Appunti Measurements

SIGNAL ANALYSIS IN TIME DOMAIN

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

1

2 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.

Most of times, we do not consider the number given by the mean square value, but we consider

the Root Mean Square value (RMS), defined as

1

2 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 insteas 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 understand

easily.

Another important parameter, is the variance, defined as

1

2 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

1

2 2 2

()

= lim [∫ ⅆ + ∫ ⅆ − 2 ∫ () ⅆ ]

→∞ 0 0 0

1

2 2 2

()

= lim [∫ ⅆ + ∫ ⅆ − 2 ∫ () ⅆ ]

→∞ 0 0 0

1 1

2 2 2

()

= lim ∫ ⅆ + lim ∙ − 2 ⋅

̅ ̅

→∞ →∞

0 2 2 2

= −

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

max|( − )|

=

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 recognise 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.

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

( + Δ) − ()

() = lim Δ

Δ→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 instanf 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

+ Δ,

between 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 an histogram, where on the

abscissa we have the value of the signal, and on the ordinate we have the probability that the

Δ,

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

−∞

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

=1

̂ =

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

3

( )

−

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.

A further coefficient is the kurtosis coefficient, defines as

4

= −3

4 4

4

( )

∑ −

1

=1

̂ = −3

4 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

was odd, moreover, now we have a -3 in the expression. The -3 is just a normalization in order to

have a kurtosis coefficient equal to 0 for guassian distribution, since the gaussian distribution is

most of the time our reference. The even power is such that each spikes we are going to have in

the signal, positive or negative, are going to increase this quantity. For each spike we are going to

increase the numerator. For this reason, we often exploit this index when we are dealing with

rotating machines vibrations, like in the previously example. A positive kurtosis coefficient means

that we are increasing the numerator, so, if we analyse the distribution, the consequence is that

we have many points thar are much bigger than the mean value, so, we have less points close to

the mean value, and much more points at the tails. So, if we have more points in the tails, the

distribution becomes sharper. In conclusion, a positive kurtosis indicates a sharper distribution

than the gaussian one. On the contrary, when we have a negative kurtosis, this means that

compared to the gaussian distribution, we have less points in the tails, and more points close to

the mean. This means that we are going to have a broader distribution.

Now we can analyse the previously example related to the gear tooth, by means of the kurtosis

and the skewness coefficients:

As we can see, the kurtosis has a huge increase, even if it is not higher like the one of the crest

factor, it is still easy to be seen. So, again, the kurtosis coefficient is an index which is able to

identificate the presence of a spike, and so, in this case, the presence of a damage.

Now we consider the case in which we do not have a single damaged tooth, but we have many

damages. We can be in the situation where we have a signal like the following

In this situation, the sine component is always the same, but we have many spikes superimposed

to the harmonic component. Due to the variation of the phase, we have many peaks, which are

not as large as the case in which there is just one peak. We can see that in this case the crest

factor is decreased. It is still able to identify a damage, but it is not as reliable as before. This is