Vibroacoustic Simulation - Alexander Peiffer

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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)

       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)

      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)

      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):

      The autocorrelation is symmetric in time, proven 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 equals upper E left-bracket f left-parenthesis t prime minus tau right-parenthesis f left-parenthesis t prime right-parenthesis right-bracket equals upper R Subscript f f Baseline left-parenthesis negative tau right-parenthesis (1.154)

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

       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 equals upper E left-bracket f left-parenthesis t prime minus tau right-parenthesis g left-parenthesis t Superscript prime Baseline right-parenthesis right-bracket equals upper R Subscript g f Baseline left-parenthesis negative tau right-parenthesis period (1.155)

      So we get finally

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