Wiener–Khinchin theorem
E158222
The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Wiener–Khinchin theorem canonical | 2 |
| Wiener–Khinchin–Einstein theorem | 1 |
| Wiener–Khintchine theorem | 1 |
| autocorrelation theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1374548 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Wiener–Khinchin theorem Context triple: [Norbert Wiener, knownFor, Wiener–Khinchin theorem]
-
A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
B.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
-
C.
Fourier
Fourier is a French surname most famously associated with Jean-Baptiste Joseph Fourier, the mathematician and physicist known for developing Fourier analysis and Fourier series.
-
D.
Wigner distribution function
The Wigner distribution function is a quasi-probability distribution used in quantum mechanics and signal processing to represent quantum states in phase space, often exhibiting non-classical features such as negative values.
-
E.
Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Wiener–Khinchin theorem Target entity description: The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
-
A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
B.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
-
C.
Fourier
Fourier is a French surname most famously associated with Jean-Baptiste Joseph Fourier, the mathematician and physicist known for developing Fourier analysis and Fourier series.
-
D.
Wigner distribution function
The Wigner distribution function is a quasi-probability distribution used in quantum mechanics and signal processing to represent quantum states in phase space, often exhibiting non-classical features such as negative values.
-
E.
Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in probability theory ⓘ theorem in signal processing ⓘ |
| alsoKnownAs |
Wiener–Khinchin–Einstein theorem
ⓘ
linked to:
Wiener–Khinchin theorem
Wiener–Khintchine theorem ⓘ
linked to:
Wiener–Khinchin theorem
autocorrelation theorem ⓘ
linked to:
Wiener–Khinchin theorem
|
| appliesTo |
complex-valued random processes
ⓘ
real-valued random processes ⓘ stationary stochastic process ⓘ wide-sense stationary random process ⓘ |
| assumes | second-order moments exist ⓘ |
| concerns | second-order statistics of random processes ⓘ |
| domain |
continuous-time processes
ⓘ
discrete-time processes ⓘ |
| field |
harmonic analysis
ⓘ
probability theory ⓘ signal processing ⓘ stochastic processes ⓘ |
| generalizationOf | relationship between convolution and multiplication in Fourier domain ⓘ |
| historicalNote | results were developed independently by Norbert Wiener and Aleksandr Khinchin in the 1930s ⓘ |
| implies |
autocorrelation function is positive semidefinite
ⓘ
power spectral density is nonnegative ⓘ |
| inverseForm | R_X(τ) = ∫_{-∞}^{∞} S_X(f) e^{j2π f τ} df ⓘ |
| mathematicalForm | S_X(f) = ∫_{-∞}^{∞} R_X(τ) e^{-j2π f τ} dτ ⓘ |
| namedAfter |
Aleksandr Khinchin
ⓘ
Norbert Wiener ⓘ |
| relatedTo |
Bochner's theorem
ⓘ
linked to:
Bochner theorem on characteristic functions
Parseval's theorem ⓘ cross-correlation theorem ⓘ power spectrum ⓘ |
| relates |
autocorrelation function
ⓘ
power spectral density ⓘ |
| requires |
existence of autocorrelation function
ⓘ
integrability conditions for Fourier transform ⓘ wide-sense stationarity ⓘ |
| statement |
the autocorrelation function of a wide-sense stationary process is the inverse Fourier transform of its power spectral density
ⓘ
the power spectral density of a wide-sense stationary process is the Fourier transform of its autocorrelation function ⓘ |
| usedIn |
communications engineering
ⓘ
image processing ⓘ optics ⓘ radio astronomy ⓘ seismology ⓘ spectral analysis of random signals ⓘ statistical signal processing ⓘ time series analysis ⓘ |
| usesTransform | Fourier transform ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Wiener–Khinchin theorem Description of subject: The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Wiener–Khinchin theorem
linked to: Wiener–Khinchin theorem
linked to: Wiener–Khinchin theorem