Keresztkorrelációban

Keresztkorrelációban, often translated as cross-correlation, is a measure of similarity of two waveforms as a function of the displacement of one relative to the other. It is a mathematical operation that takes two signals, say f(t) and g(t), and produces a third signal which expresses how much the two signals resemble each other at different time lags. The cross-correlation function R_fg(τ) is defined as the integral of the product of one signal with a time-shifted version of the other signal. Mathematically, it is expressed as R_fg(τ) = ∫ f(t)g(t-τ)dt.

The value of the cross-correlation function at a particular time lag τ indicates the degree of similarity between the two signals when one is shifted by τ units relative to the other. A high positive value suggests strong similarity, a high negative value indicates strong dissimilarity (or an inverted similarity), and a value close to zero suggests little to no correlation.

Cross-correlation has numerous applications in various fields. In signal processing, it is used for tasks such as pattern recognition, time delay estimation, and noise reduction. For example, it can be used to find a known signal within a noisy signal or to determine the time difference between two recorded versions of the same signal. In image processing, cross-correlation is employed for template matching, where a small template image is searched for within a larger image. In astronomy, it is used to compare different celestial observations. In economics and finance, it can be used to analyze the relationship between different time series. The concept extends to discrete signals where the integral is replaced by a summation.