Spectral density in the context of "Parseval's theorem"

A spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies with time. When applied to an audio signal, spectrograms are sometimes called sonographs, voiceprints, or voicegrams. When the data are represented in a 3D plot they may be called waterfall displays.

Spectrograms are used extensively in the fields of music, linguistics, sonar, radar, speech processing, seismology, ornithology, and others. Spectrograms of audio can be used to identify spoken words phonetically, and to analyse the various calls of animals.

In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

Some SDE techniques assume that a signal is composed of a limited (usually small) number of generating frequencies plus noise and seek to find the location and intensity of the generated frequencies. Others make no assumption on the number of components and seek to estimate the whole generating spectrum.

In condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or states per unit energy range. The density of states is defined as ${\displaystyle D(E)=N(E)/V}$, where ${\displaystyle N(E)\delta E}$ is the number of states in the system of volume ${\displaystyle V}$ whose energies lie in the range from ${\displaystyle E}$ to ${\displaystyle E+\delta E}$. It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation.

Generally, the density of states of matter is continuous. In isolated systems however, such as atoms or molecules in the gas phase, the density distribution is discrete, like a spectral density. Local variations, most often due to distortions of the original system, are often referred to as local densities of states (LDOSs).

Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the voltage or current noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment (such as radio receivers) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise is proportional to absolute temperature, so some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to improve their signal-to-noise ratio. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

Thermal noise in an ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum (Figure 2). When limited to a finite bandwidth and viewed in the time domain (as sketched in Figure 1), thermal noise has a nearly Gaussian amplitude distribution.

SMACS J0723.3–7327, commonly referred to as SMACS 0723, is a galaxy cluster about 4 billion light years from Earth, within the southern constellation of Volans (RA/Dec = 110.8375, −73.4391667). It is a patch of sky visible from the Southern Hemisphere on Earth and often observed by the Hubble Space Telescope and other telescopes in search of the deep past. It was the target of the first full-color image to be unveiled by the James Webb Space Telescope (JWST), imaged using NIRCam, with spectra included, showing objects lensed by the cluster with redshifts implying they are 13.1 billion years old. The cluster has been previously observed by the Hubble Space Telescope (HST) as part of the Southern MAssive Cluster Survey (SMACS), as well as Planck and Chandra.

In 2022, in the field gravitationally lensed by SMAC 0723, some of the then most ancient massive star clusters were discovered in a lensed galaxy called the "Sparkler Galaxy".