Spectral density
Distribution of signal power over frequency components.
In signal processing, the power spectrum of a continuous time signal shows how the signal's power is spread across different frequency components. Fourier analysis tells us that any physical signal can be broken down into a continuous range of frequencies, with some power possibly concentrated at specific discrete frequencies. When we look at the statistical average of a signal's energy or power—including noise—based on its frequency content, that quantity is called its spectral density.
If a signal's energy is confined to a finite time interval and its total energy is finite, we can calculate its energy spectral density. More often used is the power spectral density (PSD, or simply power spectrum), which applies to signals that exist over all time or over a period long enough—especially relative to a measurement's duration—that it might as well be infinite. In that case, the total energy over all time would generally be infinite, so the PSD describes the spectral power distribution that would be observed. Summing or integrating the spectral components gives the total power (for a physical process) or variance (for a statistical process), matching what you'd get by integrating over time, as Parseval's theorem requires.
The spectrum of a physical process often holds key information about that process. For example, a musical instrument's pitch and timbre can be found through spectral analysis. The color of a light source comes from the spectrum of its electromagnetic wave's electric field as it oscillates at very high frequencies. Getting a spectrum from time series data like these involves the Fourier transform and related methods based on Fourier analysis. In many cases, the time domain isn't directly captured—for instance, when a dispersive prism produces a light spectrum in a spectrograph, or when a sound is perceived through the inner ear's auditory receptors, each sensitive to a particular frequency.
This article, however, focuses on situations where the time series is known (at least statistically) or directly measured (such as by a computer-sampled microphone). The power spectrum is important in statistical signal processing, the statistical study of stochastic processes, and many other physics and engineering fields.
- Si unit
- watt per hertz (W/Hz)
- Alternative unit (voltage)
- V²·Hz⁻¹
- Alternative unit (displacement)
- m²/Hz
- Alternative unit (acceleration)
- g₀²·Hz⁻¹
Lore & Background
The power spectral density (PSD) applies to signals existing over all time, or over a time period large enough that it could as well have been over an infinite time interval. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating over the time domain, as dictated by Parseval's theorem. The spectrum of a physical process often contains essential information about the nature of that process; for instance, the pitch and timbre of a musical instrument can be determined from a spectral analysis, and the color of a light source is determined by the spectrum of the electromagnetic wave's electric field as it oscillates at an extremely high frequency.
Obtaining a spectrum from time series data involves the Fourier transform and generalizations based on Fourier analysis. In many cases the time domain is not directly captured in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrograph, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency. The power spectrum is important in statistical signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics and engineering.
Reader's Guide
The concept of spectral density is fundamental to understanding how energy or power is distributed across frequencies in a signal. It bridges time-domain and frequency-domain representations, allowing engineers and scientists to analyze phenomena ranging from musical tones to cosmic microwave background radiation. The energy spectral density is suitable for transient, pulse-like signals with finite total energy, while the power spectral density is used for stationary processes that extend over all time. The Wiener–Khinchin theorem establishes that the energy spectral density and the autocorrelation of a signal form a Fourier transform pair. In practice, spectral density is measured by filtering a signal through a narrow bandpass filter and measuring the energy or power delivered, then dividing by the filter bandwidth. The choice between one-sided and two-sided representations depends on the field: noise PSDs are generally one-sided in engineering and two-sided in physics. The units vary with the physical nature of the signal, from watts per hertz for physical power to V²·Hz⁻¹ for voltage signals, and even m²/Hz for displacement data.
Did You Know?
- The spectral density of a fluorescent light shows peaks at atomic transitions.
- A voice waveform over time has a broad audio power spectrum.
- The power spectral density is commonly expressed in watts per hertz (W/Hz).
- The Wiener–Khinchin theorem relates energy spectral density to the autocorrelation of a signal.
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