Appunti Measurements

Signal analysis in time domain

Key indexes for signal analysis

We introduce some important indexes which allow us to better understand the physical features of a signal: the first index is the mean value, that for continuous signals is defined as:

1 = lim ∫ () ⅆ →∞ 0

If instead we are dealing with discrete signals, the expression becomes:

1 = ∑ =1

The mean value is a parameter that expresses the central tendency of a signal.

Another important index is the mean square value, defined as:

12 2 () = lim ∫ ⅆ→∞ 0

In this case, the signal is being squared. The mean square value has an important physical meaning, since it represents the power content of the signal, and so, the power content of the physical phenomenon that we are considering.

Root mean square value (RMS)

Most of the times, we do not consider the number given by the mean square value, but we consider the Root Mean Square value (RMS), defined as:

12 2√ ()√ = = lim ∫ ⅆ→∞ 0

The RMS is again an index of the power content of the signal, but it isn’t actually a power, since we have the square root. We prefer to use the RMS instead of the mean square value, because it has the same unit of measure of the signal, so, under a physical point of view, it can be understood easily.

Variance and standard deviation

Another important parameter is the variance, defined as:

12 2(() ) = lim ∫ − ⅆ→∞ 0

In the variance, we are squaring the distance between the signal and the mean value. This parameter tells us how much the data are “distant” from the mean value. If we square the variance we obtain the standard deviation. The variance is related to the RMS and the mean square value, through the following relation:

2 2 2 = −

The expression of the variance can be re-written as:

12 2 2() = lim [∫ ⅆ + ∫ ⅆ − 2 ∫ () ⅆ ]→∞ 0 0 0

12 2 2() = lim [∫ ⅆ + ∫ ⅆ − 2 ∫ () ⅆ ]→∞ 0 0 0

1 12 2 2() = lim ∫ ⅆ + lim ∙ − 2 ⋅ ̅ ̅ →∞ →∞0

Another important parameter that must be taken into account is the crest factor, defined as:

max|( − )| =

Application of the crest factor

Let's start seeing the crest factor index and its use. A typical example of the use of the crest factor is in rotating machinery based on gears. In these machines, due to the rotation of the components, the whole machine is subject to vibrations and the frequency of the vibrations is linked to the frequency of rotation. In order to simplify the example, we suppose that the vibration is made from a singular harmonic component. On the left we can see, as an example, the vibration in the case where in the machine everything is fine, which means that there are no damages, and everything is working properly. As expected, we have a single harmonic component.

Conversely, on the right, we can see the vibration profile in the case in which one tooth of the gear is damaged. This damage results in the presence of a spike in the signal. The spike can be considered as the typical feature which can be found in the vibration profile of a machine where we have a damaged tooth. Looking at the vibration signal, it is straightforward to recognize when we have the damage. However, we have the problem that it is not possible to have a man in front of a computer looking at the signals all the time. Therefore, we must find an automatic approach for detecting the presence of the damaged tooth. This can be automatically performed by a computer. In this case, the dedicated software must continuously monitor the crest factor and when it increases it means that a damage has occurred. Thus, using the crest factor it is possible to make the process automatic without the need of the presence of humans in front of the computer screen. The crest factor is sensitive to the spike and so to the damage, and it is a very good feature to be calculated in order to automatically detect the presence of a damaged tooth, since the factors that we have previously seen are not so sensitive to the presence of spikes. It is important to notice that we cannot use just the value of the signal to detect the presence of spikes: in fact, the rotation speed typically is not constant, and so, the amplitude of the spike will decrease as the angular speed decreases. So, if we just consider the value of the signal, the software could confuse an increase of the amplitude caused by the angular speed, with the presence of a damaged tooth. Instead, with the crest factor, we normalize the amplitude with respect to the RMS, and so, we are avoiding these problems.

Probability density function

Another important index that we have to consider is the probability density function, defined as:

( + Δ) − ()() = lim ΔΔ→0

The probability density function tells us the probability that the signal assumes the value x. Where is the cumulative probability function, which represents the probability that at a given time instant t, the signal assumes a value lower or equal to x. If we consider this signal, when we calculate the probability density function for x, we are considering, by means of the cumulative probability function, the probability that the signal is between + Δ and , and we sum up all the times in which the signal is there, and then we divide for the total amount of acquired data.

∑ Δ=1( + Δ) − () = lim →∞Δ

We must remember that is a value set by us. () is a quantity which must be computed for many values, from the minimum to the maximum. At the end, the probability density function gives us a plot which is a histogram, where on the abscissa we have the value of the signal, and on the ordinate we have the probability that the signal assumes that value. The smaller is the more accurate the plot will be. (), It is possible to demonstrate that if we know the probability density function we can calculate all the indexes that we have previously seen: the mean value, the mean square value, the variance, and so on.

+∞ = ∫ () ⅆ −∞

+∞2 2 = ∫ () ⅆ−∞

+∞2 2 2 2( ) = − = ∫ − () ⅆ−∞

Skewness coefficient

The reason why we need to calculate the probability density function is because there are some important indexes, which are based on the concept of probability density function. The first one of these indexes that we introduce is the skewness coefficient, defined as:

3 =3 3 3( )∑ − 1 =1̂ =3 3

Under a statistical point of view, a negative skewness coefficient means that we have more data on the left side of the mean than on the right side, basically, the probability density function has a curve which is not symmetric and where the left tail is longer. So it means that we have many points below the mean, in fact, the values of will be mostly negative. On the contrary, if we have a positive skewness, we are in the situation in which we have more points over the mean, and so we are going to have a probability density function which has a longer right tail. Finally, a null skewness means that we have a probability density function which is completely symmetric. The skewness coefficient defines any distribution asymmetry around its sample mean value.

Kurtosis coefficient

A further coefficient is the kurtosis coefficient, defined as:

4 = −34 4 4( )∑ − 1 =1̂ = −34 4

Kurtosis is an index which expresses the distribution width. It is possible to notice that even if the formulation is pretty similar to the one of the skewness, now the power is even, while previously it was odd, moreover, now we have a -3 in the expression.