# Spectral density

The spectral density of a signal is a way of measuring the strength of the different frequencies that form the signal. For example, if we had a sound wave from a piano in which the keys of middle C and A were struck, then the pressure variations making up the sound wave would be the signal and "middle C and A" are in a sense the spectral density of the sound signal. Mathematically, if the notes were pure sinusoids (produced by something like a tuning fork; not a piano), the spectral density would be a function of sound frequency, with two spikes at 261.6 Hz and 440 Hz, corresponding to the frequencies of middle C and A.

The spectral density is a general concept applied to a signal which may have any physical dimensions or none at all. In physics, the signal is usually a wave, such as an electromagnetic wave, or an acoustic wave. The spectral density of the wave, when multiplied by an appropriate factor, will give the power carried by the wave, usually per unit frequency or per unit wavelength. 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.

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## Definition

If [itex]f(t)[itex] is a signal, the spectral density [itex]\Phi(\omega)[itex] of the signal is the square of the magnitude of the continuous Fourier transform of the signal.

[itex]\Phi(\omega)=\left|\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt\right|^2 = F(\omega)F^*(\omega)[itex]

where [itex]\omega[itex] is the angular frequency ([itex]2\pi[itex] times the cyclic frequency) and [itex]F(\omega)[itex] is the continuous Fourier transform of [itex]f(t)[itex]. If the signal is discrete with components [itex]f_n[itex], we may approximate [itex]f(t)[itex] by:

[itex]f(t)\approx\sum_n f_n \delta(t-n)\,[itex]

where [itex]\delta(x)[itex] is the Dirac delta function and the sum over n may be over a finite or infinite number of elements. If the number is infinite we have:

[itex]\Phi(\omega)=\left|\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^\infty f_n e^{-i\omega n}\right|^2=\frac{F(\omega)F^*(\omega)}{2\pi}[itex]

where [itex]F(\omega)[itex] is the discrete-time Fourier transform of [itex]f_n[itex]. If the number is finite (=N) we may define [itex]\omega=2\pi m/N[itex] and:

[itex]\Phi_m=\left|\frac{1}{\sqrt{2\pi}}\sum_{n=0}^{N-1} f_n e^{-2\pi i mn/N}\right|^2=\frac{F_mF^*_m}{2\pi}[itex]

where [itex]F_m[itex] is the discrete Fourier transform of [itex]f_n[itex]. As is always the case, the multiplicative factor of [itex]1/2\pi[itex] is not absolute, but rather depends on the particular normalizing constants used in the definition of the various Fourier transforms.

The spectral density of a signal exists if and only if the signal is stationary. If the signal is not stationary then the same methods used to calculate the spectral density can still be used, but the result cannot be called the spectral density.

## Properties

• One of the results of Fourier analysis is Parseval's theorem which states that the area under the spectral density curve is equal to the area under the square of the magnitude of the signal:
[itex]\int_{-\infty}^\infty \left| f(t) \right|^2 dt = \int_{-\infty}^\infty \Phi(\omega)\,d\omega.[itex]
The above theorem holds true in the discrete cases as well.
• Note that the signal f(t) cannot be recovered from the spectral density. The phase information from the signal is lost, as well as the temporal information.
• The spectral density of [itex]f(t)[itex] and the autocorrelation of [itex]f(t)[itex] form a Fourier transform pair.
• The spectral density is usually calculated using the Fourier transform, but other techniques such as Welch's method and the maximum entropy method can also be used.

## Related concepts

• The spectral centroid of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.
• Spectral density is a function of frequency, not a function of time. However, the spectral density of small "windows" of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a spectrogram. This is the basis of a number of spectral analysis techniques such as the short-time Fourier transform and wavelets.

## Applications

### Colorimetry

The power spectral density of a light source is a measure of the power carried by each frequency or "color" in a light source. It is usually measured at points (usually 31) along the visible spectrum. Some spectrophotometers can measure increments as fine as 1 or 2 nanometers. Values are used to calculate other specifications and then plotted to demonstrate the spectral attributes of the source. This can be a helpful tool in analyzing the color characteristics of a particular source.

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