Smoothing and Filtering a Signal

Given an input series i, 2, , and an output series, y , the cross covariance 

In this equation c (0) and c (O) are the values of the auto covariance at lag 0 for the input and the output series. 

Values of ruy(k) near to 1 or -1, imply high dependency, low values imply independence. In examining the cross correlation of a certain data set a boundary can be chosen below which independence is assumed. The value of the confidence limit for the cross correlation of process data is usually in the order of 0.15 to 0.3, depending on the number of points in the data set. 

To develop an accurate model, the signal to noise ratio should be more than five. This means that changes in a particular process variable should be at least five times the value of the process noise for that variable. If this is not the ease, one could use a smoothing or filtering technique to suppress the noise. 

The following classifications concerning signal smoothing and filtering can be made  

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