audio spectrum analyzers are available at low cost, providing easy access not only to industry professionals, but also to academics, students and the hobbyist. The acoustic spectrogram generated by the spectrum analyzer provides an acoustic signature of the musical note. In addition to revealing the fundamental frequency and its overtones, the spectrogram is also useful for analysis of the temporal ADSR envelope, ADSR envelope, ADSR envelope, and ADSR envelope of the musical note.
The classical example of a continuous spectrum, from which the name is derived, is the part of the spectrum of the light emitted by excited state of hydrogen that is due to free becoming bound to a hydrogen ion and emitting photons, which are smoothly spread over a wide range of wavelengths, in contrast to the discrete lines due to electrons falling from some bound quantum state to a state of lower energy. As in that classical example, the term is most often used when the range of values of a physical quantity may have both a continuous and a discrete part, whether at the same time or in different situations. In , continuous spectra (as in bremsstrahlung and thermal radiation) are usually associated with free particles, such as atoms in a gas, electrons in an electron beam, or conduction band electrons in a metal. In particular, the position and momentum of a free particle has a continuous spectrum, but when the particle is confined to a limited space its spectrum becomes discrete.
Often a continuous spectrum may be just a convenient model for a discrete spectrum whose values are too close to be distinguished, as in the in a crystal.
The continuous and discrete spectra of physical systems can be modeled in functional analysis as different parts in the decomposition of the spectrum of a linear operator acting on a function space, such as the Hamiltonian operator.
The classical example of a discrete spectrum (for which the term was first used) is the characteristic set of discrete seen in the emission spectrum and absorption spectrum of isolated of a chemical element, which only absorb and emit light at particular . The technique of spectroscopy is based on this phenomenon.
Discrete spectra are seen in many other phenomena, such as vibrating strings, in a microwave cavity, in a pulsating star, and resonances in high-energy particle physics. The general phenomenon of discrete spectra in physical systems can be mathematically modeled with tools of functional analysis, specifically by the decomposition of the spectrum of a linear operator acting on a functional space.
Discrete spectra are also produced by some non-linear oscillators where the relevant quantity has a non- waveform. Notable examples are the sound produced by the vocal cords of mammals.
Hannu Pulakka (2005), [https://aaltodoc.aalto.fi/bitstream/handle/123456789/982/urn007925.pdf Analysis of human voice production using inverse filtering, high-speed imaging, and electroglottography]. Master's thesis, Helsinki University of Technology.
(2025). 9780387304465, Springer New York. ISBN 9780387304465 and the stridulation organs of crickets, whose spectrum shows a series of strong lines at frequencies that are integer multiples () of the oscillation frequency.
A related phenomenon is the appearance of strong harmonics when a sinusoidal signal (which has the ultimate "discrete spectrum", consisting of a single spectral line) is modified by a non-linear filter; for example, when a pure tone is played through an overloaded amplifier,Paul V. Klipsch (1969), Modulation distortion in loudspeakers Journal of the Audio Engineering Society. or when an intense monochromatic laser beam goes through a non-linear medium. In the latter case, if two arbitrary sinusoidal signals with frequencies f and g are processed together, the output signal will generally have spectral lines at frequencies , where m and n are any integers.
Discrete spectra are usually associated with systems that are bound state in some sense (mathematically, confined to a compact space). The position and momentum operators have continuous spectra in an infinite domain, but a discrete (quantized) spectrum in a compact domain and the same properties of spectra hold for angular momentum, Hamiltonians and other operators of quantum systems.
The quantum harmonic oscillator and the hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the hydrogen atom the spectrum has both a continuous and a discrete part, the continuous part representing the ionization.
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