power spectral density — from formula anatomyPower spectral density (PSD) describes how the power of a signal or time series is distributed across different frequencies. Instead of telling you the total power, it tells you how much power sits in each frequency band. This is essential for analyzing anything from sound waves to radio transmissions to quantum noise, because it reveals the underlying periodicities and random fluctuations that a simple average would hide.
The concept was formalized in the early 20th century, with Norbert Wiener (1930) and Alexander Khinchin (1934) independently proving the autocorrelation–PSD relationship. It became a standard tool in electrical engineering and physics after the rise of spectral analysis in the 1940s.
For a signal x(t), the PSD is computed via the Fourier transform: it is the squared magnitude of the transform, often normalized by time. For stationary random processes, the Wiener–Khinchin theorem states that the PSD is the Fourier transform of the autocorrelation function. In practice, you take a time series, break it into segments, compute their Fourier transforms, and average the squared magnitudes to reduce noise.
Used in telecommunications to design filters, in astrophysics to analyze cosmic microwave background fluctuations, in acoustics to characterize noise, and in quantum optics to study photon statistics.
The PSD of white noise is flat — it has equal power at all frequencies, which is why it sounds like a hiss and looks like a straight line on a spectrum analyzer.
| S_I | variable | power spectral density of current noise — power spectral density ampere squared per hertz |
| e | variable | elementary charge — elementary charge coulomb |
| I | variable | average electric current — electric current ampere |
| \text{ASD}(f) | variable | Amplitude spectral density at frequency f — amplitude spectral density per root hertz |
| S(f) | variable | Power spectral density at frequency f — power spectral density per hertz |
| f | variable | Frequency — frequency hertz |
| \sqrt{} | operator | Mathematical operation taking the square root — square root |
| S(f) | variable | Power spectral density as a function of frequency — power spectral density watt per hertz |
| f | variable | Frequency — frequency hertz |
| β | variable | Spectral exponent, typically near 1 |
| C_{xy}(f) | variable | Magnitude squared coherence at frequency f — coherence |
| P_{xy}(f) | variable | Cross-spectral density between signals x and y at frequency f — cross spectral density watt per hertz |
| P_{xx}(f) | variable | Power spectral density of signal x at frequency f — power spectral density watt per hertz |
| P_{yy}(f) | variable | Power spectral density of signal y at frequency f — power spectral density watt per hertz |
| | \cdot | | operator | Magnitude (absolute value) of a complex number — absolute value |
