Signal analysis

Signal analysis allows on the basis of the frequency analysis, the description of the dynamic characteristics of a vibrating system of the input and output signals of the system. It is next to statistical methods such as averaging and calculation of standard deviations in the evaluation of acoustic and vibration control signals of paramount importance.

Frequently, the analyte systems are mechanical structures. Then, the input could be a stimulating force and the output variables be the resulting surface rapids ( " vibrations " ) at any point on the structure. About the signal analysis can then be described in detail, for example, with which vibration velocities, the structure responds to a particular force excitation.

Another broad area of ​​application of signal analysis is in electrical systems, particularly in four- poles. In this case, may be a current or a voltage of the input variable. The output is usually also a current or a voltage. For large electrical systems such as machinery or transformers can be by a broadband signal analysis (see transfer function or frequency response), not only electrical but also mechanical information derived ( eg via deformations ).

Basics

The general formulation of the theory of signal analysis is based on linear systems. By special extensions but also non-linear systems can be handled.

The basis of the analysis signal is the Fourier transform. It allows the transfer of time signals in the frequency domain by the decomposition of the time functions in the sum of an infinite number of harmonic functions with single infinitely finely graded frequencies (Fourier integral). This relationship can be formulated for the time signal x (t) with the corresponding Fourier spectrum X (f) by the equation

The computational representation of this transformation on digital computers is referred to as Discrete Fourier Transform ( DFT):

Xk is a finite Fourier spectrum of the discretized time function xn ( N samples ) respectively. The most commonly used algorithm for its calculation of the Fast Fourier Transform (FFT).

The numerical calculation has some special features with it that need to be considered in the signal analysis.

• The discrete-time sampling ( discretization ) of a measuring signal arising in relation to the frequency content of the signal to the small sampling distortion, called aliasing, or aliasing ( Nyquist-Shannon sampling theorem ). They can be avoided by analog low-pass filtering below half the sampling frequency ( "anti- aliasing filter ").
• The timing of scanning ( " time " or " analysis window " ) leads to the appearance of so-called side bands in the frequency domain. Corresponds to the period of observation of the period contained in the signal frequency or its integer multiple, as they affect, the discrete spectrum of the sidebands, for example, by the occurrence of additional frequency components. This phenomenon is referred to as a leak or leakage effect. By special evaluation functions in the time window (eg Hanning window ) its effects can mitigated, but not avoided entirely.
• The Frequenzdiskretisierung causes (after back-transformation ) is a periodization of the time signal, but this is usually of no significance for the analysis. From the infinitely finely graduated frequencies of the Fourier integral are equidistant frequency lines ' with? F = 1 / T.
• The digitization of the analog signal leads to a restriction of the dynamic range, the quantization noise, which presents itself as the more insignificant, the higher the resolution of the A / D converter. Because of the limited dynamics of the analog instruments, this effect generally does not need to be observed with good modulation during digitization.

When these features, the DFT ( FFT) represents a powerful tool for frequency analysis, in recent years, analog techniques ( filter banks ) has almost completely displaced. Building on her, with the help of advanced signal analysis techniques particularly simple relations of different signals to each other (typically a " system input" and more " system outputs") determined. This requires generally involves. the parallel detection of the signals.

The following picture shows the main signal analysis functions are shown in a block diagram. Based on the connecting lines can be followed in principle the calculation process for the individual functions. In the left part of the image frequency functions are the functions of time, in the right placed. Combines the two portions of the Fourier transform F and the inverse Fourier transform F- 1, for the recalculation of the time signal x ( t) by the equation

Can be described. Thus, the inverse Fourier transform of a function of time allows the determination of the Fourier transformed. Entered in the diagram of forward and reverse transformations can be carried out, therefore, if necessary, in each case in the other direction. Assigns a block to multiple inputs, this points to several calculation options.

Signal analysis functions

The individual signal analysis functions are of varying importance. Outstanding are the auto power spectrum from which the RMS spectrum is calculated, the frequency response, which describes the system behavior and is required, for example to carry out the modal analysis and the consistency with which the quality of the analysis results can be assessed. The Cepstrum is used to determine the periodic units and their orders in the signal, as well as in a limited way, the auto-correlation function. With the cross-correlation function maturities between input and output signal can be recognized. The cross power spectrum has little own significance. Therefore, it is usually used only for determining the frequency response and cross-correlation function.