Kreuzkorrelationsfunktion

The Kreuzkorrelationsfunktion, also known as the cross-correlation function, is a mathematical tool used to measure the similarity between two signals as a function of the time-lag applied to one of them. It is a fundamental concept in signal processing and time series analysis, providing insights into the relationship between two signals.

The cross-correlation function is defined as the expectation of the product of two signals, one of which is shifted in time. Mathematically, for two signals x(t) and y(t), the cross-correlation function Rxy(tau) is given by:

Rxy(tau) = E[x(t)y(t+tau)]

where E denotes the expectation operator, and tau is the time-lag. The cross-correlation function can be estimated from a finite number of samples using the sample cross-correlation function.

The cross-correlation function has several important properties. It is symmetric, meaning Rxy(tau) = Ryx(-tau), where Ryx(tau) is the cross-correlation function of y(t) and x(t). It is also linear, meaning that the cross-correlation of a linear combination of signals is the same linear combination of their cross-correlations.

The cross-correlation function is widely used in various applications, including system identification, signal detection, and time series analysis. In system identification, it is used to estimate the impulse response of a system. In signal detection, it is used to detect the presence of a known signal in noise. In time series analysis, it is used to analyze the relationship between two time series.

In summary, the Kreuzkorrelationsfunktion is a powerful tool for analyzing the relationship between two signals. It provides valuable insights into the similarity and dependency between signals, making it an essential concept in signal processing and time series analysis.