Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

upper E left-bracket g squared right-bracket plus upper K squared upper E left-bracket f squared right-bracket minus 2 upper K upper E left-bracket f g right-bracket"/> (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

upper J equals upper K squared upper A plus 2 upper B upper K plus upper C (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.

StartFraction d upper J Over d upper K EndFraction equals 2 upper A upper K plus 2 upper B equals 0 (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

StartLayout 1st Row 1st Column upper K 0 2nd Column equals negative upper B slash upper A equals StartFraction upper E left-bracket f g right-bracket Over upper E left-bracket f squared right-bracket EndFraction EndLayout (1.146)

StartLayout 1st Row 1st Column upper J 0 2nd Column equals upper C minus upper B squared slash upper A equals upper E left-bracket g squared right-bracket minus StartFraction upper E squared left-bracket f g right-bracket Over upper E left-bracket f right-bracket EndFraction EndLayout (1.147)

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

StartFraction upper J 0 Over sigma Subscript g Superscript 2 Baseline EndFraction equals 1 minus left-parenthesis StartFraction upper E squared left-bracket f g right-bracket Over sigma Subscript f Baseline sigma Subscript g Baseline EndFraction right-parenthesis equals 1 minus rho Subscript f g (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.

1.5.3 Correlation Functions for Random Time Signals

In the above considerations we have taken the values from an ensemble of random processes or signals taken at t₁. We can also define a correlation coefficient for values taken from two processes at different times t₁ and t₂:

rho Subscript f g Baseline left-parenthesis t 1 comma t 2 right-parenthesis equals StartFraction upper E left-bracket f left-parenthesis t 1 right-parenthesis g left-parenthesis t 2 right-parenthesis right-bracket Over sigma Subscript f Baseline sigma Subscript g Baseline EndFraction (1.149)

The numerator is called the cross correlation function cross correlation:

upper R Subscript f g Baseline left-parenthesis t 1 comma t 2 right-parenthesis equals upper E left-bracket f left-parenthesis t 1 right-parenthesis g left-parenthesis t 2 right-parenthesis right-bracket period (1.150)

If the two processes are stationary the value of the cross correlation function depends only on the distance between the two times, i.e. t2=t1+τ. τ is called the lag or separation between the two time samples and we can write:

upper R Subscript f g Baseline left-parenthesis tau right-parenthesis equals upper E left-bracket f left-parenthesis t right-parenthesis g left-parenthesis t plus tau right-parenthesis right-bracket (1.151)

It also makes sense to correlate the function f(t) with itself at later moments f(t+τ). This is called the autocorrelation function defined by:

upper R Subscript f f Baseline left-parenthesis tau right-parenthesis equals upper E left-bracket f left-parenthesis t right-parenthesis f left-parenthesis t plus tau right-parenthesis right-bracket (1.152)

This function will later enable us to describe the spectrum of random functions. At τ = 0 the value is known as variance of f(t) as given by Equation (1.137):

upper R Subscript f f Baseline left-parenthesis 0 right-parenthesis equals upper E left-bracket f squared left-parenthesis t right-parenthesis right-bracket (1.153)

The autocorrelation is symmetric in time, proven by:

(1.154)

In addition some useful properties can be derived for the cross correlation function

(1.155)

So we get finally

upper R Subscript f g Baseline left-parenthesis tau right-parenthesis equals upper R Subscript g f Baseline left-parenthesis negative tau right-parenthesis (1.156)

For the stationary ergodic process we can replace the ensemble averaging by the average over time

Скачать книгу