# FFT Calculations

## DIAdem 2018 Help

Edition Date: June 2018
Part Number: 370858P-01
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DIAdem executes FFT calculations for one time signal, for two time signals, and for the inverse FFT. The FFT with one time signal transforms one individual signal into the frequency domain. The Inverse FFT reverses the transformation of a signal, which is defined by a real part and an imaginary part, into the time domain. The FFT with two time signals calculates the coherence, the cross spectrum, or the transfer function by using the FFT on signal pairs. DIAdem has a number of window functions and can divide the time range into subintervals.

### Window Functions

The FFT is based on the assumption that the time signal continues periodically outside the measured range. In reality this precondition is not usually met and therefore the time-related signal is weighted by multiplying it by a window function. The window function reduces the influence of the signal at the edges of the time interval. DIAdem offers the window functions Rectangle, Hanning, Hamming, Blackman, FlatTop, Kaiser, Cos10Proz, Welch, Parzen, Bartlett, Vallee, Cauchy, Gauss, Exponential, and Riemann. Additionally you can use user-defined window functions for special purposes.

The multiplication of the time signal by the window function dampens the amplitude of the transformed signal in the frequency domain. This effect varies with different window functions and also depends on whether the signal is periodic or not. The window functions can execute an amplitude attenuation to reduce this damping effect.

### Time Intervals

The complete signal can be divided into parts, for which DIAdem calculates an FFT respectively. DIAdem can calculate with overlapping time intervals and can create an index channel for further processing of the results. The complete signal is dissected into time intervals, to inspect short, temporary effects that do not impact the complete signal.

## FFT with One Time Signal

The FFT with one time signal calculates first the real part and the imaginary part of the transformed signal. Subsequently DIAdem calculates the phase and various amplitudes of the frequency response. DIAdem can calculate the amplitude as peak amplitude, RMS amplitude, power spectrum, autospectrum, or power density spectrum. Various window functions, section-wise calculations with and without overlapping, and various averaging procedures are available for all FFT calculations. DIAdem can additionally execute a third or octave analysis of the amplitudes, in which case the individual frequencies are summated in standardized frequency intervals.

## Inverse FFT

The Inverse FFT is the reverse operation of the FFT with one time signal. The Inverse FFT reverts a signal, which is defined by the real part and the imaginary part, from the frequency domain into the time domain. Usually the results of the FFT are processed in the frequency domain and then reverted into the time domain with the Inverse FFT.

## FFT with Two Time Signals

The FFT with two time signals calculates the coherence, the cross spectrum, and various transfer functions. For the cross spectrum and the transfer functions DIAdem calculates the phase and the amplitude as peak amplitude, RMS amplitude, power spectrum, autospectrum, or power density spectrum in addition to the real part and the imaginary part.

### Coherence

DIAdem calculates the coherence as the measurement for the linear relation between two signals from the averaged cross spectrum and autospectrum of the signals. The coherence calculation only makes sense for several channel pairs or for more than one time interval, for example, after repeating the same measurement which is overlaid with distortion signals.

### Cross Spectrum

The cross spectrum is the complex product of two time signals transformed into the frequency domain. The cross spectrum represents the similarity of the signals in the frequency domain and corresponds with the cross correlation in the time domain.

### Transfer Function

The transfer function calculation regards the two time signals as input signal and output signal of a dynamic system. The calculation of the complex transfer frequency response analyzes the system in the frequency domain.

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