Fast Fourier Transformation (FFT)

You can use the Time to FFT tool in the Response Dynamics Function Toolkit to perform a Fast Fourier Transformation on a function. The FFT transforms time domain data into the frequency domain. The discrete Fourier transform is a transformation pair (forward and inverse) with implied constant, consistent discretization.

Changing the discretization violates the duality of the transformation pair. Each FFT, FFT-1 operation is correct, but appropriate pairs are not Fourier transformation pairs if the discretization is changed. The following example applies.

FFT Time Signal

therefore:

where:

• T1 = total time of period

• N1 = specified resolution of time function

• Δt1 = time increment

• fmax = maximum frequency

• Δf1 = frequency increment

FFT -1

fmax = N2Δf2

therefore:

T2 = 2N2Δt2 by Shannon's theorem

where:

N2 = specified resolution of frequency function

For consistent discretization:

T1 = T2 and Δt1 = Δt2

therefore:

T1 = N1Δt1 = T2 = 2N2Δt2

N1Δt1 = 2N2Δt2 = 2N2Δt1

N1 = 2N2

The software uses a single-sided Fast Fourier Transform (based on Singleton's method), which results in half-peak values for the amplitudes in the frequency domain. Therefore, to use the frequency function as an excitation, you must multiply the function by 2 to get results that match the time-domain results. This single-sided approach reduces the number of calculations required.

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Fast Fourier Transformation (FFT), Simcenter 3D 2021.1 Series

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