Vibroacoustic Simulation - Alexander Peiffer

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rel="nofollow" href="#fb3_img_img_a2ea5b07-701c-50a8-8159-39e0a634b1cb.png" alt="p left-parenthesis f right-parenthesis equals StartFraction d upper P left-parenthesis f right-parenthesis Over d f EndFraction"/> (1.131)

      On the other side we can reconstruct the probability to be in the range of f1 to f2 by integrating over the probability density function

       Prob left-bracket f 1 less-than f less-than-or-equal-to f 2 right-bracket equals integral Subscript f 1 Superscript f 2 Baseline p left-parenthesis f right-parenthesis d f equals upper P left-parenthesis f 2 right-parenthesis minus upper P left-parenthesis f 1 right-parenthesis (1.132)

      Figure 1.20 Probability and probability density function of a continuous random process. Source: Alexander Peiffer.

      Figure 1.21 Ensemble and time averaging of signals from random processes. Source: Alexander Peiffer.

      Consider now the mean value of an ensemble of N experiments. The mean value is defined by

       mathematical left-angle f mathematical right-angle Subscript upper E Baseline equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts f Subscript i (1.133)

      If we assume Nk discrete results fk that occur with frequency rk=nk/N we can also write

mathematical left-angle f mathematical right-angle Subscript upper E Baseline equals StartFraction 1 Over upper N Subscript k Baseline EndFraction sigma-summation Underscript k equals 1 Overscript upper N Subscript k Baseline Endscripts f Subscript k Baseline r Subscript k

      In a continuous form this can be expressed as rk=p(fk)Δfk

       mathematical left-angle f mathematical right-angle Subscript upper E Baseline equals sigma-summation Underscript k equals 1 Overscript upper N Subscript k Baseline Endscripts f Subscript k Baseline p left-parenthesis f Subscript k Baseline right-parenthesis normal upper Delta f Subscript k (1.134)

      For fk→f we get the definition of the based on ensemble averaging and expressed as the integral over the probability density.

       StartLayout 1st Row upper E left-bracket f right-bracket equals mathematical left-angle f mathematical right-angle Subscript upper E Baseline equals integral Subscript negative normal infinity Superscript normal infinity Baseline f p left-parenthesis f right-parenthesis d f EndLayout (1.135)

      Similar to the expression for time signal rms-value, we define in addition the expected mean square value

       StartLayout 1st Row upper E left-bracket f squared right-bracket equals mathematical left-angle f squared mathematical right-angle Subscript upper E Baseline equals integral Subscript negative normal infinity Superscript normal infinity Baseline f squared p left-parenthesis f right-parenthesis d f EndLayout (1.136)

      and the variance

       StartLayout 1st Row 1st Column upper E left-bracket f right-bracket 2nd Column equals upper E left-bracket f left-parenthesis t right-parenthesis right-bracket equals limit Underscript upper T right-arrow normal infinity Endscripts StartFraction 1 Over upper T EndFraction integral Subscript negative upper T slash 2 Superscript upper T slash 2 Baseline f left-parenthesis t right-parenthesis d t EndLayout (1.138)

       StartLayout 1st Row 1st Column upper E left-bracket f squared right-bracket 2nd Column equals upper E left-bracket f squared left-parenthesis t right-parenthesis right-bracket equals limit Underscript upper T right-arrow normal infinity Endscripts StartFraction 1 Over upper T EndFraction integral Subscript negative upper T slash 2 Superscript upper T slash 2 Baseline f squared left-parenthesis t right-parenthesis d t EndLayout (1.139)

      We are usually not able to perform an experiment for an ensemble of similar but distinct experimental set-ups, but we can easily record the signals over a long time and take several separate time windows out of this signal.

      1.5.2 Correlation Coefficient

      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:

       upper E left-bracket f right-bracket equals upper E left-bracket g right-bracket equals 0 (1.140)

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

dollar-sign g equals upper K f

      The error or deviation of this assumption reads

      or in terms of the mean square value

       upper J equals upper E left-bracket delta squared right-bracket equals upper E left-bracket left-parenthesis g minus upper K f right-parenthesis squared right-bracket (1.142)

      This can be rewritten as