[SAC-HELP] Re: sac-help Digest, Vol 33, Issue 8 FFT in SAC

[Robert B. Herrmann]

sac-help-request at iris.washington.edu wrote:

The simple test of the FFT is to see the effect of an impulse:

fg impulse delta 0.1 npts 1024
w impulse.sac


*1. Test of sac2000 FFT

*Sac creates a time series with a centered impulse of height 1.0 and not
a unit area pulse.

fft
writesp am

This creates a series of the amplitude spectrum, which has values of
0.1, e.g., 1 (from the time series) * DT from the fft

ifft

This creates a time series consisting of a centered impulse with height 1.0

Thus sac2000 [8/8/2001 (Version 00.59.44)]  is performing the following
operation to convert h(t) into H(f)
              N-1
H(f)   = SUM    h(t) exp ( - j 2 pi f t ) dt
             k=0
         
            N-1
h(t)  = SUM     H(f)exp(+j 2 pi f t ) df
            n=0

where  f = n df. t = k dt , and   df = 1/Ndt

*2. Constraints on testing the FFT using an analytic function*

If you wish to test an FFT, you cannot use a sine-wave since the value
of the FFT will be related to the total number of cycles in the time
window and because the Discrete Fourier Transform is an approximation to
the Fourier Transform that suffers from periodicity in both the time and
frequency domains.

A properly designed test can use the analytical  Gaussian <-> Gaussian
transform pair, or perhaps some thing from Laplace transforms using

       H(t) exp( - a t ) <->  1/(s + a)  where s = j 2 pi f

You must always worry about the fact there there is a limited time
length and a limited frequency band.  A perfect test of an analytical
transform pair is thwarted by the uncertainty principle

*3. Other FFT's

*If you use MATLAB or MATHCAD, you will always have to read the
documentation about the definition of the FFT used.
I use the electrical engineering definition but incorporate the physical
dimensions to the integral.

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> Today's Topics:
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>    1.  About FFT scaling factor in SAC (wwxu)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Tue, 11 Mar 2008 07:53:16 GMT
> From: "wwxu" <wwxu at mail.iggcas.ac.cn>
> Subject: [SAC-HELP] About FFT scaling factor in SAC
> To: sac-help at iris.washington.edu
> Message-ID: <20080311075316.5572.eqmail at mail.iggcas.ac.cn>
> Content-Type: text/plain; format=flowed; charset="gb2312"
>
> Hi, ALL 
>
> We suppose that x[n] and y[k] are a DFT pairs, i.e.
>        y[k]=1/N*sum_n(x[n]*exp(-2*pi*j*k*n/N))   n=0,1,...,N-1
> where sum_n means the sum for n=0,1,N-1(similar for sum_k),
> and dt is the sampling interval, df is the sampling frequency,
> N is the sampling number.
> To obey Parseval's theorem, we get
>        sum_n(x[n]*x[n])*dt=sum_k(y[k]*y[k])*df
> since sum_n(x[n]*x[n])=N*sum_k(y[k]*y[k])
> and df=1/(N*dt), also consider the amplitude symmetry of the real series,
> we should multiply a coefficent to y[k]:
>        sqrt(2)*N*dt
> but in SAC software, this value is N*dt/2,
> so do you have any idea? 
>
> Thanks. 
>
>
>
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> End of sac-help Digest, Vol 33, Issue 8
> ***************************************
>   


-- 
Robert B. Herrmann
Otto W. Nuttli Professor of Geophysics
Department of Earth and Atmospheric Sciences
Saint Louis University
O'Neil Hall, Room 203
3642 Lindell Boulevard 
St. Louis, MO 63108

TEL: 314 977 3120
FAX: 314 977 3117
Email: rbh at eas.slu.edu

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