Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

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)

In Figure 1.20 the examples for above defined functions are depicted. Those distinct ways of averaging reveal that the different averaging methods must be described in more detail. Until now averaging was performed over time intervals. This must not be confused with averaging over an ensemble. average averaging means averaging over an ensemble of experiments, systems, or even random signals. It will be denoted by ⟨⋅⟩E. Ensemble averaging can be similar to time averaging but this is only valid for specific time signals or random processes.

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

In Figure 1.21 the differences between time and ensemble averaging are shown. On the left hand side (ensemble) we perform a large set of experiments and take the value at the same time t₁, on the right hand side we perform one experiment but investigate sequent time intervals.

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 N_k discrete results f_k 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

(1.137)

We come back to the difference between ensemble and time averaging as shown in Figure 1.21. A process is called ergodic when the ensemble averaging can be replaced by time averaging, thus

(1.138)

(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:

The error or deviation of this assumption reads

delta equals g minus upper K f (1.141)

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

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