a problem about the amplitude of Power Spetrum

02-Analytical capabilities
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lbwhu
Posts: 3
Joined: 09 Jul 2009, 10:59

a problem about the amplitude of Power Spetrum

Post by lbwhu »

In the SeismoSignal,we could transform Fourier Spetrum to Power Spetrum. When I input several kinds of time-history of the acceleration and see the amplitude of Fourier Spetrum and Power Spetrum, I was surprised to discover that the amplitude of Fourier Spetrum is noticeably greater than that of Power Spetrum.According to lots of reference,the amplitude of Power Spetrum approximately equates the square of that of Fourier Spetrum. I do not know why? The units of the data are right.

For example: One of my computing results

Frequence the amplitude of Fourier amplitude of Power
0.0122 185.154526948498 0.0029241065
0.0183 226.713173324206 0.0043840755
0.0244 163.231213424895 0.0022726415
0.0305 197.171364388138 0.0033159823
0.0366 218.831090849341 0.0040845345
0.0427 162.294557553602 0.0022466345
0.0488 203.517239598668 0.0035328640
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seismosoft
Posts: 1184
Joined: 06 Jul 2007, 04:55

Re: a problem about the amplitude of Power Spetrum

Post by seismosoft »

In SeismoSignal Power Spectra Amplitude is computed as follows: PowerAmpl = FFTAmpl*FFTAmpl/(Pi*duration*RmsAcc^2), where 'duration' is the duration of the record, 'RmsAcc' is the acceleration RMS (see 'Ground Motion Parameters' module) and Pi is 3.14159.

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lbwhu
Posts: 3
Joined: 09 Jul 2009, 10:59

Re: a problem about the amplitude of Power Spetrum

Post by lbwhu »

Well,I truely appreciate your reply. But the Japanese scholar, OHSAKI Y., have obtained the relationship between Fourier Spectra and Power Spectra by Parseval's theorem(in the book named 'Introduction to the spectral analysis of ground motion', OHSAKI Y. gave people the transformation between the Fourier Spectra and Power Spectra). PowerAmpl[f] = FFTAmpl[f]*FFTAmpl[f]/duration, FFTAmpl[w]*FFTAmpl[w] means the square of amplitude of Fourier Spectra at each dispersed point; f means the frequency. I have checked many references and I found that many references use this method. According to the method of SeismoSignal, I found that the amplitudes of power spectra are much smaller than that of Fourier Spectra, which may lead to the problems of precision.
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seismosoft
Posts: 1184
Joined: 06 Jul 2007, 04:55

Re: a problem about the amplitude of Power Spetrum

Post by seismosoft »

Thank you for your comment and feedback, which we may certainly consider in future updates of the software.

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linyuesonr
Posts: 2
Joined: 25 Jan 2016, 10:29

Re: a problem about the amplitude of Power Spetrum

Post by linyuesonr »

I'm notecied that from SeismoSoft Support as follows,

In SeismoSignal Power Spectra Amplitude is computed as follows: PowerAmpl = FFTAmpl*FFTAmpl/(Pi*duration*RmsAcc^2), where 'duration' is the duration of the record, 'RmsAcc' is the acceleration RMS (see 'Ground Motion Parameters' module) and Pi is 3.14159.

In additional, can you tell me that the fourmular and normalization details for the FFTAmpl? As I found that the FFT amplitude is not equal to the amplitude of acceleration time histories.

For example, if the acceleration time histories is: Y=2*sin(100*Pi*X), X's number is 1024 from 0 to 0.5 sec, then the amplitude of FFT in Seismosoft is 0.38008, but not 2. Why?
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seismosoft
Posts: 1184
Joined: 06 Jul 2007, 04:55

Re: a problem about the amplitude of Power Spetrum

Post by seismosoft »

Hi linyuesonr,

As mentioned in the the help of the program, the Fourier amplitude is equal to SQRT (Re^2+Im^2), where Re and Im are the real and imaginary parts of the Fourier transform.
Regards,
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linyuesonr
Posts: 2
Joined: 25 Jan 2016, 10:29

Re: a problem about the amplitude of Power Spetrum

Post by linyuesonr »

Thank you for your reply.

However, I found that the Fourier amplitude is relevant to the duration, can you tell me more about the details? I'm interested on the approach of how the amplitude of "2" transform to "0.38008" (when Y=2*sin(100*Pi*X), X's number is 1024 from 0 to 0.5 sec).
When I change the distribution of data, the Fourier amplitude is "0.93243" (when Y=2*sin(100*Pi*X), X's number is 2048 from 0 to 1.0 sec) and the Fourier amplitude is "0.23242" (when Y=2*sin(100*Pi*X), X's number is 512 from 0 to 0.25 sec).

Therefore, whether the Fourier amplitude has any revlance to the "duration", "Pi", "RmsAcc" or any other parameters? If I figure out the details that I might know how the data amplitude "A" (e.g. Y=A*sin(2*Pi*50*X)) transform to the Fouier amplitude of the Seismosoft. :?

Thank you very much.

Best regards,
Yue LIN
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seismosoft
Posts: 1184
Joined: 06 Jul 2007, 04:55

Re: a problem about the amplitude of Power Spetrum

Post by seismosoft »

"I found that the Fourier amplitude is relevant to the duration": this should not be correct. It is however relevant to the time-step dt, when the changes in the amplitude are the same.
In any case, one cannot describe the details of Fast Fourier Transform within the context of a forum that supports the use of the programs. We suggest that you refer to one of the textbooks that are mentioned in the help system of the progam, in the Fourier and Power Spectra section. We also note that for the FFT transformation the DSPlab library is being used in our programs (http://www.teworks.com/help.htm).

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ehsan334
Posts: 1
Joined: 10 Feb 2016, 13:36

Re: a problem about the amplitude of Power Spetrum

Post by ehsan334 »

Hello.
I have a question about PSD from Seismosignal.
software computes power amplitude and it dose not compute power spectral density function. I mean PSD is a density (summation of all values will be 1, But when you sum values from power spectrum column (in tab Fourier and power spectra from Seismosignal) it is not 1.
Is it correct?
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