math concept
17 topics use this
Math concept
Spectral Analysis
Core equation
$$S(\omega) = \int_{-\infty}^\infty R(\tau)\,e^{-i\omega\tau}\,d\tau$$
Spectral analysis decomposes signals and random processes into their frequency components. The power spectral density, periodogram, and Welch's method are its main tools — connecting signal processing, time series, and quantum mechanics.
Power spectral density (PSD)
For a WSS process, the PSD is the Fourier transform of the autocorrelation:
\[S(\omega) = \mathcal{F}\{R(\tau)\}(\omega)\]Parseval’s theorem: $\int S(\omega)\,d\omega = R(0) = \mathbb{E}[X_t^2]$ (total power).
Periodogram estimation
Given $N$ samples, the periodogram estimates the PSD:
\[\hat{S}(\omega) = \frac{1}{N}\left|\sum_{n=0}^{N-1} x[n]e^{-i\omega n}\right|^2\]The periodogram is asymptotically unbiased but not consistent — variance does not decrease with $N$.
Welch’s method
Average periodograms of overlapping windowed segments to reduce variance:
- Divide signal into $K$ overlapping segments of length $M$
- Apply a window function (Hann, Hamming) to each
- Average the $K$ periodograms
Reduces variance by $\approx K$ at the cost of frequency resolution.
Fields that use this concept
Physical sciences
Astrophysics
Cosmic Microwave Background
The thermal radiation relic of the hot Big Bang, encoding the state of the universe 380,000 years after its formation.
Gravitational Waves
Ripples in spacetime curvature propagating at the speed of light, produced by accelerating asymmetric mass distributions.
Life sciences
Bioinformatics
Earth sciences
Climate modeling
Earth sciences
Geophysics
Earthquake Seismology
Moment tensors, seismic moment, and the Gutenberg-Richter law characterise earthquake sources and their statistical recurrence.
Ground-Penetrating Radar
Electromagnetic wave propagation and reflection imaging reveal shallow subsurface structure with centimetre-scale resolution.
Seismic Waves
P and S waves, the elastic wave equation, and Snell's law govern how seismic energy travels through the Earth.
Earth sciences
Meteorology
Atmospheric Waves
Rossby waves, gravity waves, and their dispersion relations that govern large-scale energy propagation in the atmosphere.
Atmospheric Boundary Layer
The turbulent lowest kilometre of the atmosphere that mediates exchanges of heat, moisture, and momentum between the surface and free troposphere.
Numerical Weather Prediction
How finite-difference and spectral methods solve the atmospheric PDEs that drive operational weather forecasts.
Weather Radar
How radar equations, reflectivity, Doppler velocity, and dual-polarisation moments are used to observe precipitation and storms.
Engineering & CS
Signal processing
Digital Filter Design
Methods for designing FIR and IIR digital filters to meet prescribed frequency-domain specifications.
Matched Filter
The optimal linear filter for detecting a known signal in additive white Gaussian noise, maximizing output signal-to-noise ratio.
Nyquist-Shannon Sampling Theorem
The fundamental theorem establishing the minimum sampling rate required to perfectly reconstruct a bandlimited continuous signal.
Power Spectral Density
Describes how the power of a random signal is distributed across frequencies, estimated via periodogram and spectral averaging methods.
Wavelet Transform
A time-frequency analysis tool that overcomes the fixed resolution of the STFT by using dilated and translated basis functions.
Z-Transform
The discrete-time counterpart to the Laplace transform, enabling algebraic analysis of digital filters and LTI systems.