Cross correlation formula discrete

1. What is the formula for cross-correlation?
2. What is cross-correlation in time series?
3. How do you find cross-correlation with FFT?

What is the formula for cross-correlation?

Cross-correlation between X_i and X_j is defined by the ratio of covariance to root-mean variance, ρ i , j = γ i , j σ i 2 σ j 2 . γ ^ i , j = 1 N ∑ t = 1 N [ ( X i t − X ¯ i ) ( X j t − X ¯ j ) ] .

What is cross-correlation in time series?

Cross correlation presents a technique for comparing two time series and finding objectively how they match up with each other, and in particular where the best match occurs. It can also reveal any periodicities in the data.

How do you find cross-correlation with FFT?

We can compute correlations using the FFT as follows: FFT the two data sets, multiply one resulting transform by the complex conjugate of the other, and inverse transform the product. The result (call it rk) will formally be a complex vector of length N.