Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 38

Imagine a random process creating signal sequences as shown in Figure 1.19. At each time the signal value f(t) may be different and has a certain continuous value. One option to characterize this signal is to define the probability that the signal value is less or equal to a specific value f_k. Thus we define a probability for f(t) to be less than or equal to f_k

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Figure 1.19 Stochastic fluctuation with time. Source: Alexander Peiffer.

Next, we are interested in the probability that the value of f(t) is in a range defined by Δf=f2−f1 meaning the probability Prob[f1<f≤f2]. Now we can define the probability density function as

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Consequently in the limit Δf→0 the probability function density p is defined by

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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

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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

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

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

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

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

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and the variance

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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

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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.