Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 39

Even more important than the key figures of one random process is the relationship between two different processes, the so-called correlation. It defines how much a random process is linearly linked to another process. Imagine two random processes f(t) and g(t). Without loss of generality we assume the mean values to be zero:

(1.140)

Let us consider that the process g is linked to f by a linear factor K:

The error or deviation of this assumption reads

(1.141)

or in terms of the mean square value

(1.142)

This can be rewritten as

(1.143)

This is a minimization problem and we search for the slope K that minimizes the sum of squared deviations J. This function has a quadratic dependence and can be rewritten as

(1.144)

where A=E[f2], B=−E[fg], and C=E[g2], with all three terms being real expected values from real random processes. Equation (1.143) is parabolic in shape, and the minimum is found by setting the first derivative with respect to K to zero.

(1.145)

Therefore, the point that minimizes J is given by K0=−B/A. In order to assure K₀ being a minimum we need d2J/dK2>0, meaning that A must be positive. This can be easily proven, as the expected value of the squared function E[f2] must be positive. If we substitute K₀ into Equation (1.143) we get the following relationships for K₀ and J₀

(1.146)

(1.147)

Using the definition of variances we can write J₀ in the case of zero mean processes in a non-dimensional form:

(1.148)

The quantity ρfg=E[fg]/σfσg is the normalized correlation coefficient correlation coefficient ! normalised between f and g. If both processes are perfectly correlated ρfg=1. If they are fully uncorrelated ρfg=0. In terms of the linear relationship from (1.141) all points would be perfectly on the line for full correlation and would be arbitrarily distributed for no correlation (Figure 1.22).

Figure 1.22 Example for correlation of random processes. No correlation (left) and different correlation values (right). Source: Alexander Peiffer.