Skip to content

Cross-spectrum

In signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain.

Version
v1 · 2026-09-28 · History
Domain-specific #
8792
Domain group
Applied Sciences & Engineering
Origin domain
Engineering & Design (beyond software)
Subdomains
Signal Processing, Spectral Analysis → Engineering & Design (beyond software)

Core Idea

Cross-spectrum is treated here as the recurring signal processing identity summarized by this source-grounded definition: In signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain. In signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain. It describes how the correlation between the two series is distributed over different frequencies.

How would you explain it like I'm…

Sounds Both Mics Heard

Imagine two microphones in the same room, each hearing lots of sounds. The Cross-spectrum looks at both recordings and finds which pitches, like a fridge's steady hum, show up in both. That helps you find sounds that come from the same place.

Shared Pitches in Two Signals

Signals like sounds can be split into different pitches, from low to high, called frequencies. The Cross-spectrum takes two signals and shows, for each frequency, how much the two signals are related. For example, if two microphones record the same room, the cross-spectrum can show that a hum at one pitch appears in both, hinting at a shared source. It helps scientists and engineers find what two signals have in common.

Frequency-by-Frequency Correlation

The Cross-spectrum is a tool in signal processing and statistics for studying how two time series are related, broken down by frequency. Instead of asking whether two signals move together overall, it shows how their correlation is spread over different frequencies. For instance, with two microphones in a room, it can highlight the frequencies of a sound, like an appliance hum, that are strong in both recordings. Technically, it is the Fourier transform of the cross-covariance function, which measures how the two signals relate at different time lags. So it takes a time-based relationship and rewrites it as a function of frequency.

 

The Cross-spectrum is a frequency-domain description of the relationship between two time series. Given series x and y with cross-covariance function gamma_xy(h), which measures how x at one time co-varies with y at a lag h, the cross-spectrum Gamma_xy is defined as the Fourier transform of gamma_xy. It therefore describes how the covariance between the two series is distributed across frequencies, rather than across time lags. Frequencies where the cross-spectrum is large indicate components prominent and co-varying in both series, as when two microphones in the same room both capture an appliance's hum, which helps identify common sources. Because the cross-covariance between two different series is generally not symmetric in the lag, the cross-spectrum is in general complex-valued, carrying both magnitude and phase information. The concept is specifically this Fourier-domain representation of two-series co-variation, not spectral analysis of a single signal.

Scope of Application

  • Definition. Let (Xt,Yt) represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions \gamma{xx} and \gamma{yy} and cross-covariance function \gamma{xy}.

  • Documented setting. In signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain.

  • Documented setting. This means it takes the relationship between the two signals over time and represents it as a function of frequency.

  • Documented setting. Technically, the cross-spectrum is the Fourier transform of the cross-covariance function.

  • Definition. Then the cross-spectrum \Gamma{xy} is defined as the Fourier transform of \gamma{xy}.

Clarity

A clear use of Cross-spectrum names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain.

Manages Complexity

Cross-spectrum compresses multiple signal processing details into a stable diagnostic relation. The source shows both the central mechanism—in signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain.—and the practical consequence—a{xy}(f)= (\Lambda{xy}(f)^2 + \Psi{xy}(f)2)\frac{1}{2} ,.

Abstract Reasoning

  1. Type the carrier. Identify the signal processing entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain.
  3. Check operation and conditions. Then the cross-spectrum \Gamma{xy} is defined as the Fourier transform of \gamma{xy}. 4.

Knowledge Transfer

Within the home domain. Knowledge about Cross-spectrum transfers literally when a new case preserves the same carrier type, relation, and recognition test. Let (Xt,Yt) represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions \gamma{xx} and \gamma{yy} and cross-covariance function \gamma{xy}. In signal processing and statistics, the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain. Beyond the home domain. No canonical parent is asserted for Cross-spectrum.

Neighborhood in Abstraction Space

Cross-spectrum sits in a moderately populated region (49th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08