keresztkorrelációs

Keresztkorreláció is a mathematical tool used to measure the similarity between two signals as a function of the displacement of one relative to the other. It is a fundamental concept in signal processing, statistics, and various scientific fields. The cross-correlation function, denoted by $R_{xy}(\tau)$, for two signals $x(t)$ and $y(t)$ is typically defined as the integral of the product of one signal with a time-shifted version of the other: $R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t)y(t+\tau)dt$. The variable $\tau$ represents the time lag or shift.

The value of the cross-correlation at a specific lag $\tau$ indicates how similar the two signals are when one is shifted by that amount. A high positive value suggests strong similarity, a high negative value indicates strong dissimilarity (or inverse relationship), and a value near zero suggests little to no correlation. For discrete-time signals, the integral is replaced by a summation.

Applications of keresztkorreláció are widespread. In engineering, it is used for pattern recognition, time delay estimation (e.g., in radar or sonar), and system identification. In image processing, it helps in image registration and object detection. In statistics, it can be used to analyze the relationship between two time series. The concept is also applied in fields like neuroscience to understand the synchronization of neural activity and in finance to study the relationship between different financial markets. The Wiener–Khinchin theorem relates cross-correlation to the cross-power spectrum, which is often more convenient for frequency-domain analysis.