# Signal analysis

The signal analysis enables the description of the dynamic properties of a vibrating system from the input and output signals of this system on the basis of frequency analysis . In addition to statistical methods such as averaging and the calculation of standard deviations, it is of outstanding importance when evaluating acoustic and vibration-related signals.

The systems to be analyzed are often mechanical structures. Then the input variable could be a stimulating force and the output variables could be the resulting surface velocities (“vibrations”) at any point on the structure. The signal analysis can then be used, for. B. describe in detail the vibration rates with which the structure reacts to a certain force excitation.

Another broad area of ​​application for signal analysis is in electrical systems, especially in quadrupole systems . In this case, the input variable can be a current or a voltage . The output variable is usually also a current or a voltage. In large electrical systems such as machines or transformers, broadband signal analysis (see transfer function or frequency response ) can be used to derive not only electrical information, but also mechanical information (e.g. about deformations).

## Basics

Representation of a dynamic system with input and output variables

The general formulation of signal analysis theory is based on linear systems. However, with special extensions, non-linear systems can also be treated.

The basis of the signal analysis is the Fourier transformation . It enables the transfer of time signals into the frequency domain by breaking down the time functions into the sum of an infinite number of harmonic individual functions with infinitely finely graduated frequencies (Fourier integral). This relationship can be formulated for the time signal x (t) with the associated Fourier spectrum X (f) using the equation

${\ displaystyle X (f) = \ int _ {- \ infty} ^ {\ infty} x (t) e ^ {- \ mathrm {j} 2 \ pi ft} \, dt}$

The computational representation of this transformation on digital computers is called Discrete Fourier Transformation (DFT):

${\ displaystyle X_ {k} = {\ frac {1} {N}} \ sum _ {n = 0} ^ {N-1} x_ {n} e ^ {- j {\ frac {2 \ pi nk} {N}}}}$ (k = 0, 1,…, N-1)

X k is called the finite Fourier spectrum of the discretized time function x n (N samples). The algorithm most frequently used to calculate it is the Fast Fourier Transformation (FFT).

Illustration of the discrete Fourier transformation for a sine signal with the frequency 3.33 kHz (analysis window length: 0.3 ms, sampling rate: 20 kHz)

The numerical calculation has some special features that must be taken into account when analyzing the signal.

• Due to the discrete-time sampling ( discretization ) of a measuring signal arising in relation to the frequency content of the signal to the small sampling distortions as a band overlapping or aliasing are designated ( Nyquist-Shannon sampling theorem ). They can be avoided by using analog low-pass filtering below half the sampling frequency ("anti-aliasing filter").
• The time limit of the sampling ("time" or "analysis window") leads to the appearance of so-called sidebands in the frequency range. If the observation period does not correspond to the period duration of frequencies contained in the signal or their integral multiples, then these sidebands influence the discrete spectrum z. B. by the appearance of additional frequency components. This phenomenon is known as the leakage effect. Using special evaluation functions in the time window (e.g. Hanning window ), its effects can be reduced, but not entirely avoided.
• The frequency discretization causes (after reverse transformation) a periodization of the time signal, which, however, is mostly irrelevant for the analysis. The infinitely finely graduated frequencies of the Fourier integral become equidistant 'frequency lines' with Δf = 1 / T.
• The digitization of the analog signal leads to a limitation of the dynamic range , the quantization noise , which is less significant the higher the resolution of the A / D converter is. Due to the limited dynamics of the analog measuring devices, this effect usually does not need to be taken into account during digitization if the modulation is good .

If these special features are taken into account, the DFT (FFT) is a powerful tool for frequency analysis that has almost completely replaced analog techniques (filter banks) in recent years. Based on this, the relationships between different signals (typically one “system input” and several “system outputs”) can be determined particularly easily with the aid of the extended signal analysis techniques. The prerequisite for this is i. General the parallel acquisition of the signals.

The following figure shows the most important signal analysis functions in a block diagram. In principle, the calculation process for the individual functions can be followed using the connecting lines. The time functions are arranged in the left part of the picture, the frequency functions in the right. The two areas are linked via the Fourier transformation F and the inverse Fourier transformation F −1 , which are used for the back calculation to the time signal x (t) using the equation

${\ displaystyle x (t) = \ int _ {- \ infty} ^ {\ infty} X (f) e ^ {\ mathrm {j} 2 \ pi ft} \, df}$

can be described. The inverse Fourier transform therefore enables a time function to be determined from its Fourier transform. The forward and backward transformations entered in the diagram can therefore also take place in the other direction if necessary. If a block has several inputs, this indicates several calculation options.

## Signal analysis functions

Block diagram with the most important signal analysis functions

The individual signal analysis functions are of different importance. Outstanding are the car power spectrum , from which the RMS spectrum is calculated, the frequency response, which describes the system behavior and z. B. is needed to carry out the modal analysis and the coherence with which the quality of the analysis results can be assessed. The cepstrum is used to determine periodic components and their orders in the signal, as is the autocorrelation function to a limited extent . With the cross-correlation function , transit times between the input and output signals can be identified. The cross-performance spectrum has little informative value of its own. It is therefore mostly only used to determine the frequency response and the cross-correlation function.