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In applied mathematics and physics, the spectral density, power spectral density, or energy spectral density is a general concept applied to a signal which may have physical dimensions such as power per Hz, or energy per Hz, or none at all.
Explanation In physics, the signal is usually a wave, such as an electromagnetic wave, random vibration, or an acoustic wave. The spectral density of the wave, when multiplied by an appropriate factor, will give the power carried by the wave, per unit frequency. This is then known as the power spectral density (PSD) or spectral power distribution (SPD) of the signal. The units of spectral power density are commonly expressed in watts per hertz (W/Hz) or watts per nanometer (W/nm) (for a measurement versus wavelength instead of frequency). Although it is not necessary to assign physical dimensions to the signal or its argument, in the following discussion the terms used will assume that the signal varies in time. Energy spectral density The energy spectral density describes how the energy (or variance) of a signal or a time series is distributed with frequency. If is a finite-energy (square integrable) signal, the spectral density of the signal is the square of the magnitude of the continuous Fourier transform of the signal. ight|^2 = rac where is the angular frequency ( times the cycle frequency) and is the continuous Fourier transform of . If the signal is discrete with components , over an infinite number of elements, we still have an energy spectral density: ight|^2= rac where is the discrete-time Fourier transform of . If the number of defined values is finite, the sequence can be treated as periodic, using a discrete Fourier transform to make a discrete spectrum, or it can be extended with zeros and a spectral density can be computed as in the infinite-sequence case. As is always the case, the multiplicative factor of is not absolute, but rather depends on the particular normalizing constants used in the definition of the various Fourier transforms. Power spectral density The above definitions of energy spectral density require that the Fourier transforms of the signals exist, that is, that the signals are square-integrable or square-summable. An often more useful alternative is the power spectral density (PSD), which describes how the power of a signal or time series is distributed with frequency. Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener–Khinchin theorem provides a simple alternative. The PSD is the Fourier transform of the autocorrelation function of the signal if the signal can be treated as a stationary random process. The power spectral density of a signal exists if and only if the signal is a wide-sense stationary process. If the signal is not stationary, then the autocorrelation function must be a function of two variables, so no PSD exists, but similar techniques may be used to estimate a time-varying spectral density. Properties ight|^2 dt = int_^infty Phi(omega),domega. The above theorem holds true in the discrete cases as well. A similar result holds for the total power in a power spectral density being equal to the corresponding mean total signal power, which is the autocorrelation function at zero lag. Related concepts Electronics engineering The concept and use of the power spectrum of a signal is fundamental in electronic engineering, especially in electronic communication systems (e.g. radio & microwave communications, radars, and related systems). Much effort had been made and millions of dollars spent on developing and producing electronic instruments called "spectrum analyzers" for aiding electronics engineers, technologists, and technicians in observing and measuring the power spectrum of electronic signals. The cost of a spectrum analyzer varies according to its bandwidth and its accuracy. The top quality instruments cost over $100,000. The spectrum analyzer measures essentially the magnitude of the short-time Fourier transform (STFT) of an input signal. If the signal being analyzed is stationary, the STFT is a good smoothed estimate of its power spectral density. Colorimetry
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