# What is the difference between cross correlation and persons r correlation for two different signal?

What is the difference between cross-correlation and pearsons r correlation for two different signals? It will be very grateful if somebody explains with an example

For example, assume two time series, $$x_t$$ and $$y_t$$, each with $$T$$ observations. Also, let $$\bar{x}$$ and $$\bar{y}$$ represent the sample means and let $$\sigma_x^2$$ and $$\sigma_y^2$$ represent the sample variance. The sample cross-correlation is obtained as follows:
$$\frac{\sum_t^T(x_t-\bar{x})\sum_s^T(y_s-\bar{y})}{\sigma_x\sigma_y}$$
Notice the subscripts, with the cross-variance I will compare $$x_t$$ to lags of $$y_{t-s}$$ and vice versa. As a result, with only two series the cross-correlation generates a $$T\text{x}T$$ matrix (i.e. the above sample cross variance will be entry ($$t$$,$$s$$) within the matrix). Pearson's correlation is actually contained within this cross-correlation matrix, i.e.
$$\frac{\sum_{t=1}^T(x_t-\bar{x})\sum_{t=1}^T(y_{t}-\bar{y})}{\sigma_x\sigma_y}$$