crosscorrelatiefuncties

Crosscorrelation functions are mathematical tools used to analyze the relationship between two signals or time series. They are widely used in signal processing, time series analysis, and various fields of engineering and science. The crosscorrelation function measures the degree to which two signals are linearly related. It is defined as the expectation of the product of the two signals at different time lags.

The crosscorrelation function Rxy(tau) between two signals x(t) and y(t) is given by:

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

where E denotes the expectation operator, mu_x and mu_y are the means of x(t) and y(t) respectively, and tau is the time lag. The crosscorrelation function can be estimated from a finite number of samples using the sample crosscorrelation function.

Crosscorrelation functions have several important properties. They are symmetric, meaning Rxy(tau) = Ryx(-tau). They also have a maximum value at tau = 0 when the two signals are perfectly correlated. Crosscorrelation functions are used in various applications, such as system identification, signal detection, and synchronization. They are particularly useful when dealing with noisy signals, as they can help to identify the underlying relationship between the signals despite the presence of noise.