## the units for power spectrum and fourier spectrum

02-Analytical capabilities
inversesugar
Posts: 3
Joined: 29 Jun 2012, 06:12

### the units for power spectrum and fourier spectrum

Dear all,I would like to ask what is the unit for power spectrum and fourier spectrum respectively.
1.For fourier spectrum.

I do read all the posts related to the fourier spectrum and found that,in this post(http://www.seismosoft.com/forum/topic.asp?TOPIC_ID=506),the seismosoft support confirmed that if the user define time history in 'g',the unit for fourier spectrum is also 'g',which is same with the input data.However, in another post(http://www.seismosoft.com/forum/topic.asp?TOPIC_ID=235),the unit for fourier spectrum is m/s^2/Hz,when the input data is in m/s^2,while Hz is 1/s, thus the unit for fourier spectrum is m/s?Do this correct?

2.For power spectrum.
In the software help posts,the power spectrum is calculated by the formula[ FFT(a(t))]^2/[T*PI*aRms^2],and the unit for it is 1/Hz(http://www.seismosoft.com/forum/topic.a ... r,spectrum) which mean that the power spectrum has been normalized?But in my calculation I could not obtain such a unit.Would you please show me how could I deduce such a unit?Or what is wrong with my deduction?
Deduction:(Define the input data in m/s^2)
1).The unit of the data after FFT is the same with the original data,thus the unit for [FFT(a(t))]^2 is m^2/s^4.
2).I think the unit for aRms is the same with the input data,m/s^2,which was deduced by the formula of it,right?
3).T, the unit is s, PI, the unit is rad.
Thus the unit for PSD would be like this:

In my some other references, the unit for the PSD should be( m^2/s^4)/Hz,how could I obtain such a series of data?
seismosoft
Posts: 1130
Joined: 06 Jul 2007, 04:55

### Re: the units for power spectrum and fourier spectrum

Dear inversugar,

1) we can confirm that the information you found on the most recent of the two Forum posts is the correct one (i.e. Fourier Amplitude units is m/s, when the input time-history is defined in m/s^2)

2) you are correct that Power Spectrum calculations in other software may differ from that used in SeismoSignal, and that in such cases often units of m^2/s^3 can be found as being employed. In SeismoSignal, the Fourier Amplitude is used in the calculations, hence, considering the units of the latter, discussed in point 1 above, one readily arrives at Power Spectrum units of 1/Hz.

Seismosoft Support
Parsaparvanehro
Posts: 1
Joined: 31 Oct 2021, 19:31

### Re: the units for power spectrum and fourier spectrum

Dear inversesugar
The PSDF calculated by Seismosignal is normalized to pi*RMSAcc^2. it mean you must multiplied it to pi*AccRMS^2 and use in your random vibration analysis, for example. where pi is 3.14.... and be aware of RMSAcc unit.
GeorgS
Posts: 1
Joined: 19 Nov 2021, 22:06

### Re: the units for power spectrum and fourier spectrum

Good day to all
I still do not understand how to go from the Fourier amplitude calculated in the Seismosignal to the real amplitude of the harmonic of a certain frequency (in the form of x = A * cos (w*t + phi), where A-real amplitude must be in meters...
seismosoft
Posts: 1130
Joined: 06 Jul 2007, 04:55

### Re: the units for power spectrum and fourier spectrum

GeorgS,
First of all please bear in mind that from the Fourier Amplitude you can deduce only an estimate of the real amplitude of a harmonic of your time-series, which estimate is highly dependent on the quality of the time series (noise etc), the length (number of points) of your time series – of course longer time series of a harmonic signal will give a better estimate of the amplitude and the time series step – time series with a high sampling frequency will give a more detailed picture of the time-series content thus a more detailed estimate of the amplitude and frequency.

Having in mind the above, the units of the Fourier amplitude are same as the units of your time-series multiplied by seconds. In order to get an estimate of the real amplitude of a harmonic contained in your signal you need to take the Fourier Amplitude corresponding with the harmonic, multiply it by 2 (due to the 2-sided shape of the Fourier Transform results), then divide the result by the number of points of your time series and finally divide the results by the step of your time-series (which is in seconds).
Seismosoft Support