Structural Dynamics: Random Vibrations Flashcards
What is the definition of a deterministic process?
A process is deterministic in as far as it represents a mathematical description whose input parameters has all been predetermined and remains unchanged.
What is a non-deterministic process?
There are cases where the excitation signal in terms of value assumed in space and time is random and can only be defined through a probability of occurence. In this case the excitation is called non-deterministic, and it is defined through a stochastic process. The physical characteristics of a stochastic process are decribed by its statistical properties.
What is the definition of realization and ensemble/stochastic(random) process?
An observation of a stochastic process is called a realization, ie a sample signal non of which can, with certainty, be repeated even if the conditions are seemingly the same, and so it must be considered random.
The collection of all possible realizations is called ensemble or stochastic (random) process.
What are some statistical properties?
*The expected value, or ensemble or statistical average, commonly known as mean. The expression is called first statistical moment
*Variance and standard deviation
What is a Gaussian process (central limit theorem)?
Central limit theorem: When the effects of several independant random variables are summed together, the resulting stochasting distribution tends to be Gaussian (or Normal). The normal distribution is fully characterized by its mean and variance.
What is autocorrelation and cross-correlation?
There is expected to be a relationship between the value assumed by the process at time t1 and assumed at the time t2. The autocorrelation is defined as the correlation of a randon variable with itself at different instants in time. It allows us to understand how statistics of the time history of a variable is related to itself shifted in time.
It is possible to do the same using two different random variables X and Y and this is called cross-correlation. It is the correlation of a random variable with another random variable at different instants of time.
What is a stationary process?
The process is stationary, if taken the signals Χ at different times plus a time shift ε, the probability density coincides with the signals X at times ti. The probability density is independent of a shift of the time origin, but also the mean and variance does not depend on time anymore.
For two stationary signals, the autocorrelation and the cross-correlation depends only on the time shift interval τ.
What is an ergodic process?
The determination of the statistics from an ensemble poses some practical problems related to the repetition of experiments. It is often difficult to have sufficient repetitions to obtain accurate estimation of probability densitities.
However, if the process is stationary, it seems reasonable to take one realization and replace ensemble averages with time averages. A stochastic process for which this exchange is possible is called ergodic.
A stochastic process X(t) is said to be ergodic, if any characteristic of the process can be obtained, with probability 1, from a single realization x(t) of the process using the temporal mean operator as the expectation operator.
What are the properties of autocorrelation for a stationary process?
*The autocorrelation is symmetric wrt to the y axis.
*The autocorrelation at zero time shift is equal or larger than the autocorrelaton at other time shifts. It is maximum at τ=0 becase the signal is autocorrelated with itself
*It decays in a monotonous or oscillatory way
What are some properties of auto-covariance and cross-covariance?
*The autocovariance is the equivalent of taking the autocorrelation and substracting the mean of the signal. The autocovariance at τ=0 is equal to the variance.
*Coherence is the cross-covariance devided by the two variances of the signals and shows the correlation between two signals. If ρxy = 0, the signals are uncorrelated.
How can autocorrelation be used to de-noise a signal?
If the autocorrelation is used to a signal y(t) = x(t) + n(t), where n is a noise and we take the autoorrelation, ultimately Ryy=Rxx, because RNN goes to 0 rapidly, while RNX goes to 0 as N and X are uncorrelated.
What is the Power Spectral Density (PSD)?
The frequency content of an ergodic signal may be found looking at the Fourier transform of the autocovariance. So the PSD ΦΧΧ is the Fourier transform of the autocovariance. Since it is related to the square of a signal, it is considered a measure of the energy content. The area below the PSD is proportional to the variance of the signal.
The more random is the signal, the wider is the area where the PSD is not zero and the narrower will be the auto-covariance. The lower is the number of frequencies of the signal, the more similar the PSD will be to a sinusoidal signal.
σ^2χ = 1/π SΦΧΧdω
What is white noise?
White noise is a process with uniform Power Spectral Density? The autocovariance is the Dirac delta. This process is not physically realizable, because there is an infinite area (power) under the spectrum).
Also band-limited white noise exists, also called ideal low-pass filter, where the PSD goes to zero outside a bound +-w0.
What is the relation between autocovariance and variance?
*The more rapidly the autocovariance decays with τ, the less the shifted time histories are similar.
*The more rapidly the autocovariance decays, the more casual or chaotic is the signal.
*The autocovariance allows to characterize exactly how a stochastic process behaves, in a determinstic way throughh a function.
